Keywords: Homotopy algebras, Lie 8-algebroids, dg-manifolds, singular foliations, algebraic geometry, singularities

This thesis breaks into two main parts. Let me describe them.

• We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra O and homotopy equivalence classes of negatively graded acyclic Lie 8-algebroids.

Therefore, this result makes sense of the universal Lie 8-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. This extends to a purely algebraic setting the construction of the universal Q-manifold of a locally real analytic singular foliation of [LLS20, Lav17]. Also, to any ideal I Ă O preserved by the anchor map of a Lie-Rinehart algebra A, we associate a homotopy equivalence class of negatively graded Lie 8-algebroids over complexes computing Tor O pA, O{Iq. Several explicit examples are given.

• The second part is dedicated to some applications of the results on Lie-Rinehart algebras.

1. We associate to any affine variety a universal Lie 8-algebroid of the Lie-Rinehart algebra of its vector fields. We study the effect of some common operations on affine varieties such as blow-ups, germs at a point, etc.

2. We give an interpretation of the blow-up of a singular foliation F in the sense of Mohsen [Moh21] in terms of the universal Lie 8-algebroid of F, in fact an almost Lie algebroid over a geometric resolution of F.

3. We introduce the notion of longitudinal vector fields on a graded manifold over a singular foliation, and study their cohomology. We prove that the cohomology groups of the latter vanish.

4. We study symmetries of singular foliations through universal Lie 8-algebroids. More precisely, we prove that a weak symmetry action of a Lie algebra g on a singular foliation F (which is morally an action of g on the leaf space M {F) induces a unique up to homotopy Lie 8-morphism from g to the Differential Graded Lie Algebra (DGLA) of vector fields on a universal Lie 8-algebroid of F (such morphim is known under the name "L 8 -algebra action" in [MZ12]). We deduce from this general result several geometrical consequences. For instance, We give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space. Last, we present the notion of bi-submersion towers over a singular foliation and lift symmetries to those.

RÉSUMÉ

Cette thèse se décompose en deux parties principales. 2. La deuxième partie est consacrée à quelques applications des résultats sur les algèbres de Lie-Rinehart.

(a) On associe à toute variété affine une algébroïde de Lie infinie universelle de l'algèbre de Lie-Rinehart de ses champs de vecteurs. Nous étudions l'effet de certaines opérations courantes sur des variétés affines telles que les éclatements, germes en un point, etc.

(b) Nous donnons une interprétation de l'éclatement d'un feuilletage singulier F au sens de Mohsen [START_REF] Mohsen | Blow-up groupoid of singular foliations[END_REF] en termes de l'algébroïde de Lie infinie universelle de F.

(c) Nous introduisons la notion de champs de vecteurs longitudinaux sur une variété graduée sur un feuilletage, et étudier leur cohomologie. Nous prouvons que les groupes de cohomologie de ce dernier sont nuls.

(d) Nous étudions les symétries de feuilletages singuliers à travers des algébroïdes de Lie infinie universels. Plus précisément, nous prouvons qu'une action par symétrie faible d'une algèbre de Lie g sur un feuilletage singulier F (qui est moralement une action de g sur l'espace des feuilles M {F) induit un unique morphisme de Lie infini à homotopie près de g vers l'algèbre de Lie différentielle graduée (DGLA) des champs de vecteurs sur un algébroïde de Lie infinie universel de F. On déduit de ce résultat général plusieurs conséquences. Par exemple, nous donnons un exemple d'action d'algèbre de Lie sur une sous-varieté affine qui ne peut s'étendre à l'espace ambiant. Enfin, nous présentons la notion de tour de bisubmersions sur un feuilletage singulier et relève des symétries à celles-ci.

Mots clés en français: Algèbres à homotopie près, Algébroïdes à homotopie près, Singularités, Géométrie différentielle et algébriques, Feuilletages, dg-variétés.

First, I would like to express my gratitude to my supervisor Camille Laurent-Gengoux for accepting me as his Ph.D. student. He was patient, attentive to me. He was a great support for me and the one on which I could lean when I feel lost in my ideas. He encouraged me in my research. He gave me confidence. He knew how to motivate me. He gave me valuable advice and suggested directions to take, articles to read in order to lead better my research. He transmitted to me knowledge and the ethics of a researcher, he is therefore my mathematic father. I also thank him for the time he devoted to me and for his comments and his suggestions throughout of this Ph.D. I feel very lucky for this opportunity I had to work with him.

My gratitude also goes to Chenchang Zhu and Marco Zambon for accepting to be reporters for my thesis. I also thank Claire Debord, Pol Vanhaecke, Robert Yuncken, and Rajan Amit Mehta for taking part in the jury.

I would like to acknowledge the full financial support for this Ph.D from Région Grand Est. Likewise, I thank the managers and staff of Université de Lorraine for their collaboration and their help in the administrative procedures. I particularly would like to warmly thank Prof. Philippe Bonneau for helping me find accommodation and settling me comfortably in Metz to begin my PhD.

Thanks to the CNRS MITI 80Prime project GRANUM also to Franco-German PHC project Procope. I would like to thank the Institut Henri Poincaré for hosting me in november 2021. I want to thank S. Lavau for his advice and comments on my paper "Symmetries of singular foliations". I would like to thank C. Ospel, P. Vanhaecke and V. Salnikov for giving the possibility to present the results on Lie-Rinehart algebras at the "Rencontre Poisson à La Rochelle, 21-22 October 2021". I was glad to meet my mathematic grand father Claude Roger at this conference and I would like to thank him for his articles on the Lie-Rinehart algebras that he gave me. Also, I thank the organizers of the Geometry Seminar at the Aristotle University of Thessaloniki, in particular Panagiotis Batakidis for inviting me in January 28th, 2022 to give a talk on my work.

I would like to thank Iakovos Androulidakis with whom I had the honor of discussing my work at the "Foliations, pseudodifferential operators and groupoids" school, Göttingen, Germany in February 28th-March 4th, 2022. I also would like to thank Leonid Ryvkin for always being ready to discuss with me when I needed it. I am also grateful to the organizers of Poisson 2022 Advanced School at CRM (Centre de Recerca Matemàtica) Barcelona, for giving this precious opportunity to be in charge of the problems sessions for the Lecture "Singular Foliations". Likewise, I would like to thank the organizers of Poisson 2022 Conference -ICMAT, Madrid for giving the possibility to present my results at the Poster session. Last, I would like to express sincere gratitude to Université d'État d'Haïti and more precisely the department of mathematics of École Normale Supérieure (ENS), for giving a golden opportunity to meet mathematics. I would like to mention a few names of ENS professors who have contributed to this achievement: Prof. Bérard Cenatus, Prof. Lesly Dejean, Prof. Dr.

Antonine Phigareau, Prof. Dr. Pierre Timothe, Prof. Dr. J.B Antenord, Prof. Dr. OLguine Yacinthe, Prof. Dr. Oscar Walguen, Prof. Steeve Germain, Prof. Dr. Aril Milce, Prof. Dr. Yvesner Marcelin and Prof. Dr. J. K Innocent and Prof. Dr. Dieuseul Predelus. I specially would like to thank Prof. Dr. Achis Chery and Prof. Dr. Pol Vanhaecke for supporting me by writing letters in my favor to obtain this grant.

Je tiens à remercier Anderson Augusma ainsi que Dor Dieunel pour m'avoir aidé à vérifier les erreurs typographiques dans le manuscrit.

Je remercie chaleureusement Wanglaise Fateon qui m'a également aidé à relire mes articles et à identifier les coquilles. La rencontrer a été l'une des meilleures choses qui me soient arrivées.

Enfin, je dédie cette thèse à mes parents, Madame et Monsieur Wisner Louis qui ont toujours été présents pour m'encourager au tout début et tout au long de ces années de thèse. Je remercie mon seul et unique frère et petit frère Benjamin Louis pour ses blagues drôles qui m'ont aidé à ne pas devenir fou dans la foulée. Je t'aime mon frère.

À tous mes proches !

Introduction

Les études récentes sur les 8-algèbres de Lie ou les 8-groupes de Lie, leurs morphismes et leurs équivalents en -oïdes (c'est-à-dire les 8-algèbroïdes de Lie [START_REF] Campos | Homotopy equivalence of shifted cotangent bundles[END_REF][START_REF] Voronov | Higher derived brackets and homotopy algebras[END_REF][START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF] et les "groupoïdes supérieurs" [START_REF] Severa | Integration of differential graded manifolds[END_REF]) est généralement justifié par leur utilisation dans divers domaines de la physique théorique et des mathématiques. Lie 8-algèbres ou -oïdes apparaissent souvent là où, à première vue, ils ne semblent pas faire partie de l'histoire, mais finissent par être nécessaires pour répondre à des questions naturelles, en particulier des questions où aucun concept de structure supérieure ne semble a priori impliqué. Parmi les exemples d'une telle situation, citons la quantification de déformation des variétés de Poisson [START_REF] Kontsevich | Deformation quantization of Poisson manifolds[END_REF] et de nombreux développements récents de la théorie des opérateurs BV, par ex. [START_REF] Campos | BV formality[END_REF], déformations de sous-variétés coisotropes [START_REF] Cattaneo | Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model[END_REF], problèmes d'intégration d'algébroïdes de Lie par stacky-groupoïdes [START_REF] Tseng | Integrating Lie algebroids via stacks[END_REF], sous-variétés complexes et classes d'Atiyah [START_REF] Chen | From Atiyah classes to homotopy Leibniz algebras[END_REF][START_REF] Kapranov | Rozansky-Witten invariants via Atiyah classes[END_REF][START_REF] Laurent-Gengoux | Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg-manioflds[END_REF]. La liste pourrait être plus longue.

Par exemple, il apparaît dans la théorie des feuilletages singuliers. Les feuilletages singuliers apparaissent fréquemment en géométrie différentielle ou algébrique. Ici, suivant [AS09, AZ14, Cer79, Deb01, LLS20] nous définissons un feuilletage singulier sur une variété analytique lisse, complexe, algébrique, réelle, M avec le faisceau de fonctions O comme sous-faisceau F : U ÝÑ FpUq du faisceau de champs de vecteurs X, qui est stable par crochet de Lie et est localement fini engendré en tant qu'un O-module. D'après le théorème d'Hermann [START_REF] Hermann | The differential geometry of foliations, II[END_REF], cela suffit pour induire une partition de la variété M en sous-variétés de dimensions éventuellement différentes, appelées feuilles du feuilletage singulier. Les feuilletages singuliers apparaissent par exemple comme des orbites d'actions de groupe de Lie avec des dimensions éventuellement différentes ou comme des feuilles symplectiques d'une structure de Poisson. Lorsque toutes les feuilles ont la même dimension, on retrouve les "feuilletages réguliers" [START_REF] Diederich | Introduction to the Geometry of Foliations, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy[END_REF][START_REF] Laurent-Gengoux | Geometry of singular foliations: a draft of an introduction[END_REF].

Dans [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]- [START_REF] Lavau | Lie 8-algebroids and singular foliations[END_REF], il est prouvé que "derrière" certains feuilletages singuliers F il existe une classe naturelle d'homotopie 8-algébroïdes de Lie, appelée 8-algebroid de Lie universelle de F, et que ce dernier répond à des questions fondamentales naturelles sur l'existence de "good" générateurs et relations pour un feuilletage singulier. La première partie de cette thèse est principalement une algébrisation de [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]- [START_REF] Lavau | Lie 8-algebroids and singular foliations[END_REF], qui permet d'élargir largement les classes d'exemples. Plus précisément, les théorèmes 4.2.1 et 4.2.4 sont similaires aux théorèmes principaux Théorème 2.8. et Théorème 2.9 dans [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] :

1. Le théorème 4.2.1 équipe toute O-résolution libre d'une algèbre de Lie-Rinehart A avec une structure 8-algébroïde de Lie (Théorème 2.8. dans [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] était que pour les résolutions géométriques du feuilletage singulier analytique localement réel sur un sous-ensemble ouvert avec adhérence compacte). C'est une sorte de théorème de transfert d'homotopie, sauf qu'aucun théorème de transfert d'homotopie existant ne s'applique dans le contexte des O-modules générique (par exemple, [START_REF] Campos | Homotopy equivalence of shifted cotangent bundles[END_REF] ne traite que des projectifs O-modules). La difficulté est que nous ne pouvons pas appliquer les formules de transfert explicites qui apparaissent dans le lemme de perturbation homologique, car il n'y a en général pas de section O-linéaire de A à ses résolutions projectives.

2. Theorem 4.2.4 dit que toute structure 8-algébroïde de Lie qui se termine dans A est équipée d'un unique morphisme 8-algébroïde de Lie à homotopie à n'importe quelle structure comme dans le premier item (le théorème 2.8. dans [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] était une similaire pour Lie 8-algébroïdes dont l'ancre prend des valeurs dans un feuilletage singulier donné).

Comme dans [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF], un corollaire immédiat du résultat est que deux 8-algébroïdes de Lie comme dans l'item 1 sont homotopiques équivalents d'une manière unique à l'homotopie près, définissant ainsi une classe canoniquement associée à l'algèbre de Lie-Rinehart qui mérite au vu de l'item 2 d'être appelée "universelle."

Cependant :

1. Alors que [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] traitait des 8-algèbres de Lie sur des résolutions projectives de longueur et de dimension finies, nous travaillons ici avec des 8-algèbrroïdes de Lie sur n'importe quelle résolution libre -même celles de longueur infinie et de dimension infinie à chaque degré.

2. En particulier, comme on est dans un contexte où prendre deux fois le dual ne ramène pas l'espace initial, on ne peut pas travailler avec des Q-variétés (celles-ci étant les "dual" 8-algébroïdes de Lie) : il est beaucoup plus compliqué de traiter les morphismes et les homotopies.

Ce faisant, plusieurs limitations de [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] sont surmontées. Alors que [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] ne s'appliquait qu'aux feuilletages singuliers qui étaient analytiques, algébriques ou localement réels sur un sous-ensemble ouvert relativement compact, la présente thèse associe une classe d'homotopie naturelle 8-algébroïdes de Lie à toute algèbre de Lie-Rinehart, et en particulier a) à tout feuilletage singulier sur une variété lisse, (finiment engendré ou non). Cette construction fonctionne toujours avec des feuilletages singuliers au sens de Stefan-Sussmann par exemple, ou avec n'importe quel module involutif C 8 pM q dans ΓpAq.

b) à toute variété affine, à laquelle on associe son algèbre de Lie-Rinehart des champs de vecteurs), et plus généralement aux dérivations de toute algèbre commutative, c) aux algébroïdes de Lie singulière au sens d'Androulidakis et de Zambon [START_REF] Androulidakis | Integration of Singular Algebroids[END_REF], d) à divers contextes inattendus, par ex. Champs de vecteurs de Poisson d'une variété de Poisson, vue comme une algèbre de Lie-Rinehart sur des fonctions de Casimir, ou symétries d'un feuilletage singulier, vue comme une algèbre de Lie-Rinehart sur des fonctions constantes sur les feuilles.

Ces 8-algébroïdes de Lie sont construits sur des résolutions par des O-modules libres de l'algèbre de Lie-Rinehart initiale sur O. Ils sont universels dans un certain sens (voir Section 4.2), et ils sont aussi en particulier uniques à homotopie près. Une algébrisation similaire des principaux résultats de [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF], utilisant la catégorie des semimodèles, apparue récemment dans Yaël Frégier et Rigel A. Juarez-Ojeda [START_REF] Frégier | Homotopy theory of singular foliations[END_REF]. Il y a de fortes similitudes entre nos résultats et les leurs, mais les morphismes et les homotopies dans [START_REF] Frégier | Homotopy theory of singular foliations[END_REF] ne correspondent pas aux nôtres. Il est cependant fort possible que le théorème 4.2.1 puisse être récupéré à l'aide de leurs résultats. Luca Vitagliano [START_REF] Vitagliano | On the strong homotopy associative algebra of a foliation[END_REF] a également construit des structures 8-algèbres de Lie à partir de feuilletages réguliers, qui sont bien sûr un cas particulier de l'algèbre de Lie-Rinehart. Ces constructions n'ont pas les mêmes finalités. Pour les feuilletages réguliers, notre structure 8algébroide de Lie est triviale dans le sens où c'est un algébroide de Lie, alors que ses structures deviennent triviales lorsqu'une bonne sous-variété transverse existe. Lars Kjeseth [START_REF] Kjeseth | Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs[END_REF][START_REF] Kjeseth | A homotopy Lie-Rinehart resolution and classical BRST cohomology[END_REF] a aussi une notion de résolution des algèbres de Lie-Rinehart. Mais sa construction est davantage dans l'esprit de la résolution de Koszul-Tate : la définition 1. dans [START_REF] Kjeseth | Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs[END_REF] définit les résolutions des algèbres de Lie-Rinehart comme des résolutions de leur coalgèbre de Chevalley-Eilenberg, et non de l'algèbre de Lie-Rinehart elle-même en tant que module. Il répond à une autre catégorie de questions, liées à BRST et à la recherche de modèle cohomologique pour la cohomologie de l'algèbre de Lie-Rinehart.

Ces résultats peuvent être utilisés pour comprendre la géométrie des feuilletages singuliers tels que leurs symétries. Plus précisément, Soit pM, Fq une variété feuilletée. Une symétrie globale d'un feuilletage singulier F sur M est un difféomorphisme ϕ : M ÝÑ M qui préserve F, soit ϕ ˚pFq " F. L'image d'une feuille à travers une symétrie globale est à nouveau une feuille (pas nécessairement la même feuille). Pour G un groupe de Lie, une action stricte par symétrie de G sur une variété feuilletée pM, Fq est une action lisse G ˆM ÝÑ M qui agit par symétries globales [START_REF] Garmendia | Quotients of singular foliations and Lie 2-group actions[END_REF]. Infinitésimalement, il correspond à un morphisme d'algèbre de Lie g ÝÑ XpM q entre l'algèbre de Lie pg, r¨, ¨sg q de G et l'algèbre de Lie des symétries de F.

Une action de symétrie stricte de G sur M passe au quotient sur l'espace feuille M {F, même si ce dernier espace n'est pas une variété. La direction opposée est plus sophistiquée, puisqu'une action de symétrie stricte de G sur M {F n'induit pas d'action stricte sur M en général. Cependant, il est logique de considérer des applications linéaires ϱ : g ÝÑ XpM q qui satisfont rϱpxq, Fs Ă F pour tout x P g, et qui sont des morphismes d'algèbre de Lie à F près, à savoir, ϱprx, ys g q ´rϱpxq, ϱpyqs P F pour tous les x, y P g. Ces dernières applications linéaires sont appelées "actions faibles par symétrie". Ces actions induisent une "action stricte"sur l'espace des feuilles, c'est-à-dire un morphisme d'algèbre de Lie g ÝÑ XpM {Fq, chaque fois que M {F est une variété.

Insistons sur l'observation suivante : Une action infinitésimale d'une algèbre de Lie g sur une variété différentielle M est un morphisme d'algèbre de Lie g ÝÑ XpM q. En remplaçant M par une Q-variété pE, Qq, on s'attend à les définir comme des morphismes de 8-algèbres de Lie g ÝÑ XpE, Qq, ce dernier espace étant un DGLA. Divers résultats à ce sujet sont donnés dans Mehta-Zambon [START_REF] Rajan | L 8 -algebra actions[END_REF]. En particulier, ces auteurs en donnent plusieurs définitions et interprétations équivalentes.

Au vu de [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF][START_REF] Lavau | Lie 8-algebroids and singular foliations[END_REF], derrière chaque feuilletage singulier ou plus généralement toute algèbre de Lie-Rinehart (voir [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF]) il existe une structure 8-algébroïde de Lie qui est unique à homotopie près appelée l'8-algébroïde de Lie universelle. Voici une question naturelle : qu'induit une symétrie d'un feuilletage singulier F sur une 8-algébroïde de Lie universelle de F ? Le théorème 8.3.1 de cette thèse donne une réponse à cette question. Il dit que toute action de symétrie faible d'une algèbre de Lie sur un feuilletage singulier F peut être relevée en un 8-morphisme de Lie à valeur dans le DGLA des champs de vecteurs sur un 8-algébroïde de Lie universel de F. Dans cette thèse, un tel 8-morphisme de Lie est appelé un relèvement de l'action par symétrie. De tels 8-morphismes de Lie ont été étudiés par Mehta et Zambon [START_REF] Rajan | L 8 -algebra actions[END_REF] comme "actions de L 8 -algèbres". Cela va dans le même sens que [START_REF] Garmendia | Quotients of singular foliations and Lie 2-group actions[END_REF] qui soulignait déjà les structures à 2-groupes de Lie associées à l'action stricte par symétrie de groupes de Lie. De plus, le théorème 8.3.1 dit que ce relevé est une équivalence à homotopie près.

Ce résultat donne plusieurs conséquences géométriques. Voici une question élémentaire : une action d'algèbre de Lie g Ñ XpW q sur une variété affine W Ă C d peut-elle être étendue à une action d'algèbre de Lie g Ñ XpC d q sur C d ? En d'autres termes : il est trivial que tout champ de vecteurs sur W s'étende à C d , mais cette extension peut-elle être faite de telle manière qu'elle préserve le crochet de Lie ? Bien qu'aucun "8-oïds" n'apparaisse dans la question, qui semble être une pure question de géométrie algébrique, nous affirmons que la réponse passe par les 8-algebroïds de Lie et les feuilletages singuliers. Plus précisément, l'idée est alors de dire que toute g-action sur W induit une action faible par symétrie sur le feuilletage singulier I W XpC d q de tous les champs de vecteurs s'annulant sur W (ici I W est l'idéal qui définit W ). Par le théorème 8.3.1, nous savons qu'il est possible de relever toute action faible par symétrie de feuilletage singulier en un 8-morphisme de Lie sur l'universelle. Mais est-il possible de construire un tel 8-morphisme de Lie où le degré polynomial ´1 du coefficient de Taylor du second ordre est nul ? Il existe des obstructions cohomologiques. Dans certains cas spécifiques, des classes d'obstruction apparaissent sur certaines cohomologies, bien qu'en général l'obstruction soit plutôt une équation de type Maurer-Cartan qui peut ou non avoir des solutions. Nous montrons que les deux questions sont en fait liées.

Voici une autre question intéressante à laquelle nous aimerions appliquer nos résultats : Peuton désingulariser une variété affine singulière W Ď C d en utilisant l'8-algébroïde de Lie universelle du feuilletage singulier F " XpW q de champs de vecteurs tangents à W ? Dans la section 7.2.2, nous utilisons la résolution géométrique du feuilletage singulier (c'est-à-dire la résolution sur laquelle est construit le 8-algébroïde de Lie universel) pour retrouver plusieurs notions de résolution des singularités : l'une étant due à Nash [START_REF] Lê | Geometry of Tangents on Singular Spaces and Chern Classes[END_REF] et une seconde à Mohsen [START_REF] Mohsen | Blow-up groupoid of singular foliations[END_REF]. Mais la signification de ces espaces n'est pas claire. Nous voudrions les mettre en relation avec les crochets supérieurs de l'8-algébroïde de Lie universelle, par ex. comprendre le rôle du 3-crochet dans cette procédure.

Dans le dernier chapitre de la thèse, nous introduisons la notion de "tours de bi-submersions" sur des feuilletages singuliers que nous notons T B . Cette dernière notion, comme son nom l'indique, est une famille de "bi-submersions" qui se construisent les unes sur les autres. Le concept de bi-submersion sur des feuilletages singuliers a été introduit dans [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF] et il est utilisé dans la K-théorie [START_REF] Androulidakis | A Baum-Connes conjecture for singular foliations[END_REF] ou la géométrie différentielle [START_REF] Androulidakis | Pseudodifferential calculus on a singular foliation[END_REF][START_REF] Androulidakis | Holonomy transformations for singular foliations[END_REF][START_REF] Garmendia | Hausdorff Morita equivalence of singular foliations[END_REF]. Nous montrons qu'une telle tour de bi-submersion sur un feuilletage singulier F existe si F admet une résolution géométrique. Sous réserve qu'elle existe, on montre dans le théorème 10.1.26 que toute action infinitésimale d'une algèbre de Lie g sur le feuilletage singulier F se relève sur la tour de bi-submersions. Ce relevé ressemble au début d'une sorte d' 8-morphisme de Lie. Nous nous demandons si nous pouvons continuer la construction du théorème 10.1.26 vers un 8-morphisme de Lie.

La thèse est organisée comme suit. Au chapitre 1, nous rappelons quelques bases sur les structures de co-algèbres graduéegraduées, sur l'algèbre de symétrie graduée et leurs morphismes. Nous passons également en revue la notion de co-dérivations et leurs propriétés. Au chapitre 2, nous introduisons la notion de Lie 8-algebroid sur une algèbre unitaire commutative arbitraire O sur un corps de caractéristique nulle, et définissons également leurs morphismes en termes de co-dérivations. Nous définissons la notion d'homotopie entre eux. Quelques Lemmes et Propositions techniques sont donnés. Au chapitre 3, nous fixons les notations et passons en revue les définitions, exemples, et donnent les principales propriétés des algèbres de Lie-Rinehart. De plus, nous construisons une structure 2-algébroïde de Lie pour toute algèbre de Lie-Rinehart qui admet une résolution libre de longueur inférieure ou égale à 2. Au chapitre 4, nous énonçons et prouvons les principaux résultats de la première partie de la thèse, c'est-à-dire l'équivalence de catégories entre les algèbres de Lie-Rinehart et les classes d'homotopie d'8-algébroïdes de Lie acycliques libres, ce qui justifie le nom 8-algèbroïde de Lie universelle d'une algèbre de Lie-Rinehart. Aussi, nous décrivons les 8-algébroïdes de Lie universelle de plusieurs algèbres de Lie-Rinehart. La complexité atteinte par le les crochets supérieure dans ces exemples devraient nous convaincre qu'il ne s'agit pas d'une structure triviale, même pour des algèbres de Lie-Rinehart relativement simples. Le chapitre 5 est consacré aux applications des résultats des chapitres précédents aux variétés affines. Nous rappelons quelques définitions et théorèmes de base sur les variétés affines W . Nous présentons trois constructions principales de Lie-Rinehart associées à W et relions ensemble leurs 8-algébroïdes de Lie universelle. De plus, quelques exemples d'8-algébroïdes de Lie universels sont donnés, tels que les éclatements, des champs de vecteurs qui s'annulent sur une intersection complète, etc. Au chapitre 6, nous rappelons la notion de Q-variétés et appliquer les résultats sur les algèbres de Lie-Rinehart pour retrouver les 8-algébroïdes de Lie universelles d'un feuilletage singulier [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]. Ce chapitre se termine par un résultat sur la cohomologie des champs de vecteurs longitudinaux. Au chapitre 7, on rappelle la définition de l'isotropie d'Androulidakis et de Skandalis Algèbre de Lie d'un feuilletage singulier en un point et rappelons de [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] sa relation avec les 8-algébroïdes de Lie universelle d'un feuilletage singulier. Nous terminons par une procédure d'éclatement d'un feuilletage singulier inspiré de O. Mohsen. Dans le chapitre 8, nous étudions les symétries des feuilletages singuliers. Nous présentons quelques définitions et faits sur les actions faibles par symétrie des algèbres de Lie sur les feuilletages singuliers et donnons quelques exemples. Nous énonçons les résultats principaux de la deuxième partie de la thèse et présentons leurs preuves. Dans le chapitre 9, nous définissons une classe d'obstruction pour étendre une action d'algèbre de Lie sur une variété affine à l'espace ambiant et donnons également quelques exemples. Dans le chapitre 10, nous nous intéressons aux symétries des bi-submersions. Ensuite, nous introduisons la notion de tours de bi-submersions sur un feuilletage singulier et soulignons quelques observations liées à leurs symétries.

Enfin, en annexe A, nous rappelons la définition de l'algèbre tensorielle et fixons les notations. En annexe B, nous rappelons quelques faits généraux sur l'algèbre homologique et donnons quelques constructions géométriques.

Part I

Lie-Rinehart algebras " Acyclic Lie

8-algebroids

CHAPTER 1

Preliminaries

This chapter sets the ground for the whole thesis, especially chapters 2, 3 and 4. It fixes terminologies, conventions and notations. For more details, we also refer the reader to Appendix A and B. Throughout this thesis, K is a field of characteristic zero, and O is an associative commutative unital K-algebra unless otherwise mentioned. Also, an O-module E is seen as K-vector space in the natural way, λ¨e :" pλ¨1 O q¨e, where 1 O " 1 is the unit of O. In the sequel, we will drop the notation "¨".

Geometrically, O can be understood as the algebra of smooth functions on a manifold M , or on an open subset U Ă M of a complex manifold, or the coordinate ring of an affine variety W .

Graded symmetric algebras

For E " ' iPZ E i be a Z-graded module, we denote by |x| P Z the degree of a homogeneous element

x P E.

• We denote by

Ä ' E and call (reduced) graded symmetric algebra of E over O the quotient

T ' O E{xx b O y ´p´1q |x||y| y b O xy
of the tensor algebra (see Appendix A) over O of E, namely

T ' O E :" 8 à k"1 E b O ¨¨¨b O E looooooomooooooon k times
by the ideal generated by x b O y ´p´1q |x||y| y b O x, with x, y arbitrary homogeneous elements of E. This quotient is a graded commutative algebra. We denote its product by d.

• Similarly, we denote by S ' K pEq and call (reduced) graded symmetric algebra of E over the field K the quotient of the tensor algebra (over K) of E, i.e.,

T ' K E :" 8 à k"1 E b K ¨¨¨b K E looooooomooooooon k times
1.1. GRADED SYMMETRIC ALGEBRAS by the ideal generated by x b K y ´p´1q |x||y| y b K x, with x, y arbitrary homogeneous elements of E. We denote by ¨the product in S ' K pEq.

The algebras Ä ' E and S ' K pEq come equipped with two different "grading" . Let us make them explicit.

1. We define the degree of x " x 1 d ¨¨¨d x n P Ä n E or x " x 1 ¨¨¨¨¨x n P S n K pEq by

|x 1 ¨¨¨¨¨x n | " |x 1 d ¨¨¨d x n | " |x 1 | `¨¨¨`|x n |
for any homogeneous x 1 , . . . , x n P E. With respect to this degree, Ä ' E and S ' K pEq are graded commutative algebras.

2. The second grading is called "polynomial-degree". The polynomial-degree of x 1 d ¨¨¨d x n P

Ä n E or x 1 ¨¨¨¨¨x n P S n K pEq is defined to be n. We have the following polynomial-degree decomposition Ä ' E " ' kě1

Ä k E and S ' K pEq " ' kě1 S k K pEq, where Ä k E and S k K pEq stand for the O-module of elements of polynomial-degree k and the K-vector space of elements of polynomial-degree k, respectively.

Convention 1.1.1. For E a graded O-module, the set of elements of polynomial-degree k and degree

d in Ä ' E (resp. S k K pEq) shall be denoted by Ä k E | d (resp. S k K pEq | d ). For example, k ä E | d " à i 1 `¨¨¨`i k "d E i 1 b O ¨¨¨b O E i k .
• For any homogeneous elements x 1 , . . . , x k P E and σ P S k a permutation of t1, . . . , ku, the Koszul sign ϵpσ; x 1 , . . . , x k q is defined by:

x σp1q d ¨¨¨d x σpkq " ϵpσ; x 1 , . . . , x k q x 1 d ¨¨¨d x k .

We often write ϵpσq for ϵpσ; x 1 , . . . , x k q.

• For i, j P N, a pi, jq-shuffle is a permutation σ P S i`j such that σp1q ă . . . ă σpiq and σpi `1q ă . . . ă σpi `jq, and the set of all pi, jq-shuffles is denoted by Spi, jq. More generally, a pi 1 , . . . , i k qshuffle is a permutation σ P S i 1 `¨¨¨`i k such that σp1q ă ¨¨¨ă σpi 1 q σpi 1 `1q ă ¨¨¨ă σpi 1 `i2 q, . . . σpi 1 `i2 `¨¨¨`i j´1 `1q ă ¨¨¨ă σpi 1 `i2 `¨¨¨`i j q, . . . σpi 1 `i2 `¨¨¨`i k´1 `1q ă ¨¨¨ă σpi 1 `i2 `¨¨¨`i k q

The set of all pi 1 , . . . i k q-shuffles is denoted by Spi 1 , . . . , i k q.

Graded symmetric co-algebras and their morphisms

Graded co-algebra structures are the dual version of graded algebra structures, i.e., they are obtained by reversing all arrows and permutes the order of composition. Let us recall the definition of a coalgebra structure on an O-module. We refer the reader to, e.g [START_REF] Vallette | Algebraic Operads[END_REF][START_REF] Kassel | Quantum Groups[END_REF][START_REF] Manetti | Lie Methods in Deformation Theory[END_REF] or Chapter 8 of [START_REF] Manetti | Lectures on deformations of complex manifolds[END_REF] for more details on this topic. Equivalently, the following diagrams commute

C ∆ ∆ / / C b C ∆bid C b C idb∆ / / C b C b C C ∆ ∆ $ $ C b C τ / / C b C.
The pair pC, ∆q is called graded O-co-algebra. We say that pC, ∆q is counital if there is a morphism of graded vector spaces u : C Ñ K such that pu b idq ˝∆ " pid b uq ˝∆ " id i.e. the diagram below commutes

C id * * ∆ ∆ / / C b C idbu C b C ubid / / K b C » C » C b K.
Proposition 1.2.2. Let pC, ∆q and pC 1 , ∆ 1 q be two co-algebras over O. Then, the tensor product

pC b C 1 , pid C b τ b id C 1 q ˝∆ b ∆ 1 q is a co-algebra over O.
Example 1.2.3. The polynomial ring Krts in one indeterminate t is a co-algebra with the coproduct ∆ given on monomials t n , n ě 0 by:

∆pt n q " n ÿ k"0 ˆn k ˙tk b t n´k
and extends by linearity.

The graded symmetric algebra is also an example of co-algebra.

1 Note that this condition is automatically satisfied when the module is concentrated in positive degree.

Lemma 1.2.4. Let E be a O-module. Both Ä ' E and S ' K pEq admit natural co-algebra structures with respect to the deconcatenation ∆ defined by: Remark 1.2.6. According to Definition 1.2.1, the coproduct ∆ is graded with respect to the positive grading given by the "polynomial-degree" of monomials in the symmetric algebras Ä ' E and S ' K pEq.

∆px 1 d ¨¨¨d x n q " n´1 ÿ i"1 ÿ σPSpi,n´iq
Example 1.2.7. Consider M " R d with global coordinates px 1 , . . . , x d q. The algebra of differential forms pΩpM q, ^q over M has a co-algebra structure ∆ Ω : ΩpM q Ñ ΩpM q b ΩpM q given as in Lemma 1.2.4. Namely,

∆pdx 1 ^¨¨¨^dx n q " n´1 ÿ i"1 ÿ σPSpi,n´iq ϵpσqdx σp1q ^¨¨¨^dx σpiq b dx σpi`1q ^¨¨¨^dx σpnq .
This matches (1.1) if we consider E " Ω 1 pM q is concentrated in degree `1.

Co-morphisms and co-derivations

We recall the definitions of co-morphisms, co-derivations of graded co-algebra structures.

Definition 1.2.8. Let pC 1 , ∆ 1 q and pC, ∆q be two graded co-algebras.

1. A morphism of co-algebras or co-morphism from pC 1 , ∆ 1 q to pC, ∆q is a linear map Φ : C 1 ÝÑ C of degree 0 such that the following diagram commutes

C 1 ∆ 1 Φ / / C ∆ C 1 b C 1 ΦbΦ / / C b C (1.2)
In formula: ∆ ˝Φ " pΦ b Φq ˝∆1 .

2. Let Φ : C 1 ÝÑ C be a co-morphism. A Φ-co-derivation of degree l is a linear map H : C 1 ÝÑ C of degree l such that the following diagram commutes,

C 1 ∆ 1 H / / C ∆ C 1 b C 1 HbΦ`ΦbH / / C b C (1.3)
that is, the so-called co-Leibniz identity is satisfied:

∆ ˝H " pH b Φ `Φ b Hq ˝∆1 . (1.4)
When C 1 " C and Φ " id we speak of a co-derivation (of degree l).

Remark 1.2.9. It is easily checked that the composition of two co-morphisms

C 2 Ψ Ñ C 1 Φ Ñ C is again a co-morphism.
Proposition 1.2.10. We denote by CoDerpCq the linear space of co-derivations of pC, ∆q. The set CoDerpCq is a graded Lie sub-algebra of HompC, Cq when equipped with the graded commutator.

Proof. For two co-derivations H 1 , H 2 P CoDerpCq one has,

∆ ˝H1 ˝H2 " pH 1 b id `id b H 1 q ˝pH 2 b id `id b H 2 q ˝∆ " pH 1 ˝H2 b id `H1 b H 2 `p´1q |H 1 ||H 2 | H 2 b H 1 `id b H 1 ˝H2 q ˝∆.
Also, we obtain a similar equation for p´1q

S ' K pE 1 q Ñ S ' K pEq is said to be of polynomial-degree/degree r P Z, if it sends polynomials of S k K pEq to those of S k´r K pEq/elements of S ' K pEq | d to S ' K pEq | d`r . In formulas, for very k ě 0, (resp. d P Z) one has Φ ´Sk K pEq ¯Ď S k´r K pEq, resp. Φ `S' K pEq | d ˘Ď S ' K pEq | d`r
. Remark 1.2.12. Any linear map Φ : S ' K pE 1 q Ñ S ' K pEq of degree l can be decomposed with respect to the polynomial-degree as formal sum:

Φ "

ÿ rPZ Φ prq (1.5)
where, for all k P N 0 , Φ prq : S ' K pE 1 q Ñ S '´r K pEq is a linear map of polynomial-degree r and of degree l.

• It should be understood that Φ prq pvq " 0 for all v P S k K pE 1 q with k ď r. Also, the decomposition (1.5) makes sense in particular when the polynomial-degrees of Φ are lower bounded that is, there is an integer N P Z, such that Φ prq " 0 for all r ă N . In that case, for every v P S k K pEq,

ÿ rPZ Φ prq pvq " ÿ N ďrăk Φ prq pvq, is a finite sum. Hence, Equation (1.5) becomes Φ " ÿ rěN Φ prq ,
for some N P Z, since the l.h.s of (1.5) is finite. Notice that a linear map Φ : S ' K pE 1 q Ñ S ' K pEq is of polynomial-degree N if and only if Φ pN q is the unique non-zero term, namely Φ prq " 0, for r ‰ N .

• Also, for the composition of two linear maps

S ' K pE 2 q Ψ Ñ S ' K pE 1 q Φ Ñ S ' K pEq, we have pΦ ˝Ψq prq " ÿ i`j"r Φ piq ˝Ψpjq .
(1.6) for every r P Z. Here, although the sum is infinite, it becomes finite when applied to a given element in S ' K pE 1 q.

Co-morphisms on symmetric algebras

Let us have a discussion on co-morphisms from the symmetric graded co-algebras pS ' K pE 1 q, ∆ 1 q and pS ' K pEq, ∆q, where ∆ 1 and ∆ are the respective coproducts like in Lemma 1.2.4. Given any linear map F : S ' K pE 1 q Ñ E. Denoting by F r : S r`1 K pE 1 q Ñ E for r P N 0 , the restriction of F to S r`1 K pE 1 q. The linear map F can be extended to a unique co-morphism F : pS ' K pE 1 q, ∆ 1 q Ñ pS ' K pEq, ∆q by taking for k P N the component on S k K pEq to be, for any homogeneous x 1 , . . . ,

x n P E 1 ÿ i 1 `¨¨¨`i k "n ÿ σPSpi 1 ,...,irq ϵpσq 1 k! k ź j"1 F i j px σpi 1 `¨¨¨`i j´1 `1q , . . . , x σpi 1 `¨¨¨`i j q q. (1.7)
where Spi 1 , . . . , i k q is the set of pi 1 , . . . , i k q-shuffles, with i 1 , . . . , i k P N.

Example 1.2.13. Let us compute F : `Sk K pE 1 q, ∆ 1 ˘ÝÑ pS ' K pEq, ∆q for k " 1, 2, 3. For x, y, z P E 1 ,

• F pxq " F 0 pxq.

• F px ¨yq " F 1 px ¨yq `F0 pxq ¨F0 pyq.

• F px ¨y ¨zq " F 2 px ¨y ¨zq `F0 pxq ¨F1 py ¨zq `p´1q |x||y| F 0 pyq ¨F1 px ¨zq p´1q p|x|`|y|q|z| F 0 pzq ¨F1 px ¨yq `F0 pxq ¨F0 pyq ¨F0 pzq.

We can easily check that ∆ ˝F " p F b F q ˝∆1 , e.g • ∆ ˝F pxq " ∆ ˝F0 pxq " 0 " p F b F q ˝∆1 pxq • On one side, ∆ ˝F px ¨yq " ∆ ˝F1 px ¨yq `∆pF 0 pxq ¨F0 pyqq " F 0 pxq b F 0 pyq `p´1q |x||y| F 0 pyq b F 0 pxq.

On the other side,

p F b F q ˝∆1 px, yq " p F b F qpx b y `p´1q |x||y| y b xq " p F b F qpx b yq `p´1q |x||y| p F b F qpy b xq " F 0 pxq b F 0 pyq `p´1q |x||y| F 0 pyq ¨F0 pxq.
The following proposition claims that every co-morphism from S ' K pE 1 q to S ' K pEq is of the form described in (1.7). For a proof, see [START_REF] Vallette | Algebraic Operads[END_REF][START_REF] Kassel | Quantum Groups[END_REF][START_REF] Manetti | Lectures on deformations of complex manifolds[END_REF]. Proposition 1.2.14. Let E 1 , E be two graded O-modules. For any morphism of graded vector space F : S K pE 1 q ÝÑ E there exists an (unique) morphism of graded co-algebras F : pS K pE 1 q, ∆ 1 q ÝÑ pS K pEq, ∆q that satisfies pr ˝F " F .

Here pr is the canonical projection onto the term of polynomial-degree 1, i.e., pr :

S ' K pEq Ñ S 1 K pEq » E.
Remark 1.2.15. The following facts are crucial, and will be used in the sequel.

1. By Proposition 1.2.14, a co-morphism Φ : S ' K pE 1 q Ñ S ' K pEq is entirely determined by the collection indexed by r P N 0 of maps called its r-th Taylor coefficients:

Φ r : S r`1 K pE 1 q Φ ÝÑ S ' K pEq pr ÝÑ E, (1.8)
Notice that the component Φ prq of polynomial-degree r ě 0 of Φ coincides with r-th Taylor coefficient Φ r on S r`1 K pE 1 q. In fact we have, Φ r " ppr ˝Φprq q |S r`1 K pE 1 q .

2. By item 1, a co-morphism Φ : S ' K pE 1 q Ñ S ' K pEq is completely determined by its components of polynomial-degree r ě 0, hence it admits a polynomial decomposition of the form:

Φ " ÿ rě0 Φ prq .
(1.9)

3. Similar results as in Proposition 1.2.14 hold for Φ-co-derivations or for co-derivations, e.g. see Corollary VIII.34 in [START_REF] Manetti | Lectures on deformations of complex manifolds[END_REF] or [JLL]. Given a co-morphism Φ : S ' K pE 1 q Ñ S ' K pEq a Φ-coderivation H of degree l is entirely determined by the Taylor coefficients defined as in (1.8)

H r : S r`1
K pE 1 q Ñ E, r P N 0 . For k P N 0 and x 1 , . . . , x n P E 1 , the component on S k K pEq of Hpx 1 ¨¨¨¨¨x n q takes the form

ÿ i 1 `¨¨¨`i k "n ÿ σPSpi 1 ,...,i k q ϵpσq 1 k! H i 1 px σp1q , ¨¨¨, x σpi 1 q q ¨k´1 ź j"1 Φ i j ´1px σpi 1 `¨¨¨`i j´1 `1q , . . . , x σpi 1 `¨¨¨`i j q q.
(1.10)

Also, H has a decomposition into polynomial-degree of the form :

H " ÿ rě0 H prq . (1.11)
Example 1.2.16. Every linear map H : S ' K pEq Ñ E of degree l admits an unique extension to a co-derivation H : S ' K pEq Ñ S ' K pEq of degree l. The latter is given explicitly by the formula

Hpx 1 ¨¨¨¨¨x n q " n ÿ i"1 ÿ σPSpi,n´iq
ϵpσqHpx σp1q ¨¨¨¨¨x σpiq q ¨xσpi`1q ¨¨¨¨¨x σpnq .

(1.12)

The following lemma-definition is helpful in the sequel.

Lemma 1.2.17 (Definition). We say that a co-algebra morphism Φ : S ' K pE 1 q Ñ S ' K pEq is O-multilinear when the equivalent conditions below are satisfied:

(i) For every n ě 0, the n-th Taylor coefficient Φ pnq : S n`1 K pE 1 q ÝÑ E of Φ is O-multilinear (ii) There exists an induced co-algebra morphism Φ O : Ä ' E 1 ÝÑ Ä ' E making the following diagram commutative : S ' K pE 1 q Φ / / S ' K pEq Ä ' E 1 Φ O / / Ä ' E
When it is the case we still write Φ for Φ O .

Proof. The equivalence is straightforward.

Ä ' E 1 ÝÑ Ä ' E.

Richardson-Nijenhuis bracket and co-derivations

We need to use the Richardson-Nijenhuis bracket to express our results and explain some proofs in the coming chapters. This bracket was defined mainly to understand Lie algebra structures on a vector space and their deformations [START_REF] Nijenhuis | Richardson: Cohomology and deformations in graded Lie algebras[END_REF] as well as Poisson structures and their generalizations. Let us recall the definition in our context.

Let E be an O-module. For k P N 0 , homomorphisms of degree j from S k`1 K pEq to E shall be, by definition, the space

Hom j K ´Sk`1 K pEq , E ¯:" ' mPZ Hom K ´Sk`1 K pEq | m´j , E m ¯.
We refer the reader for example to [START_REF] Kolář | Natural operations in differential geometry[END_REF][START_REF] Kosmann-Schwarzbach | Derived Brackets[END_REF] for the following definition.

i K ´Sp`1 K pEq , E ¯b Hom j K ´Sq`1 K pEq , E ¯Ñ Hom i`j K ´Sp`q`1 K pEq , E
which is defined on homogeneous elements

P P Hom ' K ´Sp`1 K pEq , E ¯and R P Hom ' K ´Sq`1 K pEq , E ¯by rP, Rs RN " P ˝R ´p´1q |P ||R| R ˝P, (1.13)
with the understanding that P ˝R is defined on x 0 ¨¨¨¨x p`q P S p`q`1 K pEq by pP ˝Rqpx 0 ¨¨¨¨x p`q q :" ÿ σPSpq`1,pq ϵpσqP pRpx σp0q ¨¨¨x σpqq q ¨xσpq`1q ¨¨¨¨x σpp`qq q.

(1.14)

The bracket is extended by bilinearity.

Remark 1.2.20. The bracket r¨, ¨sRN should not be confused with the graded commutator r¨, ¨s of the graded vector space Hom K pS ' K pEq, S ' K pEqq. In fact, the graded commutator of two elements P, R P Hom ' K pS ' K pEq, S ' K pEqq is rP, Rs " P ˝R´p´1q |P ||R| R˝P , but here P ˝R is the usual composition of homomorphism.

It is easy to check that Lemma 1.2.21. The graded vector space Hom ' K pS ' K pEq , Eq together with the Richardson-Nihenhuis bracket r ¨, ¨sRN is a graded Lie algebra structure.

For a given i P Z, and a given P " ř kě0 P k with P k P Hom i K ´Sk`1 K pEq, E ¯, we denote by P P CoDerpS ' K pEqq the unique co-derivation of degree i with Taylor coefficients pP k q kPN 0 . The relation between the Richardson-Nijenhuis bracket an co-derivations is described in the following lemma: Lemma 1.2.22. The map P Þ Ñ P is a graded Lie algebra morphism that is, for every P, R of degrees l, r as above, we have Ğ rP, Rs RN " r P , Rs " P ˝R ´p´1q lr R ˝P .

Proof. Linearity follows from uniqueness of the extension P described in Proposition 1.2.14. Indeed, for P, R as above, one should have pr ˝p P `Qq " P `R " pr ˝p Ğ P `Rq. By unicity, we must have Ğ P `R " P `R. Likewise, one has for any λ P K, Ď λP " λ P .

To show that it is a graded Lie algebra morphism, it suffices to check that pr ˝r P , Rs " rP, Rs RN . For

x " x 0 ¨¨¨¨x r P S r`1 K pEq `pr ˝r P , Rs ˘r pxq " P ˝Rpxq ´p´1q |P ||R| R ˝P pxq (1.15) By using Formula (1.12), we obtain for example

P ˝Rpxq " k ÿ j"0 ÿ σPSpj`1,k´jq
ϵpσqP pR j px σp0q ¨¨¨¨¨x σpjq q ¨xσpj`1q ¨¨¨¨¨x σpkq q "

ÿ i `j " k i, j ě 0 ÿ σPSpj`1,iq
ϵpσqP i pR j px σp0q ¨¨¨¨¨x σpjq q ¨xσpj`1q ¨¨¨¨¨x σprq q " ÿ i `j " r i, j ě 0 pP i ˝Rj qpxq with the understanding that P i px 0 ¨¨¨¨x m q " 0 for m ‰ i. As a consequence, the right-hand side of Equation (1.15) becomes

ÿ i `j " r i, j ě 0 ´pP i ˝Rj qpxq ´p´1q |P ||R| pR j ˝Pi qpxq ¯" ÿ i `j " r i, j ě 0 rP i , R j s RN pxq
" prP, Rs RN q r pxq.

Conclusion:

In this chapter, we recalled the co-algebra language, in particular comorphisms, Richardson-Nijenhuis bracket. An important point is not to confuse the tensor products b K and b O , hence the graded symmetric algebra S ' K pEq and S ' O pEq, which is denoted by S ' pEq. This introduction is completed in Appendix A.

CHAPTER 2

Lie 8-algebroids and their morphisms

Definitions

In this section, we present the notion Lie 8-algebroid over the algebra O. We also give geometrical examples. We refer the reader to Appendix B for some generalities on graded modules. By definition, Lie 8-algebras on a graded vector space V are co-derivations of degree ´1 squaring to 0 of the graded symmetric algebra SpV q (e.g see [START_REF] Lada | Introduction to SH Lie algebras for physicists[END_REF][START_REF] Ryvkin | Observables and Symmetries of n-Plectic Manifolds[END_REF]). Lie 8-algebroids generalize Lie 8-algebras and Lie algebroids, but the situation is more sophisticated, because the 2-ary bracket is not O-linear. Let us make some preparations before introducing the definition.

Definition 2.1.1. A derivation of O is a K-linear map X : O Ñ O that satisfies the so-called Leibniz identity Xpf gq " Xpf qg `f Xpgq for all f, g P O.
The set of all derivations of O inherits a Lie K-algebra structure from the Lie algebra End K pOq, whose Lie bracket is the commutator, that is, rX, Y s " X ˝Y ´Y ˝X, for all X, Y derivations of O.

We denote by DerpOq the Lie algebra of derivations of O. Also, for X P DerpOq, Xrf s stands for the derivation X applied to f P O.

Example 2.1.2. Geometrically, derivations of O can be understood as follows:

1. when O " C 8 pM q is the algebra of functions of a smooth manifold M , as vector fields on a smooth manifold M ;

2. when O is the algebra of holomorphic functions on an open subset U of C d , as vector fields on U;

3. when O " O W is the coordinate ring of an affine variety W , as vector fields on the affine variety W .

dg-almost

pE ' , ℓ 1 , ℓ 2 , ρq over O is a complex pE, ℓ 1 , ρq : ¨¨¨ℓ 1 ÝÑ E ´3 ℓ 1 ÝÑ E ´2 ℓ 1 ÝÑ E ´1 ρ ÝÑ DerpOq.
of projective O-modules equipped with a graded symmetric degree `1 K-bilinear bracket ℓ 2 " r¨, ¨s :

d 2 E Ñ E such that:
1. ℓ 2 satisfies the Leibniz identity with respect to ρ : E ´1 ÝÑ DerpOq, i.e.

ℓ 2 px, f yq " f ℓ 2 px, yq `ρpxqrf sy (2.1)
for all x P E ´1, y P E and f P O.

2. ℓ 1 is degree `1-derivation of ℓ 2 , i.e. for all x P E ´i, y P E: ℓ 1 pℓ 2 px, yqq `ℓ2 pℓ 1 pxq, yq `p´1q i ℓ 2 px, ℓ 1 pyqq " 0, 3. ρ is a morphism, i.e. for all x, y P E ´1 ρpℓ 2 px, yqq " rρpxq, ρpyqs.

The O-linear map ρ is called the anchor map, and ℓ 1 the differential. Case 3: for x, y, z P E ´1, the proof is identical to case 2.

Example 2.1.6. A differential graded Lie algebra (DGLA) (see e.g [START_REF] Manetti | Deformation theory via differential graded lie algebras[END_REF]) is a graded vector space g " à jPZ g j over K together with a bilinear map called graded Lie bracket r¨, ¨s : g i ˆgj Ñ g i`j and a differential map d : g i Ñ g i´1 such that for all homogeneous elements v 1 , v 2 , v 3 P g has the properties

• graded commutativity:

rv 1 , v 2 s " ´p´1q |v 1 ||v 2 | rv 2 , v 1 s,
• Jacobi's identity:

p´1q |v 1 ||v 3 | rv 1 , rv 2 , v 3 ss `p´1q |v 2 ||v 1 | rv 2 , rv 3 , v 1 ss `p´1q |v 3 ||v 2 | rv 3 , rv 1 , v 2 ss " 0, • Leibniz's identity: drv 1 , v 2 s " rdv 1 , v 2 s `p´1q |v 1 | rv 1 , dv 2 s.
DGLAs are examples of almost differential graded Lie algebroid (ADGLA) with O " K (and g j " 0 for j ě 0). We take E ´i :" g i r1s for i ě 1, ℓ 1 pxq :" ´dpxq, and ℓ 2 px, yq :" p´1q |x| rx, ys. Of course, here ρ " 0. Note that ADGLAs are more general than DGLAs, since it does not impose Jacobi identity.

Example 2.1.7. [START_REF] Huebschmann | Higher homotopies and Maurer-Cartan algebras: Quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras[END_REF] An almost-Lie algebroid over a manifold M is a triple pA Ñ M, r¨, ¨sA , ρ A q made of a vector bundle A Ñ M , a skew-symmetric bracket r¨, ¨sA : ΓpAq ˆΓpAq Ñ ΓpAq, fulfilling the Leibniz identity, i.e. for all a, b P ΓpAq, f P C 8 pM q ra, f bs A " f ra, bs A `ρA paqrf sb, and a vector bundle morphism ρ : A Ñ T M , that satisfies, ρpra, bs A q " rρ A paq, ρ A pbqs.

We say that ρ A is the anchor of pA Ñ M, r¨, ¨sA , ρ A q. When r¨, ¨sA satisfies Jacobi's identity, we speak of a Lie algebroid over M [START_REF] Kirill | General theory of Lie groupoids and Lie algebroids[END_REF].

Almost Lie algebroids over a manifold M give examples of almost differential graded Lie algebroid over O " C 8 pM q with E ´1 " ΓpAq, E ´i " 0 for i ‰ 1, ℓ 2 " r¨, ¨sA , and the anchor is ρ A .

Lie 8-algebroids over O

Lie 8-algebroids over manifolds were introduced (explicitly or implicitly) by various authors, e.g. [START_REF] Sati | Twisted differential string and fivebrane structures[END_REF], [START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF], and [Še01]. We refer to Giuseppe Bonavolontà and Norbert Poncin for a complete overview of the matter [START_REF] Bonavolontà | On the category of Lie n-algebroids[END_REF]. It extends the notion of almost differential graded Lie algebroids. piq the higher Jacobi identity:

n ÿ i"1 ÿ σPS i,n´i
ϵpσq ℓ n´i`1 pℓ i px σp1q , . . . , x σpiq q, x σpi`1q , . . . , x σpnq q " 0, (2.3) for all n ě 1 and homogeneous elements x 1 , . . . , x n P E,

piiq for i ‰ 2, the bracket ℓ i is O-linear, while for i " 2, ℓ 2 px, f yq " ρpxqrf s y `f ℓ 2 px, yq for all x, y P E, f P O ,
where, by convention, ρ E is extended by zero on E ´i for all i ě 2,

piiiq ρ ˝ℓ1 " 0 on E ´2,
pivq ρ is a morphism of brackets, i.e., ρpℓ 2 px, yqq " rρpxq, ρpyqs for all x, y P E ´1.

We denote it by pE ' , ℓ ' , ρq.

Convention 2.1.9. From now on, we simply say "Lie 8-algebroid" for "negatively graded Lie 8algebroid".

Let us explain these axioms.

Remark 2.1.10. It follows from Definition 2.1.8 that:

1. The higher Jacobi identity is equivalent to

n ÿ i"1 rℓ i , ℓ n`1´i s RN " 0,
for all positive integer n. See Definition 1.2.19 a definition of r¨, ¨sRN .

2. For n " 1, higher Jacobi identity yields ℓ 1 ˝ℓ1 " 0. Thus,

¨¨¨ℓ 1 ÝÑ E ´3 ℓ 1 ÝÑ E ´2 ℓ 1 ÝÑ E ´1 (2.4)
is a complex of projective O-modules.

3. Higher Jacobi identity for n " 2 reads ℓ 1 pℓ 2 px, yqq `ℓ2 pℓ 1 pxq, yq `p´1q |x| ℓ 2 px, ℓ 1 pyqq " 0.

4. The third and the fourth axiom are consequences of item piq, and piiq if O has no zero divisors1 on E ´1. Indeed, for all x P E ´2 and y P E ´1 and f P O. Higher Jacobi identity specialized at n " 2 and Leibniz identity read

ℓ 1 pℓ 2 px, f yqq " ℓ 2 pℓ 1 pxq, f yq " f ℓ 2 pℓ 1 pxq, yq `ρpℓ 1 pxqqrf sy
Using Leibniz identity again, the left-hand side of the equation above also reads ℓ 1 pℓ 2 px, f yqq " f ℓ 1 pℓ 2 px, yqq. Hence: ρpℓ 1 pxqqrf sy " 0.

If O has no zero divisors on E ´1, then ρpℓ 1 pxqqrf s " 0. Since x and f are arbitrary, we have

ρ ˝ℓ1| E ´2 " 0.
Also, by writing higher Jacobi for n " 3 on elements x, y, z P E ´1 and f P O while using Leibniz identity we get 0 " ℓ 1 pℓ 3 px, y, f zqq `ℓ2 pℓ 2 px, yq, f zq ´ℓ2 pℓ 2 px, f zq, yq `ℓ2 pℓ 2 py, f zq, xq

" f ℓ 1 pℓ 3 px, y, zqq `f ℓ 2 pℓ 2 px,

Lie 8-algebroids as homological co-derivations

When it comes to manipulating morphisms of Lie 8-alegbroids, it quickly becomes quite tedious. Therefore, it is very useful to dualize in order to make the notion of morphisms clearer.

In the finite dimensional case [START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF], it is usual to see it as a derivation of the symmetric algebra of the dual, i.e. as a Q-manifold (see Chapter 6.1). The duality finite rank Lie 8-algebroids and Q-manifolds is especially efficient to deal with morphisms.

In infinite dimension case, we cannot dualize Lie 8-algebroids in the sense of T. Voronov [START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF][START_REF] Bonavolontà | On the category of Lie n-algebroids[END_REF] anymore, since the identification of SpV q ˚with SpV ˚q does not hold. What I do to deal with this problem in the infinite case, is to stay in the world of squared to zero co-derivations and impose some particular additionnal properties on their Taylor coeficients, related to O-linearity and the Leibniz rule. Now we give an alternative description of Lie 8-algebroids in terms of co-derivation (extending the usual [START_REF] Voronov | Higher derived brackets and homotopy algebras[END_REF][START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF][START_REF] Bonavolontà | On the category of Lie n-algebroids[END_REF] correspondence between Lie 8-algebroids and Q-manifolds in the finite rank case). Definition 2.2.1. A co-derivation Q P CoDerpS K pEqq of degree `1 is said to be an homological coderivation or co-differential when Q 2 " 0.

Given such a co-derivation Q, the triplet pS K pEq, ∆, Qq is then called a differential graded co-algebra.

The following lemma is important. Lemma 2.2.2. A co-derivation Q P CoDerpS K pEqq of degree `1 is an homological co-derivation if and only if pr ˝Q2 " 0.

Proof. By Proposition 1.2.10, the bracket

rQ, Qs " Q ˝Q ´p´1q |Q||Q| Q ˝Q " 2Q 2
is a co-derivation, since co-derivations of a graded co-algebra is closed under the graded commutator bracket r¨, ¨s. Therefore, Q 2 " 1 2 rQ, Qs is a co-derivation, and it is completely determined by pr Q2 .

Remark 2.2.3. The composition of two co-derivations is not a co-derivation in general. In particular, the composition of a co-derivation with itself is not a co-derivation unless it is of odd degree.

For a good understanding of the next proposition, see notations of Taylor coefficients in Equation (1.8).

Proposition 2.2.4. Given a collection E " pE ´iq iě1 of projective O-modules, there is a one-toone correspondence between Lie 8-algebroid structures pE ' , ℓ ' , ρq on E and pairs pQ E , ρq made of an homological co-derivation

Q E : S ' K pEq Ñ S ' K pEq and a O-linear morphism, ρ : E ´1 Ñ DerpOq called the anchor, such that 1. for k ‰ 1 the k-th Taylor coefficient Q pkq E : S k`1 K pE 1 q ÝÑ E of Q E is O-multilinear,
2. for all x, y P E and f P O, we have,

Q p1q E px ¨f yq " f Q p1q E px ¨yq `ρpxqrf s y, 3. ρ ˝Qp0q E " 0 on E ´2,

ρ ˝Qp1q

E px ¨yq " rρpxq, ρpyqs, for all x, y P E ´1.

The correspondence consists in assigning to a Lie 8-algebroid pE ' , ℓ ' , ρq " pℓ 1 , ℓ 2 , ℓ 3 , ¨¨¨q the coderivation Q E whose k-th Taylor coefficient is the k-ary bracket ℓ k`1 for all k P N 0 .

Proof. Take the i-th Taylor coefficient of co-derivation Q E that satisfies the requirements 1. 2. 3. and 4. of Proposition 2.2.4 as, pr

˝Qpiq E " ℓ i`1 : S i`1 K pEq Ñ E. By Lemma 1.2.22 we have 2Q 2 E " rQ E , Q E s " ÿ ně1 ÿ i`j"n`1 Ğ rℓ i , ℓ j s RN .
Thus, by uniqueness of the extension as co-derivation,

Q 2 E " 0 is equivalent to 0 " n ÿ i"1 rℓ i , ℓ n`1´i s RN P Hom 2 K pS n K pEq , Eq .
For all integer n ě 1. ℓ 1 and the binary bracket ℓ 2 are obtained by adding a sign to the differential map d and the graded skew-symmetric Lie bracket r¨, ¨s respectively, and the other brackets ℓ k for k ě 3 vanish. For k ě 3, the k-th Taylor coefficient of the corresponding co-derivation Q g is zero, hence it can be written as

Q g " Q p0q g `Qp1q g . For every homogeneous monomial x 1 ¨¨¨x k P S k g Q p0q g px 1 ¨¨¨x k q " k ÿ i"1 p´1q p|x 1 |`¨¨¨`|x i´1 |q|x i | ℓ 1 px i q ¨x1 ¨¨¨p x i ¨¨¨x k , Q p1q g px 1 ¨¨¨x k q " ÿ 1ďiăjďk p´1q p|x 1 |`¨¨¨`|x j´1 |q|x j |`p|x 1 |`¨¨¨`|x i´1 |q|x i | ℓ 2 px i , x j q ¨x1 ¨¨¨r x i ¨¨¨p x j ¨¨¨x k .
Proposition 2.2.4 and Remark 2.2.5 say that pg " ' PZ g i , r¨, ¨s , dq is a DGLA if and only if

• Q 2 g " 0; or • pQ p0q g q 2 " pQ p1q g q 2 " 0 and Q p0q g ˝Qp1q g `Qp1q g ˝Qp0q g " 0.

Morphisms of Lie 8-algebroids and homotopies

This section extends Section 3.4 of [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] to the infinite dimensional setting.

Morphisms of Lie 8-algebroids

, Q E 1 , ρ 1 q to a Lie 8-algebroid pE, Q E , ρq, is a co-morphism Φ : S ' K pE 1 q ÝÑ S ' K pEq such that Φ ˝QE 1 " Q E ˝Φ
(2.6) and 1. all Taylor coefficients are O-multilinear,

2. which satisfies ρ ˝Φ0 " ρ 1 on E 1 ´1. Above, Φ 0 " Φ p0q | E 1 : E 1 Ñ E is the 0-th Taylor coefficient of Φ. Notice that item 1. is equivalent to saying Φ is O-multilinear in the sense of Lemma 1.2.17.
Remark 2.3.2. Recall from [START_REF] Lada | Introduction to SH Lie algebras for physicists[END_REF] that Lie 8-algebra morphisms from pS ' K pEq, Q E q to pS ' K pE 1 q, Q E 1 q are defined to be co-algebra morphisms Φ : S ' K pE 1 q ÝÑ S ' K pEq that satisfy (2.6). Definition 2.3.1 adds two additional assumptions to turn a Lie 8-algebra morphism into a Lie 8-algebroid morphism.

Remark 2.3.3. For a Lie 8-algebroid morphism Φ : S ' K pE 1 q ÝÑ S ' K pEq, the 0-th Taylor coefficient Φ 0 : pE 1 , ℓ 1 1 q ÝÑ pE, ℓ 1 q of Φ is the chain map, that is, the following diagram commutes

¨¨¨/ / E 1 ´3 Φ 0 ℓ 1 1 / / E 1 ´2 Φ 0 ℓ 1 1 / / E 1 ´1 Φ 0 ρ 1 / / DerpOq id ¨¨¨/ / E ´3 ℓ 1 / / E ´2 ℓ 1 / / E ´1 ρ / / DerpOq (2.7) namely, Φ 0 ˝ℓ1 1 " ℓ 1 ˝Φ0 .
Also, by going a bit further, it satisfies for all x, y P E 1

Φ 0 ˝ℓ1 2 px, yq `Φ1 ˝ℓ1 1 px d yq " ℓ 1 ˝Φ1 px d yq `ℓ2 pΦ 0 pxq, Φ 0 pyqq. (2.8)
Example 2.3.4. Let pA, r¨, ¨sA , ρ A q and pB, r¨, ¨sB , ρ B q be two Lie algebroids over a manifold M . A Lie algebroid morphism ϕ : A ÝÑ B over the identity [START_REF] Kirill | General theory of Lie groupoids and Lie algebroids[END_REF] induces a Lie 8-algebroid morphism S ' pϕq : S ' pΓpAqq Ñ S ' pΓpBqq in the section level: where corresponding co-algebra morphism S ' pϕq is defined as

a 1 ¨¨¨¨a n Þ Ñ ϕpa 1 q ¨¨¨¨ϕpa n q,
for all a 1 , . . . , a n P A.

Example 2.3.5. Lie 8-morphisms of Differential Graded Lie Algebras. Let us consider pg, r¨, ¨sg , d g q and ph, r¨, ¨sh , d h q two differential graded Lie algebras (see Example 2.1.6), where Q g and Q h denote their respective associated co-derivations. Let Φ : pS ' pgr1sq, Q g q ÝÑ pS ' phr1sq, Q h q be a Lie 8-morphism, i.e. one has Φ ˝Qg " Q h ˝Φ.

(2.9)

In particular, Φ 0 ˝dg " d h ˝Φ0 . Let us write down what the restriction of Equation (2.9) to low Taylor coefficients: Let x, y P g be two homogeneous elements. One has, Φpx ¨yq " Φ 1 px ¨yq `Φ0 pxq ¨Φ0 pyq,

Φpxq " Φ 0 pxq.

A direct computation of the LHS of Equation (2.9) applied to x ¨y gives, Φ ˝Qg px ¨yq " Φ ´´d g pxq ¨yq ´p´1q p|x|´1qp|y|´1q d g pyq ¨x `p´1q |x| rx, ys g " ´Φ1 pd g pxq ¨yq ´Φ0 pd g pxqq ¨Φ0 pyq ´p´1q p|x|´1qp|y|´1q pΦ 1 pd g pyq ¨xq ´Φ0 pd g pyqq ¨Φ0 pxqq `p´1q |x| Φ 0 prx, ys g q Also, the RHS of Equation (2.9) applied to x ¨y gives,

Q h ˝Φpx ¨yq " Q h pΦ 1 px ¨yq `Φ0 pxq ¨Φ0 pyqq
" ´dh pΦ 1 pxqq ¨y ´dh pΦ 0 pxqq ¨Φ0 pyq ´p´1q p|x|´1qp|y|´1q d h pΦ 0 pyqq ¨Φ0 pxq `p´1q |x| rΦ 0 pxq, Φ 0 pyqs h Since both sides are equal, and Φ 0 ˝dg " d h ˝Φ0 we obtain, Φ 0 prx, ys g q ´rΦ 0 pxq, Φ 0 pyqs h " p´1q p|x|´1q|y|q Φ 1 pd g pyq ¨xq ´p´1q |x| pΦ 1 pd g pxq ¨yq ´dh pΦ 1 px ¨yqqq .

Homotopies

In this section, we define homotopy between Lie 8-morphisms, and between Lie 8-algebroids. This extends [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] from finite dimensional Q-manifolds to arbitrary Lie 8-algebroids.

A practical definition

Let us consider pΩ ' pra, bsq, ^, d dR q the differential graded algebra made of the forms on ra, bs together with the wedge product and the Rham differential. If we denote by t the coordinate on ra, bs, we have Ω ' pra, bsq " C 8 pra, bsq ' C 8 pra, bsq dt. We equip Ω ' pra, bsq with a co-associative co-algebra structure ∆ : Ω ' pra, bsq Ñ Ω ' pra, bsq b Ω ' pra,bsq Ω ' pra, bsq, given by ∆ Ω p1q " 1 b 1, where we extend by Ω ' pra, bsqlinearity.

Let pE 1 , Q E 1 , ρ 1 q and pE, Q E , ρq be Lie 8-algebroids over O.

Lemma 2.3.6. The triplet

`S' K pEq b K Ω ' pra, bsq, ∆ " pid b τ b idq ˝∆ b ∆ Ω , B " Q E 1 b id `id b d dR ȋs a differential graded co-algebra.
Proof. One should understand that for α P C 8 pra, bsq and for an homogeneous element v P S ' K pEq,

pQ E b id `id b d dR qpv b αq " Q E pvq b α `p´1q |v| v b α 1 dt also, pQ E b id `id b d dR qpv b αdtq " Q E pvq b αdt `v b d dR pαdtq loooomoooon "0 " Q E pvq b αdt.
Clearly, B 2 " 0, see the definition of tensor product of complexes, Appendix B. Let us check that B is indeed a co-derivation w.r.t the co-product ∆. Take α P C 8 pra, bsq and a homogeneous element v P S ' K pEq, we will use the Sweedler notation, ∆pvq " v p1q b v p2q , to avoid a long useless text. On one hand,

∆ ˝Bpv b αq " pid b τ b idq ˝∆ b ∆ Ω pQ E pvq b α `p´1q |v| v b α 1 dtq " id b τ b id ´∆ ˝QE pvq b α b 1 `p´1q |v| ∆pvq b α 1 dt b 1 " pid b τ b idq ˝´pQ E b id `id b Q E q ˝∆pvq b α b 1 `p´1q |v| ∆pvq b α 1 dt b 1 " Q E pv p1q q b α b v p2q b 1 `p´1q |v p1q | v p1q b α b Q E pv p2q q b 1p ´1q |v|`|v p2q | v p1q b pα 1 dtq b v p2q b 1.
On the other hand, Remark 2.3.7. Elements of degree k of S ' K pEq b K Ω ' pra, bsq are K-linear combinations of elements of the form v b α and w b βdt, with α, β P C 8 pra, bsq and v P S ' K pEq |k and w P S ' K pEq |k´1 . The latter can be seen as maps, t P ra, bs Þ Ñ αptqv b 1 and t P ra, bs Þ Ñ βptqw b dt. Hence, elements of degree k of this complex can be considered as element of S K pEq b K Ω ' pra, bsq that depend on t, i.e. as

pB b id `id b Bq ˝∆pv b αq " pB b id `id b Bq ˝pid b τ b idqp∆pvq b α b 1q " pB b id `id b Bqpv p1q b α b v p2q b 1q " Bpv p1q b αq b v p1q b 1 `p´1q |v p1q | v p1q b α b Bpv p2q b 1q " ´QE pv p1q b α `p´1q |v p1q | v p1q b α 1 dtq ¯b v p2q b 1p ´1q |v p1q | v p1q b α b Q E pv p2q q b 1.
t P ra, bs Þ Ñ v t b 1 `wt b dt, with v t P S ' K pE 1 q |k and w t P S ' K pE 1 q |k´1 .
The following is a temporary definition. It will be generalized later to another more practical for gluing homotopies. Definition 2.3.8. A homotopy that joins two Lie 8-algebroid morphisms Φ, Ψ : S ' K pE 1 q Ñ S ' K pEq is the datum made of an interval ra, bs Ă R and a chain map

pS ' K pE 1 q, Q E 1 q H ÝÑ pS ' K pEq b K Ω ' pra, bsq, Q E b id `id b d dR q v Þ ÝÑ ´t P ra, bs Þ Ñ J t pvq b 1 ´p´1q |v| H t pvq b dt 1.
which is a co-algebra morphism, 2. and that coincides with Φ and Ψ at t " a and b respectively, i.e. for all v P S ' K pE 1 q, one has Hpvqpaq " Φpvq b 1 and Hpvqpbq " Ψpvq b 1.

Remark 2.3.9. In the definition above, the map H induces for every t P ra, bs two different O-

multilinear maps $ & % J t : S ' K pE 1 q ÝÑ S ' K pEq H t : S ' K pE 1 q ÝÑ S ' K pEq.
. Since H is of degree 0, one of the maps is of degree zero and the other one of degree ´1 respectively. By using respectively the property of co-algebra morphisms and chain map property, we obtain the following for every t P ra, bs:

1. Let ∆ be the co-product on S ' K pEq b K Ω ' pra, bsq like in Lemma 2.3.6. We have, ∆ ˝Hpvq " H b H ˝∆1 pvq.

Let v " v 1 ¨¨¨v n P S ' K pE 1 q, a direct computation of gives us ∆ b ∆ Ω ˝Hpvqptq " ∆ ˝Jt pvq b 1 b 1 ´p´1q |v| ∆ ˝Ht pvq b dt b 1.
Let us use the Sweedler notation just like in Lemma 2.3.6 to compute H b H ˝∆1 pvqptq. We have, By equating both sides, we obtain equations that say that J t is a co-morphism and H t a J t -coderivation.

H b H ˝∆1 pvqptq " H b Hpv p1q b v p2q qptq " Hptqpv p1q q b Hpv p1q qptq " ´Jt pv p1q q b 1 ´p´1q |v p1q | H t pv p1q q b dt ¯b ´Jt pv p2q q b 1 ´p´1q |v p2q | H t pv p2q q b dt ¯ Hence, pid b τ b idq ´1 ˝pHbHq ˝∆1 pvq " pJ t b J t q ˝∆1 pvq b 1 b 1 `((
2. and for any homogeneous element v P S ' K pE 1 q we have: in one side

H ˝QE 1 pvqptq " J t ˝QE 1 pvq b 1 ´p´1q |v|`1 H t pvq ˝QE 1 pvq b dt.
to the other side2 ,

pQ E b id `id b d dR q ˝Hpvqptq " pQ E b id `id b d dR q ˝´J t pvq b 1 ´p´1q |v| H t pvq b dt " Q E ˝Jt pvq b 1 `p´1q |v| dJ t dt pvq b dt ´p´1q |v| Q E ˝Ht pvq b dt
By equating both sides we see that J t and H t satisfy the following condition

$ & % J t ˝QE 1 pvq " Q E ˝Jt pvq dJt dt pvq " Q E ˝Ht pvq `Ht ˝QE 1 pvq.
(2.10)

In This implies that ρ ˝Jp0q t pvq " ρ ˝Φp0q pvq " ρ 1 pvq, since ρ ˝ℓ1 " 0.

Gluing homotopies as defined in Definition 2.3.8 is however not so easy. We can reformulate the definition of homotopies between Lie 8-morphisms in the following manner. Before we do this, we would like to fix some vocabulary.

Let us recall some facts on vector-valued functions. Let V be a vector space. Unless a topology on V is chosen, the notion of V -valued continuous or differentiable or smooth function, and the concept of a limit on an interval I " ra, bs Ă R do not make any sense. However, we can always define the following notion Definition 2.3.10. A vector-valued map γ : I ÝÑ V is said to be a piecewise rational path on I if there exists a finite increasing sequence a " t 0 ď ¨¨¨ď t N " b of gluing points, such that for all i " 0, . . . , N ´1 the restriction γ i of γ to rt i , t i`1 s is of the form

γ i ptq " n ÿ j"1 β i j ptqv j , for some n P N
where for every j " 1, . . . , n, v j P V and β i j : I Ñ R a real rational function on rt i , t i`1 s that has no pole on rt i , t i`1 s.

1. We say that γ is continuous on I, if for all i " 1, . . . , N ´1, γ i and γ i`1 coincide at the gluing point t i`1 .

2. When V is a space of linear maps between the vector spaces S and T , we shall say that a V -valued map Ξ : I Ñ V is a piecewise rational (continuous) if the map pt P I Þ Ñ Ξ t psqq is a piecewise rational (continuous) T -valued path for all s P S.

3. We define the limit of γ at u P rt i , t i`1 s to be equal to

n ÿ j"1 lim tÑu β i j ptqv j
when, for every j " 1, . . . , n, β i j admits a limit a u. One can assume that tv 1 , v 2 , v 3 . . . u is a basis of V .

Let us recall the following.

Remark 2.3.11. It is a very classical fact that the integral of a piecewise-C 1 function β : I ÝÑ R on a compact interval I " ra, bs Ă R which is subordinate to a subdivision a " t 0 ă ¨¨¨ă t N " b admits primitives which are piecewise-C 1 on the same subdivision a " t 0 ă ¨¨¨ă t N " b. An important fact is that continuous piecewise-C 1 functions β : I ÝÑ R, i.e. piecewise-C 1 functions which are also continuous (even at the junction points) admit piecewise continuous derivatives β 1 ptq, and βpbq ´βpaq " ş b a β 1 ptqdt.

Definition 2.3.10 has this important consequence.

Lemma 2.3.12. A piecewise rational continuous function γ : I Ñ V is differentiable at every point which is not a gluing point, and the latter is again piecewise rational on I.

Conversely, every piecewise rational functions admit a piecewise rational continuous primitive, unique up to a constant.

Proof. The derivative of γ can be defined in the usual way using the item 3 of Definition 2.3.10. Likewise, for their primitives. The proof is then immediate.

We can now give the following definition. Definition 2.3.13. A family pJ t q tPI of co-algebra morphisms are said to be piecewise rational continuous if its Taylor coefficients J pnq t are piecewise rational continuous for all n P N. For such a family pJ t q tPI , a family pH t q tPI made of J t -co-derivations is said to be piecewise rational if all its Taylor coefficients are.

We are now ready to define formulate the definition of homotopies as follows and extend Definition 3.53 in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] to the infinite rank case. Definition 2.3.14. Let Φ and Ψ be Lie 8-algebroid morphisms from pE 1 , Q E 1 , ρ 1 q to pE, Q E , ρq. A homotopy between or that joins Φ and Ψ is a pair pJ t , H t q tPra,bs consisting of:

1. a piecewise rational continuous path t Þ Ñ J t valued in Lie 8-algebroid morphisms between S ' K pE 1 q and S ' K pEq satisfying the boundary condition:

Φ a " Φ and Φ b " Ψ, 2. a piecewise rational path t Þ Ñ H t , with H t a J t -co-derivations of degree ´1 from S ' K pE 1 q to S ' K pEq, such that the following equation: for k " 0, . . . , n.

dJ t dt " Q E ˝Ht `
When these conditions are satisfied, we say that Φ and Ψ are homotopy equivalent, and we write Φ " Ψ.

Remark 2.3.15. This clearly extends (2.10).

Remark 2.3.16. In the above definitions, it is not required that the gluing points of the various Taylor coefficients J pnq t or H pnq t to be the same for all n P N 0 .

The following Proposition shows that the notion of homotopy given in Definition 2.3.14 implies the usual notion of homotopy between chain maps (see Appendix B).

Proposition 2.3.17. Let Φ and Ψ be Lie 8-algebroid morphisms from pE 1 , Q E 1 , ρ 1 q to pE, Q E , ρq which are homotopic. Then,

Ψ ´Φ " Q E ˝H `H ˝QE 1 (2.14)
for some O-linear map H : S ' K pE 1 q ÝÑ S ' K pEq of degree ´1.

Proof. Take pJ t , H t q tPra,bs a homotopy between Φ and Ψ as in Definition 2.3.14. By definition, t Þ Ñ J t is piecewise rational continuous, therefore it is continuous on ra, bs (even at the junctions points), we can use Remark 2.3.11 and write

Φ b ´Φa " ż b a dJ t dt dt Ψ ´Φ " ż b a pQ E ˝Ht `Ht ˝QE 1 q dt " Q E ˝ˆż b a H t dt ˙`ˆż b a H t dt ˙˝Q E 1 .
Thus, the O-multilinear map H :" ş b a H t dt satisfies Equation (2.14).

Definition 2.3.18. We say that two Lie 8-algebroids pE 1 , Q E 1 , ρ 1 q to pE, Q E , ρq over O are homotopy equivalent or homotopic if there exists two Lie 8-algebroid morphisms does not appear in the pn `1q-th Taylor coefficient of Q E ˝Ht `Ht ˝QE 1 by Equation (1.2.16). By Lemma 2.3.12, there exists a unique piecewise rational continuous solution J pn`1q t such that J pn`1q a " Φ pn`1q . The construction of the Taylor coefficients of the co-algebra morphisms J t then goes by recursion. Recursion formulas also show that J t is unique.

pE 1 , Q E 1 , ρ 1 q Φ / / pE, Q E , ρq Ψ o o such that Φ˝Ψ : pE, Q E , ρq Ñ pE, Q E , ρq and Ψ˝Φ : pE 1 , Q E 1 , ρ 1 q Ñ pE 1 , Q E 1 ,
Let us show item 2. i.e. that J t is a O-multilinear chain map for all t P ra, bs: For the same reason as in Equation (2.11), Equation (2.16) implies that ρ ˝Jp0q t | E 1 " ρ 1 . The function given by Λ k ptq " pQ E ˝Jt ´Jt ˝QE 1 q pkq for all t P ra, bs, k P N 0 are differentiable at all points t except for a finitely many t P ra, bs and are piecewise rational continuous. The map dJt dt is a Lie 8-morphism because Q 2 E " 0 and Q 2 E 1 " 0, hence Λ 1 ptq " 0. By continuity, Λ k ptq is constant over the interval ra, bs. Since J a " Φ is a Lie 8-algebroid morphism, we have Λ k paq " 0. Thus, Λ k ptq " 0 and, Q E ˝Jt " J t ˝QE 1 , for all t P ra, bs.

Let us show that homotopy in the sense above defines an equivalence relation " between Lie 8-morphisms. We have the following lemma.

Lemma 2.3.20. A pair pJ t , H t q is a homotopy between Lie 8-algebroid morphisms J a and J b if and only if for all rational function, g : ra, bs Ñ rc, ds without poles on ra, bs, the pair pJ gptq , g 1 ptqH gptq q is a homotopy between J gpaq and J gpbq .

Proof. Let g : ra, bs Ñ rc, ds be a rational function without poles on ra, bs. A straightforward computation gives: " Q E ˝`g 1 ptqH gptq ˘``g 1 ptqH gptq ˘˝Q E 1 (by multiplying both sides by g 1 ptq).

dJ t dt " Q E ˝Ht `
The last equation means that pJ gptq , g 1 ptqH gptq q is a homotopy between J gpaq and J gpbq . The backward implication is obvious, it suffices to consider a " c, b " d and g " id.

Proposition 2.3.21. Homotopy between Lie 8-morphisms is an equivalence relation. In addition, it is compatible with composition, that is, if Φ, Ψ : S ' K pE 1 q Ñ S ' K pEq are homotopic Lie 8-algebroid morphisms and Φ, Ψ : S ' K pEq Ñ S ' K pE 2 q are homotopic Lie 8-algebroid morphisms, then, so are their compositions Φ ˝Φ and Ψ ˝Ψ.

Proof. We first show that this notion of homotopy is an equivalence relation. Let Φ, Ψ and Ξ : S ' K pE 1 q ÝÑ S ' K pEq be three Lie 8-morphisms of algebroids. ' Reflexivity: The pair pJ t " Φ, H t " 0q tPr0,1s defines a homotopy between Φ and Φ.

' Symmetry: Let pJ t , H t q tPr0,1s be a homotopy between Φ to Ψ. By applying Lemma 2.3.20 with gptq " 1 ´t, we obtain a homotopy between Ψ and Φ via the pair pΦ 1´t , ´H1´t q tPr0,1s .

' Transitivity: Assume Φ"Ψ and Ψ"Ξ and let pJ t , H 1,t q tPr0, 1 2 s be a homotopy between Φ and Ψ and let p Jt , H 2,t q tPr 1 2 ,1s be a homotopy between Ψ and Ξ. By gluing J t and Jt , respectively H 1t and H 2,t we obtain a homotopy p Jt , H t q tPr0,1s between Φ and Ξ.

We then show it is compatible with composition. Let us denote by pJ t , H t q the homotopy between Φ and Ψ, and p Ĵt , Ĥt q the homotopy between Φ and Ψ . We obtain,

d Ĵt ˝Jt dt " d Ĵt dt ˝Jt `Ĵ t ˝d Ĵt dt " Q E 2 ˝´Ĥ t ˝Jt `Ĵ t ˝Ht ¯`´Ĥ t ˝Jt `Ĵ t ˝Ht ¯˝Q E 1 .
Hence, Φ ˝Φ and Ψ ˝Ψ are homotopic via the pair p Ĵt ˝Jt , Ĥt ˝Jt `Ĵ t ˝Ht q which is easily checked to satisfy all axioms. This concludes the proof.

We conclude this section with a lemma that will be useful in the sequel. Notice that this is the lemma that forces to extend Definition 2.3.8, for it would not be true anymore in the smooth setting.

Lemma 2.3.22. Let pJ t , H t q tPrc,`8r be a homotopy such that for all n P N 0 and for every t ě n, H pnq t " 0. Then the n-th Taylor coefficient J pnq t is constant on rn, `8r and the co-algebra morphism J 8 whose n-th Taylor coefficient is J pnq t for any n P N 0 and t P rn, `8r is a Lie 8-algebroid morphism. Moreover, for g : ra, brÑ rc, `8r a rational function with no pole on ra, br and such that lim tÑb gptq " `8, the pair pJ gptq , g 1 ptqH gptq q is a homotopy between J c and J 8 .

Proof. Since the n-th Taylor coefficient of the J t -co-derivation ) on ra, b n s. By gluing with a constant function J 8 (resp. with 0), we see that all Taylor coefficients of J gptq (resp. g 1 ptqH gptq ) are piecewise rational continuous (resp. piecewise rational) with finitely many gluing points. This completes the proof.

Remark 2.3.23. Lemma 2.3.22 explains how to glue infinitely many homotopies, at least when for a given n P N 0 , only finitely of them affects the n-th Taylor coefficient. This would not be possible using only Definition 2.3.8.

More technical lemmas and propositions

Let us state and prove these technical assertions for later use.

Let pE 1 , Q E 1 , ρ 1 q and pE, Q E , ρq be a Lie 8-algebroid over O.

Proposition 2.3.24. Let Φ : S ' K pE 1 q Ñ S ' K pEq be a Lie 8-algebroid morphism. For H :

S ' K pE 1 q Ñ S ' K pEq a O-multilinear Φ-co-derivation of degree k ď ´1, H ˝QE 1 ´p´1q k Q E ˝H is a O-multilinear Φ-co-derivation of degree k `1. Proof. We first check that H ˝QE ´p´1q k Q E 1 ˝H is a Φ-co-derivation: ∆ ˝H ˝QE 1 " pH b Φ `Φ b Hq ˝∆1 ˝Q1 E , by definition of H " pH b Φ `Φ b Hq ˝pQ E 1 b id `id b Q E 1 q ˝∆1 , by definition of Q 1 E " pH ˝QE 1 b Φ `p´1q k Φ ˝QE 1 b H `H b Φ ˝QE 1 `Φ b H ˝QE 1 q ˝∆1
Subtracting a similar equation for p´1q k ∆ ˝QE ˝H and using (2.6), one obtains the Φ-co-derivation property for

H ˝QE 1 ´p´1q k Q E 1 ˝H. We now check that H ˝QE 1 ´p´1q k Q E ˝H is O-multilinear,
for which it suffices to check that its Taylor coefficients are O-multilinear by Lemma 1.2.17. Let x 1 , . . . , x n P E 1 be homogeneous elements. Assume x i P E 1 ´1 (if we have more elements of degree ´1 the same reasoning holds). To verify O-multilinearity it suffices to check that for all f P O:

pr ˝pH ˝QE 1 ´p´1q k Q E ˝Hqpx 1 , . . . ,x i , . . . , f x j . . . , x n q " f pr ˝pH ˝QE 1 ´p´1q k Q E ˝Hqpx 1 , . . . , x i , . . . , x j . . . , x n q.
Only the terms where the 2-ary bracket with a degree ´1 element on one-side and f on the other side may forbid f to go in front. There are two such terms:

ϵpx i , x j , x I ij qH pn´1q pℓ 1 2 px i , f x j q, x I ij q and ´p´1q k ϵpx i , x I i qℓ 2 pΦ 0 px i q, f H pn´1q px I i qq
where x I i and x I ij stand for the list x 1 , . . . , x n where x i and x i , x j are missing, respectively, and H pn´1q is the pn ´1q-th Taylor coefficient of H. Since ρ ˝Φ0 " ρ 1 , in both terms ρpx i qrf s appears, and these two terms add up to zero.

Remark 2.3.25. If the degree of H is non-negative, then H ˝QE 1 ´p´1q k Q E ˝H may not be Omultilinear anymore, since there may exist extra terms where the anchor map appears, e.g. terms of the form ℓ 2 pHpx I j q, f Φ p0q px j qq.

Lemma 2.3.26. Let Φ : S ' K pE 1 q Ñ S ' K pEq be a co-algebra morphism such that

1. Φ is O-multilinear, 2. ρ ˝Φ0 " ρ 1 on E 1 ,
If for every3 n P N 0 , pΦ ˝QE 1 ´QE ˝Φq piq " 0 for each 0 ď i ď n, then the map S K pE 1 q Ñ S K pEq given by:

pΦ ˝QE 1 ´QE ˝Φq pn`1q 1. is a Φ p0q -co-derivation of degree `1, 2. is O-multilinear,

and the induced

Φ p0q -co-derivation ´Ä' E 1 , Q p0q E 1 ¯ÝÑ ´Ä' E, Q p0q E ¯satisfies: Q p0q E ˝pΦ ˝QE 1 ´QE ˝Φq pn`1q " pQ E ˝Φ ´Φ ˝QE 1 q pn`1q ˝Qp0q E 1 .
Proof. A straightforward computation yields:

∆pΦ ˝QE 1 ´QE ˝Φq " pΦ b Φq ˝∆1 ˝QE 1 ´pQ E b id `id b Q E q ˝∆ ˝Φ " ppΦ ˝QE 1 ´QE ˝Φq b Φ `Φ b pΦ ˝QE 1 ´QE ˝Φqq ˝∆1 .
Now, ∆ preserves polynomial-degree, i.e. ∆ : S n K pEq ÝÑ ' i`j"n S i K pEq b S j K pEq and so does ∆ 1 . Taking into account the assumption pΦ ˝QE 1 ´QE ˝Φq piq " 0 for every 0 ď i ď n, we obtain:

∆ ˝pΦ ˝QE 1 ´QE ˝Φq pn`1q " ´pΦ ˝QE 1 ´QE ˝Φq pn`1q b Φ p0q `Φp0q b pΦ ˝QE 1 ´QE ˝Φq pn`1q ¯˝∆ 1 .
All the other terms disappear for polynomial-degree reasons Hence, pΦ ˝QE 1 ´QE ˝Φq pn`1q is a Φ p0q -co-derivation.

Let us prove that it is O-linear. It suffices to check O-linearity of T Φ :" Φ ˝QE 1 ´QE ˝Φ. Let us choose homogeneous elements x 1 , . . . , x N P E 1 and let us assume that x i P E 1 ´1 is the only term of degree ´1: The proof in the case where there is more than one such an homogeneous element of degree ´1 is identical. We choose j ‰ i and we compute T Φ px 1 , . . . , x i , . . . , f x j , . . . , x N q for some f P O. The only terms in the previous expression which are maybe non-linear in f are those for which the 2-ary brackets of a term containing f x j with x i or Φ 0 px i q appear (since Φ and all other brackets are O-linear). There are two such terms. The first one appears when we apply Q E 1 first, and then Φ: this forces Φ pℓ 1 2 px i , f x j q, x I ij q to appear, and the non-linear term is then:

ϵpσ i qρ 1 px i qrf s Φpx I i q (2.17)
with σ i the permutation that let i goes in front and leave the remaining terms unchanged. There is a second term that appears when one applies Φ first, then Q E . Since it is a co-morphism, Φpx 1 . . . x i . . . , f x j . . . x N q is the product of several terms among which only one is of degree ´1, namely the term ϵpx, σ i qΦ 0 px i qΦpf x I i q.

Applying Q E to this term yields the non-linear term

ϵpσ i qρpΦ 0 px i qqrf s Φpx I i q, (2.18)
where I i and I ij are as in Proposition 2.3.24. Since ρ ˝Φ0 " ρ 1 , we see that the terms (2.17) and (2.18) containing an anchor add up to zero. Let us check that pΦ ˝QE 1 ´QE ˝Φq pn`1q is a chain map, in the sense that it satisfies item 3). Considering again T Φ :" Φ ˝QE 1 ´QE ˝Φ, we have that T pkq Φ " 0, for all k " 0, . . . , n. Since

T Φ ˝QE 1 " Q E ˝TΦ , one has 0 " pT Φ ˝QE 1 `QE ˝TΦ q pn`1q " T pn`1q Φ ˝Qp0q E 1 `Qp0q E ˝T pn`1q Φ `ÿ i`j"n`1 i,jě1 ´T piq Φ ˝Qpjq E 1 `Qpjq E ˝T piq Φ loooooooooooooooomoooooooooooooooon 0
By consequent, the O-linear map pΦ ˝QE 1 ´QE ˝Φq pn`1q satisfies item 3.

Lemma 2.3.27. Let Φ, Ξ : S ' K pE 1 q ÝÑ S ' K pEq be O-linear Lie 8-algebroid morphisms. Let n P N 0 . If Ξ piq " Φ piq for every 0 ď i ď n, then pΞ ´Φq pn`1q :

S ' K pE 1 q ÝÑ S ' K pEq 1. is a Φ p0q -co-derivation 2. is O-multilinear 3. and the induced Φ p0q -co-derivation ´Ä' E 1 , Q p0q E 1 ¯ÝÑ ´Ä' E, Q p0q E ¯satisfies: Q p0q E ˝pΞ ´Φq pn`1q " pΞ ´Φq pn`1q ˝Qp0q E 1 .
Proof. For all x 1 , . . . , x k P E 1 , one has:

∆pΞ ´Φqpx 1 d ¨¨¨d x k q " k ÿ j"1 ÿ σPS pj,k´jq ϵpσqΞpx σp1q d ¨¨¨d x σpjq q b Ξpx σpj`1q d ¨¨¨d x σpkq q ´k ÿ j"1 ÿ σPS pj,k´jq ϵpσqpΦpx σp1q d ¨¨¨d x σpjq q b Φpx σpj`1q d ¨¨¨d x σpkq q " ppΞ ´Φq b Φ `Ξ b pΞ ´Φqq ˝∆1 px 1 d ¨¨¨d x k q.
Since ∆ has polynomial-degree 0 and pΞ ´Φq piq " 0 for all 0 ď i ď n, we obtain ∆pΞ´Φq pn`1q px 1 d¨¨¨dx k q " ´pΞ ´Φq pn`1q b Φ p0q `Φp0q b pΞ ´Φq

Conclusion:

This chapter recapitulates classical definition of Lie 8-algebroids, and in particular describes it as co-derivation, which is not usual. Morphisms and homotopies are described. About homotopies, two versions are given: one uses smooth maps, and is way easier (see Definition 2.3.8). Unfortunately, to allow infinitely many gluings, we have to introduce a more complicated notion of homotopy, which uses continuous locally C 1 -maps (see Definition 2.3.14).

CHAPTER 3

Lie-Rinehart algebras and their morphisms Except for Remark 3.2.1, this section is essentially a review of the literature on the subject, see, e.g. [START_REF] Huebschmann | Lie-Rinehart algebras, descent, and quantization[END_REF][START_REF] Huebschmann | Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras[END_REF].

Definitions

Lie-Rinehart algebras are the algebraic encoding of the notion of Lie algebroids over a manifold. We owe this concept to [START_REF] Rinehart | Differential Forms on General Commutative Algebras[END_REF]. Definition 3.1.1. A Lie-Rinehart algebra over an algebra O is a triple pA, r¨, ¨sA , ρ A q with A an Omodule, r¨, ¨sA a Lie algebra bracket on A, and ρ A : A ÝÑ DerpOq an O-linear Lie algebra morphism called anchor map, satisfying the so-called Leibniz identity: ra, f bs A " ρ A paqrf s b `f ra, bs A for all a, b P A, f P O.

1. Let η : O ÝÑ O 1 be an algebra morphism. A Lie-Rinehart algebra morphism over η is a Lie algebra morphism ϕ : A ÝÑ A 1 such that for every a P A and f P O:

(a) ϕpf aq " ηpf qϕpaq (b) ηpρ A paqrf sq " ρ A 1 pϕpaqrηpf qs,
When O " O 1 and η " id, we say that ϕ is a Lie-Rinehart algebra morphism over O. Also, the Jacobi identity holds, since ℓ 2 pℓ 2 px, yq, yq`ö px, y, zq " ´ℓ1 pℓ 3 px, y, zqq @x, y, z P E ´1.

A submodule

In addition, the anchor map ρ goes to quotient to a Lie algebra morphism E ´1 ℓ 1 pE ´2q Ñ DerpOq, since ρ ˝ℓ1 " 0. This Lie-Rinehart algebra is called the basic Lie-Rinehart algebra of pE ' , ℓ ' , ρq.

To summarize, every Lie 8-algebroid induces a Lie-Rinehart algebra. The opposite direction, that is, wondering whether a Lie-Rinehart algebra pA, r¨, ¨sA , ρq over O is the basic Lie-Rinehart algebra of some almost differential graded Lie algebroid or more generally a Lie 8-algebroid over O is part of the questions that I discussed in this thesis (see Chapter 4). This extends the main results of [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] from locally real analytic finitely generated singular foliations (see Example 3.2.4) to arbitrary Lie-Rinehart algebras.

Let us fix some vocabulary that will be used in the sequel. Definition 3.1.4. We say that an almost differential graded Lie algebroid pE ' , ℓ 1 , ℓ 2 , ρq or a Lie 8-algebroid pE ' , ℓ ' , ρq over O 1. covers (through a hook π) a Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q, if there exists a morphism of brackets π : E ´1 Ñ A such that π ˝ρA " ρ and πpE ´1q " A, 2. terminates in A through the hook π, when πpE ´1q Ď A.

According to Remark 3.1.3, any Lie 8-algebroid pE ' , ℓ ' , ρq covers its basic Lie-Rinehart algebra through the hook π which is given by the projection π : E ´1 ÝÑ E ´1{ℓ 1 pE ´2q, since π respects the brackets and πpE ´1q " E ´1{ℓ 1 pE ´2q. Definition 3.1.5. Let pE 1 ' , ℓ 1 ' , ρ 1 q and pE ' , ℓ ' , ρq and be Lie 8-algebroids over O that terminate in a Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q through the hooks π 1 and, π respectively. We say that a Lie 8-algebroid morphism Φ :

S ' K pE 1 q Ñ S ' K pEq is over A, if π ˝Φ0 " π 1 .
We will need the following lemma.

Lemma 3.1.6. Let pE 1 ' , ℓ 1 ' , ρ 1 q and pE ' , ℓ ' , ρq be Lie 8-Lie algebroids that terminate in some Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q through hooks π 1 and π. Let pJ t , H t q tPra,bs be a homotopy that joins J a and J b . If J a is Lie 8-algebroid morphism that terminates at A (i.e., π ˝Jp0q a | E 1 " π 1 ), then so is the 8-algebroid morphism J t for all t P ra, bs.

Proof. This is a direct consequence of Equation (2.16), since

Q p0q E |E " ℓ 1 : E ´2 Ñ E ´1 and Q p0q E 1 |E 1 " ℓ 1 1 : E 1 ´2 Ñ E 1 ´1
are valued in the kernels of π and, π 1 respectively. Last, O-multilinearity of J t follows from the O-multilinearity of Q E ˝Ht `Ht ˝QE 1 , which is granted by Proposition 2.3.24. This completes the proof.

Algebraic and geometric examples

Example 3.2.1. For every commutative K-algebra O, the Lie algebra A " DerpOq of derivations of a commutative algebra O is a Lie-Rinehart algebra over O, with the identity as an anchor map. Notice that for every Lie-Rinehart pA, r¨, ¨sA , ρ A q the anchor A ρ A ÝÑ DerpOq is obviously a Lie-Rinehart algebra morphism over O.

In particular, vector fields on a smooth or Stein manifold or an affine variety are examples of Lie-Rinehart algebras over their respective natural algebras of functions.

Example 3.2.2. Let M be a smooth manifold. By Serre-Swan Theorem, Lie algebroids over M are precisely Lie-Rinehart algebras over C 8 pM q of the form pΓpAq, r¨, ¨s, ρq where A is a vector bundle over M and ρ : A Ñ T M is a vector bundle morphism. • Locally finitely generateness: every m P M admits an open neighborhood U together with a finite number of vector fields X 1 , . . . , X r P XpUq such that for every open subset V Ď U the vector fields X 1| V , . . . , X r| V generates F on V as a C 8 pVq-module.

There are several other ways to define singular foliations on a manifold M [AZ13, Cer79, Daz85, Deb01]. All these definitions have in common to define then as Lie-Rinehart sub-algebras F of the Lie-Rinehart algebra XpM q of vector fields on M (or compactly supported vector fields X c pM q on M ). Here are some important consequences of the above definition.

1. Singular foliation admits leaves: there exists a partition of M into submanifolds called leaves such that for all m P M , the image of the evaluation map F Ñ T m M is the tangent space of the leaf through m. When F coincides with the space of vector fields tangent to all leaves at all points, we shall speak of a "Stefan-Sussman singular foliation".

2. Singular foliations are self-preserving : the flow of vector fields in F, whenever defined, preserves

F [AS09, GY18].
Notice that in Remark 2.1.13, F :" ρpΓpE ´1qq is a singular foliation in the sense above, called the basic singular foliation of pE, pℓ k q kě1 , ρq. We say, then the Lie 8-algebroid pE, pℓ k q kě1 , ρq is over F.

Example 3.2.5. [AZ18, Zam18, ZA18] Let pA, r¨, ¨sA , ρ A q be a Lie algebroid over a manifold M . A singular subalgebroid B of A, is a C 8 pM q-submodule of ΓpAq that is locally finitely generated and involutive i.e. stable under the Lie bracket r¨, ¨sA . This notion is a generalization of singular foliations in the sense of Example 3.2.4, since singular foliations on M are singular subalgebroids of T M .

A singular subalgebroid B is an example of Lie-Rinehart algebra: its bracket and anchor are the restrictions of r¨, ¨sA and ρ A to B respectively. Example 3.2.6. For a singular foliation F on a manifold M , consider S :" tX P XpM q | rX, Fs Ď Fu (i.e. infinitesimal symmetries of F) and C :" tf P C 8 pM q | Y rf s " 0, for all Y P Fu (that can be thought of as functions constant along the leaves of F). The quotient S F is Lie-Rinehart algebra over C.

Example 3.2.7 (Poisson manifold). [LGPV13]

We recall that a Poisson manifold is a manifold M together with a biderivation t¨, ¨u on its algebra of smooth functions C 8 pM q that satisfies Jacobi's identity, i.e. 1. pC 8 pM q, t¨, ¨uq is Lie algebra.

2. The biderivation t¨, ¨u is compatible with the product of functions in the following sense: for all f, g, h P C 8 pM q tf.g, hu " f.tg, hu `g.tf, hu. This is equivalent to giving a bivector field π P Γp^2T M q such that the Schouten-Nijenhuis bracket with itself vanishes, that is, rπ, πs SN " 0.

For every a Poisson manifold pM, πq, the operator δ π : Γp^'T M q Ñ Γp^' `1T M q, P Þ Ñ ´rP, πs SN defines a complex

¨¨¨/ / Γp^p ´1T M q δ p´1 π / / Γp^pT M q δ p π / / Γp^p `1T M q / / ¨¨¨,
since rπ, πs SN " 0 implies δ π ˝δπ " 0. For every p P N 0 , the quotient H p π pM q :" ker δ p π Imδ p´1 π is called the p-th Poisson cohomology of pM, πq. We define A :" H 1 π pM q to be the first Poisson cohomology of π and O :" H 0 π pM q " Caspπq to be the algebra of Casimir functions. The bracket of vector fields makes A a Lie-Rinehart algebra over Caspπq.

Basic constructions

Let pA, r¨, ¨sA , ρ A q a Lie-Rinehart algebra over an algebra O and I Ă O be an ideal. The submodule IA is a Lie-Rinehart subalgebra of A and its anchor is given by the restriction of ρ A over IA Ă A. This follows easily from rf a, gbs A " f gra, bs A `f ρ A paqrgs b ´gρ A pbqrf s a for all a, b P A, f, g P I.

1. Restriction. Consider a Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q over O. For every Lie-Rinehart ideal I Ă O, i.e. any ideal such that ρ A pAqrIs Ă I the quotient space A{IA inherits a natural Lie-Rinehart algebra structure over O{I. We call this Lie-Rinehart algebra the restriction w.r.t the Lie-Rinehart ideal I. In the context of affine varieties, when I is the ideal of functions vanishing on an affine sub-variety W , we shall denote

A IA by i W A.
2. Localization. Localizing w.r.t a multiplicative subset S Ă O (i.e. S Ď Ozt0u is closed under multiplication and 1 P S) is a very powerful tool in commutative algebra and in algebraic geometry to deal with "global" problems by reducing them to "local" ones. Let us recall the construction (e.g. see [START_REF]The Stacks project authors[END_REF], Section 10.9 or [And] for details).

(a) Let V be a O-module. The localization of V at S is defined as follows: consider the equivalence relation on S ˆV that is given by ps, vq " ps 1 , v 1 q ðñ Du P S, ups 1 v ´sv 1 q " 0.

The class of an element ps, vq P S ˆV by v s , and by S ´1V :" S ˆV{ " the set of equivalence classes.

In particular, the localization of O at S is the localization of O as a O-module. In that case, S ´1O is a K-algebra together with the addition and multiplication defined by

f s `g s 1 :" f s 1 `sg ss 1 and f s g s 1 :" f g ss 1 .
Similarly, S ´1V is a S ´1O-module with addition and scalar product multiplication defined in an obvious way. Also, we have

S ´1V » S ´1O b O V.
(b) A derivation D P DerpOq admits a unique extension to a derivation S ´1D P DerpS ´1Oq which given for ps, f q P S ˆO by the classical formula

pS ´1Dq ˆf s ˙:" Dpf qs ´Dpsqf s 2 .
Let pA, r¨, ¨sA , ρq be a Lie-Rinehart algebra over O. The localization module S ´1A " S ´1Ob O A comes equipped with a natural structure of Lie-Rinehart algebra over the localization algebra S ´1O. The new anchor map is defined by

a s P S ´1A Þ Ñ 1 s S ´1pρ A paqq P DerpS ´1Oq,
and the new Lie algebra bracket is given by

" 1 s a, 1 u b ȷ S ´1A " 1 su ra,
O U i generated by x 1 x i , . . . , x i´1 x i , x i , x i`1 x i , . . . , x N
x i satisfies the previous condition, and O U i b O A comes equipped with a Lie-Rinehart algebra. Geometrically, this operation corresponds to taking the blow-up of C N at the origin, then looking at the i-th natural chart U i on this blow-up: O U i are the polynomial functions on U i . The family O U i b O A (for i " 1, . . . , N ) is therefore an atlas for a sheaf of Lie-Rinehart algebras (in the sense of [START_REF] Villatoro | On sheaves of Lie-Rinehart algebras[END_REF]) on the blow-up of C N at the origin, referred to as the blow-up of A at the origin.

On free resolutions of length ď 2 and Lie-Rinehart algebras

In this section, we discuss the case when a Lie-Rinehart algebra A admits a free resolution of length 1 and 2. In those cases, we claim that there are Lie algebra-like structures on them. See B.1 for the notion of free resolution of modules. These results will be soon generalized.

I start with the following remark (owed to Marco Zambon).

Remark 3.2.8. Any Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q is the image of an almost differential graded Lie algebroid pE ´1, ρq concentrated in degree ´1: to see this, let ta i P A | i P Iu be a set of generators of A (take all elements of A if necessary). There exists elements u k ij P O, such that for given indices i, j, the coefficient u k ij is zero except for finitely many indices k, together with

ra i , a j s A " ÿ kPI u k ij a k @i, j P I. (3.1)
The coefficients u k ij can be chosen to satisfy the skew-symmetry condition u k ij " ´uk ji : by skewsymmetry of the bracket r¨, ¨sA on has

ra i , a j s A " 1 2 pra i , a j s A ´ra j , a i s A q " ÿ kPI 1 2 pu k ij ´uk ji qa k . Thus, one can replace u k ij by 1 2 pu k ij ´uk ji q if necessary.
Now, choose E ´1 to be the free O-module generated by the symbols pe i q iPI together with the surjective map

π : E ´1 Ñ A, e i Þ Ñ a i .
We now define:

1. an anchor map by ρpe i q :" ρ A pa i ), for all i P I, i.e. ρ " ρ A ˝π, 2. a skew-symmetric operation r¨, ¨sE ´1 on E as follows:

re i , e j s E ´1 " ÿ kPI u k ij e k for all i, j P I.
We extend by O-linearity and Leibniz identity. By construction pE ´1, r¨, ¨sE ´1, ρ) is an almost Lie algebroid over O whose image through the anchor map is A.

Remark 3.2.9. In general, this bracket does not satisfy the Jacobi identity. If E ´1 » A, this bracket is a Lie algebroid.

On free resolutions of length 1

Lie-Rinehart algebra A admits a free resolution of length 1 if and only if A is free. In that case, the almost Lie algebroid bracket r ¨, ¨sE ´1 is a Lie algebroid bracket. In conclusion: free resolutions of length 1 admit a Lie algebroid structure.

Free resolutions of length 2

Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra that admits a free resolution of length 2, namely an exact sequence of the form

0 / / E ´2 ℓ 1 / / E ´1 ρ < < < < π / / / / A ρ A / / DerpOq (3.2)
with E ´1, E ´2 free modules. Since Equation (3.2) is a free resolution of A, the map π is in particular surjective. By Remark 3.2.8, E ´1 can be endowed with an almost Lie algebroid bracket, such that π is a morphism of brackets. We can extend this bracket to sections of degree ´2 to obtain an almost differential graded Lie algebroid.

Let us compute it:

Let pe 1 i q iPI 1 and pe j q iPI be a basis of E ´2 and E ´1, respectively. For all i, j, k P I Y I 1 we have 1.

πprℓ 1 e 1 i , e j s E ´1 q " rπ ˝ℓ1 pe 1 i q, πpe j qs " 0, (since π ˝ℓ1 " 0).

In other words, rℓ 1 pe 1 i q, e j s E ´1 P ker π. By exactness of the complex (3.2) there exists an element ∇ e 1 i e j P E ´2 such that πp∇ e 1 i e j q " rℓ 1 pe 1 i q, e j s E ´1 .

(3.3) Equation (3.3) allows defining a bilinear map:

E ´1 b E ´2 Ñ E ´2 px, yq Þ Ñ ∇ x y
by extending the ∇ e 1 i e j 's by linearity and Leibniz identity with the understanding that the anchor map ρ vanishes on E ´2 in order to have

(a) ℓ 1 p∇ x yq " rℓ 1 pxq, ys E ´1 , @x E ´2, y P E ´1, (b) for all f P O: ∇ x f y " f ∇ x y `ρpxqrf s y and ∇ f x y " f ∇ x y, for all x P E ´1, y P E ´2,

Remember that

Jacpe i , e j , e k q :" re i , re j , e k s 2 s 2 `re j , re k , e i s 2 s 2 `re k , re i , e j s 2 s 2 P ker π.

By using again exactness of the complex (3.2) there is an element that we denote by re i , e j , e k s E ´1 P E ´2 that satisfies ℓ 1 `re i , e j , e k s E ´1 ˘" Jacpe i , e j , e k q.

(3.4) Thus, we can define a skew-symmetric trilinear map:

r¨, ¨, ¨sE ´1 : E ´1 ^E´1 ^E´1 ÝÑ E ´2
such that ℓ 1 prx, y, zs E ´1 q " rx, ry, zs 2 s 2 `ry, rz, xs 2 s 2 `rz, rx, ys 2 s 2 , @x, y, z P E ´1.

The following Proposition concludes the discussion above.

Proposition 3.2.10. Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra that admits a free resolution of length 2 as in (3.2)

1. E ´1 admits an almost Lie algebroid structure pE ´1, r¨, ¨sE ´1 , ρq.

2. There is a bilinear map:

E ´1 b E ´2 Ñ E ´2 px, yq Þ Ñ ∇ x y
and a skew-symmetric trilinear map:

r¨, ¨, ¨sE ´1 : E ´1 ^E´1 ^E´1 ÝÑ E ´2
such that for all f P O:

(a) ∇ x f y " f ∇ x y `ρpxqrf s y and ∇ f x y " f ∇ x y, for all x P E ´1, y P E ´2, (b) rf x, y, zs E ´1 " f rx, y, zs E ´1 for all x, y, z P E ´1, such that the 2-ary bracket on E ´1 ' E ´2 defined by:

rx, ys 2 " $ ' ' ' ' & ' ' ' ' % rx, ys E ´1 for x, y P E ´1 ∇ x y for x P E ´1, y P E ´2 ∇ y x for x P E ´2, y P E ´1 0 for x, y P E ´2
together with the 3-ary bracket on E ´1 ' E ´2 defined by rx, y, zs 3 " rx, y, zs E ´1 if x, y, z P E ´1 and zero otherwise, satisfies (a) for all x P E ´2, y P E ´1, ℓ 1 prx, ys 2 q `rℓ 1 pxq, ys 2 " 0, (3.5) (b) for all x, y, z P E ´1 ℓ 1 prx, y, zs 3 q `rx, ry, zs 2 s 2 `ry, rz, xs 2 s 2 `rz, rx, ys 2 s 2 " 0 (c) for all x, y P E ´1 and z P E ´2 rx, y, ℓ 1 pzqs 3 `rx, ry, zs 2 s 2 `ry, rz, xs 2 s 2 `rz, rx, ys 2 s 2 " 0.

The structure pE ' , ℓ 1 , ρ, r¨, ¨s2 , r¨, ¨, ¨s3 q described in Proposition 3.2.10 is called Lie 2-algebroid [BC03, Lea17, SZ11], i.e. a Lie 8-algebroid with E ´i " 0 for i ě 3.

A generalization of this construction on free resolutions of higher (even infinite) length is the object of the next Chapter.

Conclusion:

We describe Lie-Rinehart algebras, give examples and several constructions. Section 3.2.2 is a pedagogical section, which presents an elementary case of the general case that we will study.

CHAPTER 4

Main results of Part I

Presentation of the problem

In order to understand the geometry of affine varieties or some similar problems related to singular foliations using the Lie algebra of their vector fields which is in fact Lie-Rinehart algebras, have motivated us to understand Lie-Rinehart algebras in general from another point of view.

We have seen in the Chapter 3, Remark 3.1.3 that any Lie 8-algebroid over O induces a Lie-Rinehart algebra which we call its basic Lie-Rinehart algebra. In this chapter, we are investigating the opposite direction, i.e., we study the following questions: given a Lie-Rinehart algebra A over O, can we find a Lie 8-algebroid over O whose basic Lie-Rinehart algebra is A? Also, in case where it exists, do we have uniqueness? Now let us formalize that in a categorical language. We denote by Lie-8-alg-oids/O the quotient category where

ÐÝ

The question now turn out to be: when does F admit a left/right inverse?

In the next section, we present in detail the main results related to this question.

Main results

An existence Theorem

The results below appeared in my first article [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF] entitled "Lie-Rinehart algebra » acyclic Lie 8-algebroid" co-written with my supervisor C. Laurent-Gengoux. This section extends the main results of [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] from locally real analytic finitely generated singular foliations to arbitrary Lie-Rinehart algebras.

Here is our first main result. It states that universal Lie 8-algebroids over a given Lie-Rinehart algebra exist. It extends Theorem 2.8 in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]. We are convinced that it may be deduced using the methods of semi-models categories as in Theorem 4.2 in [START_REF] Frégier | Homotopy theory of singular foliations[END_REF], but does not follow from a simple homotopy transfer argument.

Theorem 4.2.1. Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra over O. Any resolution of A by free

O-modules ¨¨¨ℓ 1 ÝÑ E ´3 ℓ 1 ÝÑ E ´2 ℓ 1 ÝÑ E ´1 π ÝÑ A (4.1)
comes equipped with a Lie 8-algebroid structure whose unary bracket is ℓ 1 and that covers A through the hook π.

Since any module admits free resolutions (see Proposition B.2.5), Theorem 4.2.1 implies that:

Corollary 4.2.2. Any Lie-Rinehart algebra A over O is the basic Lie-Rinehart algebra of an acyclic

Lie 8-algebroid over O.

While proving Theorem 4.2.1, we will see that if E ´1 can be equipped with a Lie algebroid bracket (i.e. a bracket whose Jacobiator is zero), then all k-ary brackets of the universal Lie 8-algebroid structure may be chosen to be zero on E ´1: Proposition 4.2.3. Let pE, ℓ 1 , πq be a free resolution of a Lie-Rinehart algebra A. If E ´1 admits a Lie algebroid bracket r¨, ¨s such that π : E ´1 Ñ A is a Lie-Rinehart morphism, then there exists a structure of universal Lie 8-algebroid pE ' , ℓ ' , ρq that covers A whose 2-ary bracket coincides with r¨, ¨s on E ´1 and such that for every k ě 3 the k-ary bracket ℓ k vanishes on Ä k E ´1.

Universality of Theorem 4.2.1 and its corollaries

Here is our second main result. It is related to Proposition 2.1.4 in [START_REF] Frégier | Homotopy theory of singular foliations[END_REF] (but morphisms are not the same), and extends Theorem 2.9 in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF].

Theorem 4.2.4. Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra over O. Given, a) a Lie 8-algebroid pE 1 ' , ℓ 1 ' , ρ 1 q that covers A through a hook π 1 , and b) any acyclic Lie 8-algebroid pE ' , ℓ ' , ρq that covers A through a hook π, then 1. there exists a morphism of Lie 8-algebroids from pE 1 ' , ℓ 1 ' , ρ 1 q to pE ' , ℓ ' , ρq over A.

and any two such morphisms are homotopic.

Here is an immediate corollary of Theorem 4.2.4.

Corollary 4.2.5. Any two acyclic Lie 8-algebroids that cover a given Lie-Rinehart algebra are homotopy equivalent. This homotopy equivalence, moreover, is unique up to homotopy.

We will prove that the morphism that appears in Theorem 4.2.4 can be made trivial upon choosing a "big enough" universal Lie 8-algebroid: Proposition 4.2.6. Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra over O. Given a Lie 8-algebroid Moreover, for any two universal Lie 8-algebroids of A, defined on E, E 1 the homotopy equivalences Φ : E 1 Ñ E and Ψ : E Ñ E 1 , whose existence is granted by Corollary 4.2.5, go to the quotient and induce an homotopy equivalences between

structure pE 1 ' , ℓ 1 ' , ρ 1 q that terminates in A through a hook π 1 ,
O{I b O E ' » E ' {I and O{I b O E 1 ' » E 1 ' {I.
The following corollary is then an obvious consequence of Theorem 4.2.4. When, in addition to being a Lie-Rinehart ideal, I is a maximal ideal, then K :" O{I is a field and Lie 8-algebroids of I are a homotopy equivalence class of Lie 8-algebras. In particular their common cohomologies, which is easily seen to be identified to Tor ' O pA, Kq comes equipped with a graded Lie algebra structure. In particular, Tor ´1 O pA, Kq is a Lie algebra, Tor ´2 O pA, Kq is a representation of this algebra, and the 3-ary bracket defines a class in the third Chevalley-Eilenberg cohomology of Tor ´1 O pA, Kq valued in Tor ´2 O pA, Kq. This class does not depend on any choice made in its construction by the previous corollaries, and trivially extends the class called NMRLA-class in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]. If it is not zero, then there is no Lie algebroid of rank r equipped with a surjective Lie-Rinehart algebra morphism onto A, where r is the rank of A as a module over O. All these considerations can be obtained by repeating verbatim Section 4.5.1 in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] (where non-trivial examples are given).

A categorical approach of the results

Theorem 4.2.4 means that acyclic Lie 8-algebroids that covers Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q are terminal objects in the subcategory of Lie-8-alg-oids/O whose objects are Lie 8-algebroids that terminate in A. Whence, acyclic Lie 8-algebroids over O that cover pA, r¨, ¨sA , ρ A q are deserved to be called "universal 8-algebroids of A". From now on, we call them by this name.

Let us re-state Corollary 4.2.5 differently. By associating to any Lie 8-algebroid its basic Lie-Rinehart algebra one obtains therefore a natural functor: This corollary justifies the title of the first part of the thesis. Remark 4.2.15. In the language of categories, Corollary 4.2.12 means that there exists a functor from Lie-Rinehart ideals of a Lie-Rinehart algebra over O, to the category of Lie 8-algebroids, mapping a Lie-Rinehart ideal I to an equivalence class of Lie 8-algebroids over O{I.

Proof of main results

A crucial bi-complex: Page

pnq pE 1 , Eq Description of Page pnq pE 1 , Eq
Let V be an O-module, and let pE, d, πq and pE 1 , d 1 , π 1 q be complexes of projective O-modules that terminates at V:

¨¨¨d Ý Ñ E ´2 d Ý Ñ E ´1 π Ý Ñ V, ¨¨¨d 1 Ý Ñ E 1 ´2 d 1 Ý Ñ E 1 ´1 π 1 Ý Ñ V. (4.2)
For every k ě 1, the pk `1q-th graded symmetric power

Ä k`1 E 1 of E 1 over O is a projective
O-module, and comes with a natural grading induced by the grading on E 1 .

Definition 4.3.1. Let k P N 0 . We call page number k of pE, d, πq and pE, d 1 , π 1 q the bicomplex of O-modules on the upper left quadrant Z ´ˆN 0 defined by:

Page pkq pE 1 , Eq j,m :" Hom O ˜k`1 ä E 1 | ´k´m´1 , E j ¸, for m ě 0 and j ď ´1 (4.3) Page pkq pE 1 , Eq 0,m :" Hom O ˜k`1 ä E 1 | ´k´m´1 , V ¸, for m ě 0, (4.4)
together with the vertical differential D v defined for any one of the two O-modules (4.3) or (4.4) by

D v : Page pkq pE 1 , Eq j,m ÝÑ Page pkq pE 1 , Eq j,m`1 Φ Þ ÝÑ D v pΦq : Ä k`1 E 1 ÝÑ E j x 1 d . . . d x k`1 Þ Ñ Φ ˝d1 px 1 d ¨¨¨d x k`1 q where d 1 acts as an O-derivation on x 1 d . . . d x k`1 P Ä k E 1 (and is 0 on E 1 ´1). The horizontal differential, D h : Page pkq pE 1 , Eq j,m ÝÑ Page pkq pE 1 , Eq j`1,m , is given by Φ Þ Ñ d ˝Φ or Φ Þ Ñ π ˝Φ
depending on whether Φ is of type (4.3) with j ď ´2 or the type (4.3) with j " ´1. It is zero on elements of type (4.4). We denote by ´Page pkq ' pE 1 , Eq, D ¯its associated total complex. When E 1 " E we shall write Page

. . . . . . . . Ò Ò Ò ¨¨¨Ñ Hom O ´Äk`1 E 1 | ´k´3 , E ´2¯D h Ñ Hom O ´Äk`1 E 1 | ´k´3 , E ´1¯D h Ñ Hom O ´Äk`1 E 1 | ´k´3 , V ¯Ñ 0 D v Ò D v Ò D v Ò ¨¨¨Ñ Hom O ´Äk`1 E 1 | ´k´2 , E ´2¯D h Ñ Hom O ´Äk`1 E 1 | ´k´2 , E ´1¯D h Ñ Hom O ´Äk`1 E 1 | ´k´2 , V ¯Ñ 0 D v Ò D v Ò D v Ò ¨¨¨Ñ Hom O ´Äk`1 E 1 | ´k´1 , E ´2¯D h Ñ Hom O ´Äk`1 E 1 | ´k´1 , E ´1¯D h Ñ Hom O ´Äk`1 E 1 | ´k´1 , V ¯Ñ 0 Ò Ò Ò 0 0 0 "-2 column" "-1 column" "last column" (4.5)
For later use, we spell out the meaning of being D-closed.

Remark 4.3.2. An element P P Page

pkq j pE 1 , Eq in ' iě1 Hom O ´Äk`1 E 1 | ´j´i , E ´i¯i s D-closed if and only if: 1. the component P ´1 : Ä k`1 E 1 | ´j´1 Ñ E ´1 is valued in the kernel of π : E ´1 Ñ V,
2. the following diagram commutes:

Ä k`1 E 1 | ´j´i p´1q j d 1 / / P ´i Ä k`1 E 1 | ´j´i`1 P ´i`1 E ´i d / / E ´i`1 with P ´i being the component of P in Hom O ´Äk`1 E 1 | ´j´i , E ´i¯.
For pE, dq " pE 1 , d 1 q, the second condition above also reads rd, P s RN " 0. Here is an important technical result.

Proposition 4.3.3. Let pE, d, πq be a resolution of V in the category of O-modules. Then, for every

k ě 0,
1. the cohomology of the complex pPage pkq ' pEq, Dq for the total differential

D ' :" D h ´p´1q ' D v is zero in all degrees;

Moreover, a D-closed element whose component on the "last column" of the diagram above is

zero is the image through D of some element whose two last components are also zero.

3. More generally, for all n ě 1, for a D-closed element P P Page pkq j pEq of the form

à iěn Hom O ˜k`1 ä E 1 | ´j´i , E ´i¸,
one has P " DpRq and R P Page pkq j´1 pEq can be chosen in

À iěn`1 Hom O ´Äk`1 E 1 | ´j´i`1 , E ´i ¯. Proof. Since Ä k E 1 | j`m´k is a projective O-

Interpretation of Page pnq pE 1 , Eq in our context

We denote by

Q p0q E and Q p0q E 1 the differentials of polynomial-degree 0 on Ä ' E and Ä ' E 1 induced by d and d 1 . Whence, pHom O p Ä ' E 1 , Ä ' Eq, Bq with B : H Þ Ñ Q p0q E ˝H ´p´1q |H| H ˝Qp0q E 1 is a complex of O-modules (see Lemma B.1.13).
Lemma 4.3.4. Let ϕ : pE 1 , d 1 q Ñ pE, dq be a chain map, and let Φ p0q :

Ä ' E 1 Ñ Ä ' E be its extension
to a co-algebra morphism, namely:

Φ p0q px 1 ¨¨¨¨¨x n q :" ϕpx 1 q ¨¨¨¨¨ϕpx n q.

1. We have,

Φ p0q ˝Qp0q E 1 " Q p0q E ˝Φp0q . 2. Φ p0q -co-derivations form a sub-complex of pHom O p Ä ' E 1 , Ä ' Eq, Bq.
Proof. The first item can be easily checked. The second item follows exactly the same pattern as Lemma 2.3.24.

The following proposition is very important.

Proposition 4.3.5. For every k P N 0 , and ϕ : pE 1 , d 1 q Ñ pE, dq be a chain map as in Lemma 4.3.4. The sub-complex of Φ p0q -co-derivations of polynomial-degree k is isomorphic to the complex z Page pkq pE 1 , Eq obtained from Page pkq pE 1 , Eq by crossing its "last column", see diagram (6.16).

Proof. The chain isomorphism δ : Hom

O p Ä ' E 1 , Ä ' Eq Ñ Hom O p Ä ' E 1 ,
Eq consists in mapping a Φ p0q -co-derivation H of polynomial-degree k and degree j to its unique Taylor coefficient H k P Page pkq j pE 1 , Eq, i.e. δpHq " pr ˝H. Let us check that this map is indeed a chain map: for every

x " x 1 d ¨¨¨d x k`1 P Ä k`1 E 1 one has, δ ˝pQ p0q E ˝H ´p´1q |H| H ˝Qp0q E 1 qpxq " ℓ 1 ˝Hk pxqp´1q |H| H k ˜k`1 ÿ i"1 ´p´1q p|x 1 |`¨¨¨`|x i´1 |q|x i | ℓ 1 1 px i q d x 1 d ¨¨¨d x k`1 " D h pH k qpxq ´p´1q |H k | D v pH k qpxq, by definition of D h and D v " DpH k qpxq " D ˝δpHqpxq, by definition of δ.
Here is another type of interpretation for z Page ' pE 1 , Eq involving the Richardson-Nijenhuis bracket.

Proposition 4.3.6. [START_REF] Frölicher | Theory of vector-valued differential forms. I: Derivations in the graded ring of differential forms[END_REF] When E 1 " E, z Page ' pE 1 , Eq with no 0-column is the bi-graded complex of exterior forms on E and the differential D of z Page ' pE 1 , Eq is Dp¨q " rd, ¨sRN .

We finish the section with the following lemma that will be important to prove Proposition 4.2.6. It uses the consequence of the cone construction (see Appendix B). Lemma 4.3.7. Let pR, d R , π R q be an arbitrary complex of projective O-module that terminates in a O-module V. There exists a projective resolution pE, d E , π E q of V, which contains pR, d R , π R q as a sub-complex. Moreover, we can assume that R admits a projective submodule in E in direct sum.

Proof. Resolutions of an O-modules V are universal objects in the category of complexes of projective O-modules. In particular, for every projective resolution pF, d F , π F q of V, there exist a (unique up to homotopy) chain map:

ϕ : pR, d R , π R q Ñ pF, d F , π F q.
We apply the cone construction (see, e.g. [START_REF] Charles | An introduction to homological algebra / Charles A. Weibel[END_REF], Section 1.5) to:

1. the complex pR, d R , π R q 2. the direct sum of the complexes pR, d R q and pF, d F q namely, `R ' F, d R ' d F , π R ' π F 3.
the chain map obtained by mapping any x P R to px, ϕpxqq P R ' F.

The differential is given by

d E px, y, zq " p´d R x, d R y ´x, d F z ´φpxqq (4.6)
for all px, y, zq

P E ´i " R ´i`1 'R ´i 'F ´i, i ě 2.
Since the chain given in item 3 is a quasi-isomorphism, its cone is an exact complex. We truncate the latter at degree ´1 without destroying its exactness by replacing the cone differential at degree ´1 as follows:

π E : R ´1 ' F ´1 Ñ V, pr, eq Þ Ñ π F peq ´πR prq.
For a visual description, see Equation (4.7) below: the resolution of V described in Lemma 4.3.7 is defined by:

¨¨¨/ / F ´3 d F / / F ´2 d F / / F ´1 π F / / V ¨¨¨/ / R ´3 d R / / R ´2 d R / / R ´1 π R 8 8 ¨¨¨/ / R ´2 id 7 7 d R / / ϕ D D R ´1 id 7 7 ϕ D D (4.7)
The proof of the exactness of this complex is left to the reader. The henceforth defined complex pE, d E , π E q is a resolution of V, and obviously contains pR, d R , π R q as a sub-chain complex of O-modules.

Let pE, d E , π E q be a free resolution of V and pR, d R , π R q a subcomplex of projective O-modules, as in Lemma 4.3.7. We say that P P Page pkq j pEq of the form

' iěn Hom O ´Äk`1 E | ´j´i , E ´i¯p reserves R if Ä k`1 R | ´j´i
j pEq to Ä ' R (a priori it is not valued in R but in E).
Lemma 4.3.8. Let pE, d E , π E q be a free resolution of V. Let R Ă E be a subcomplex made of free sub-O-modules such that there exists a graded free O-module V such that E " R ' V. pEq any extension of Q R (e.g. define Q to be 0 as soon as one element in V is applied to it). Then P ´DE p Qq :

Ä k`1 E ÝÑ E is zero on Ä k`1 R. We have to check
that it is a D E -coboundary of a map with the same property. Put κ " P ´DE p Qq. By Proposition 4.3.3, item 1, there exists τ P Page pkq j´1 pEq such that D E pτ q " κ. The equation D E pτ q " κ is equivalent to the datum of a collection of equations

D E v pτ i q `DE h pτ i`1 q " κ i`1 , i ě 1, and D E h pτ 1 q " κ 1 , (4.8) with, τ i P Hom O ´Äk`1 E | ´j´i`1 , E ´i¯a nd κ i P Hom O ´Äk`1 E | ´j´i , E ´i¯f or every i ě 1. Since
ι Rκ 1 " 0, we have that D E h pι Rτ 1 q " ι R `DE h pτ 1 q ˘" 0, (with the understanding that ι Rτ 1| V " 0). Using the exactness of the horizontal differential D E h , there exists C 1 P Page pkq pEq such that D E h pC 1 q " ι Rτ 1 . We now change τ 1 to τ 1 1 and τ 2 to τ 1 2 by putting τ 1 1 :" τ 1 ´ιR τ 1 and τ 1 2 :" τ 2 `DE v pC 1 q. One can easily check that Equation (4.8) still holds under these changes, i.e.,

D E v pτ 1 1 q `DE h pτ 1 2 q " κ 2 and D E h pτ 1 2 q `DE v pτ 3 q " κ 3 .
We can therefore choose τ such that ι Rτ 1 " 0. We then iterate this procedure, which allows us to choose τ P Page pkq j´1 pEq such that ι Rτ " 0 and D E pτ q " κ. By construction, Q :" τ `Q preserves R, while ι RQ " Q R , and D E pQq " P . The second item follows from the first one.

Proof on the existence

In this section, we prove Theorem 4.2.1.

Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra. Consider pE, d " ℓ 1 , πq a resolution of A by free O-modules: such resolutions always exist, see Proposition B.2.5. To start, we define a binary bracket ℓ 2 . The pair pd, ℓ 2 q will obey the axioms of the object that we now introduce. Definition 4.3.9. [START_REF] Lavau | Lie 8-algebroids and singular foliations[END_REF] An almost differential graded Lie algebroid of a Lie-Rinehart algebra pA, ρ A , r¨, ¨sA q is a complex q iPI k a family of generators of the free O-module E ´k. By construction ta i " πpe p´1q i q P A | i P I 1 u is a set of generators of A. In particular, there exists elements u k ij P O, such that for given indices i, j, the coefficient u k ij is zero except for finitely many indices k, and satisfying the skew-symmetry condition u k ij " ´uk ji together with

¨¨¨d ÝÑ E ´3 d ÝÑ E ´2 d ÝÑ E ´1 π ÝÑ A of projective O-
ra i , a j s A " ÿ kPI u k ij a k @i, j P I 1 (4.9)
We now define:

1. an anchor map by ρpe (c) we extend l2 to E using O-bilinearity and Leibniz identity with respect to the anchor ρ.

p´1q i q " ρ A pa i ) for all i P I,
By construction, l2 satisfies the Leibniz identity with respect to the anchor ρ E . Also, ρ ˝d " 0 on we obtain a 2-ary bracket ℓ 2 of degree +1 which satisfies all items of Definition 4.3.9.

E ´2.
Proof (of Theorem 4.2.1). Lemma 4.3.10 gives the existence of an almost differential graded Lie algebroid with differential ℓ 1 " d and binary bracket ℓ 2 . We have to construct now the higher brackets ℓ k for k ě 3.

Step 1: Construction of the 3-ary bracket ℓ 3 . (Its construction being different from the one of the higher brackets, we put it apart). We first notice that the graded Jacobiator defined for all 1. Since π is a morphism from pE ´1, ℓ 2 q to pA, r¨, ¨sA , ρ A q, and since r¨, ¨sA satisfies the Jacobi identity, one has for all x, y, z P E ´1:

Jacpx, y, zq P ker π.

2. Furthermore, a direct computation of rJac, ds RN gives in view of item 2 of Definition 4.3.9: dJacpx, y, zq " Jacpdx, y, zq `p´1q |x| Jacpx, dy, zq `p´1q |x|`|y| Jacpx, y, dzq for all x, y, z P E.

Thus, DpJacq " 0. By Proposition 4.3.3, item 2, Jac is a D-coboundary, and, more precisely, there exists an element ℓ 3 " ř jě2 ℓ j 3 P z Page p2q 1 pEq with ℓ j 3 P Homp Ä 3 E | ´j´1 , E ´j q such that Dpℓ 3 q " ´Jac i.e. rd, ℓ 3 s RN " ´Jac. (4.10)

We choose the 3-ary bracket to be ℓ 3 .

Step 2: Recursive construction of the k-ary brackets ℓ k for k ě 4. Let us recapitulate: ℓ 1 " d , ℓ 2 and ℓ 3 are constructed and the lowest polynomial-degree terms of rℓ 1 `ℓ2 `ℓ3 , ℓ 1 `ℓ2 `ℓ3 s RN satisfy 1. rℓ 1 , ℓ 1 s RN " 0 (since d 2 " 0), 2. rℓ 1 , ℓ 2 s RN " 0 (since d " ℓ 1 and ℓ 2 define an almost Lie algebroid structure).

3. rℓ 2 , ℓ 2 s RN `2rℓ 3 , ℓ 1 s RN " 2pJac `rℓ 3 , ℓ 1 s RN q " 0 by definition of ℓ 3 , and because rℓ 2 , ℓ 2 s RN " 2Jac.

However, the following term of degree `2 and polynomial-degree 3 may not be equal to zero:

rℓ 3 , ℓ 2 s RN P à Hom O ˜4 ä E j`1 , E ´j ¸" z Page p3q 1 pEq. (4.11)
Let us check that this term is indeed a O-multilinear map: For x 1 P E ´1, x 2 , x 3 , x 4 P E ď´2 and f P O, the only terms of pℓ 3 ˝ℓ2 `ℓ2 ˝ℓ3 qpx 1 , f x 2 , x 3 , x 4 q where the anchor shows up are:

$ ' ' ' & ' ' ' % ℓ 3 pℓ 2 px 1 , f x 2 q, x 3 , x 4 q " ρpx 1 qrf sℓ 3 px 2 , x 3 , x 4 q `f pℓ 3 pℓ 2 px 1 , x 2 q, x 3 , x 4 qq p´1q |x 2 |`|x 3 |`|x 4 | ℓ 2 pf ℓ 3 px 2 , x 3 , x 4 q, x 1 q " ´ρpx 1 qrf sℓ 3 px 2 , x 3 , x 4 q `f pp´1q |x 2 |`|x 3 |`|x 4 | ℓ 2 pf ℓ 3 px 2 , x 3 , x 4 q, x 1 qq
The terms containing the anchor map add up to zero. When there are more elements in E ´1, the computation follows the same line. Moreover, by graded Jacobi identity of the Richardson-Nijenhuis bracket:

rrℓ 1 `ℓ2 `ℓ3 , ℓ 1 `ℓ2 `ℓ3 s RN , ℓ 1 `ℓ2 `ℓ3 s RN " 0
The term of polynomial-degree 4 in the previous expression gives rrℓ 3 , ℓ 2 s RN , ℓ 1 s RN " 0. Hence, by Proposition 4.3.6, rℓ 3 , ℓ 2 s RN is a D-cocycle in the complex Page p3q pEq, whose components on the last column and the column ´1 are zero. It is therefore a coboundary by Proposition 4.3.3 item 3: we can continue a step further and define ℓ 4 P ' jě3 Hom ´Ä4 E | ´j´1 , E ´j ¯such that:

´rℓ 2 , ℓ 3 s RN " rℓ 1 , ℓ 4 s RN " rd, ℓ 4 s RN . (4.12)
We choose the 4-ary bracket to be ℓ 4 . We now proceed by recursion. We assume that we have constructed all the k-ary brackets, ℓ k such as :

rd, ℓ k s RN " ´ÿ i`j"k`1 iďj rℓ i , ℓ j s RN " ´1 2 ÿ i`j"k`1 i,jě1
rℓ i , ℓ j s RN (4.13) for every k " 1, . . . , n with n ě 4. The (n`1)-ary bracket is constructed as follows. First, the operator

ř i`j"k`1 i,jě1
rℓ i , ℓ j s RN is checked to be O-linear as before. Now, we have

ÿ i`j"n`2 i,jě1 " d, rℓ i , ℓ j s RN ‰ RN " ´2 ÿ i`j"n`2 i,jě1 " ℓ i , rd, ℓ j s RN ‰ RN
(by graded Jacobi identity).

Since ℓ j satisfies Equation (4.13) up to order n, we obtain

ÿ i`j"n`2 i,jě1 " d, rℓ i , ℓ j s RN ‰ RN " ÿ i`j`k"n`3 i,j,kě1 " ℓ i , rℓ j , ℓ k s RN ‰ RN " 0,
where we used the graded Jacobi identity of the Nijenhuis-Richardson bracket in the last step. Therefore, We finish this section with a proof of Proposition 4.2.6.

ř i`j"n`2 i,jě1 rℓ i , ℓ j s RN ,
Proof (of Proposition 4.2.6). We prove this Proposition in two steps.

1. Lemma 4.3.7 guarantees the existence a free resolution pE, d, πq of the Lie-Rinehart algebra A such that E contains E 1 and such that there exists a graded free module V with E 1 ' V " E.

2.

Let D E and D E 1 be as in the proof of Lemma 4.3.8. We construct the n-ary brackets on E by extending the ones of pE 1 , pℓ 1 k q kě1 , ρ E 1 , π 1 q in the following way:

(a) We first construct an almost Lie algebroid bracket l2 on E ´1 that extends the 2-ary bracket of E 1 ´1. Since the 2-ary bracket is determined by its value on a basis, the existence of a free module V ´1 such that E 1 ´1 ' V ´1 " E ´1 allows to construct l2 on E such that its restriction to E 1 is ℓ 1 2 and such that it satisfies the Leibniz identity. As in the proof of Theorem 4.2.1 (to be more precise: Lemma 4.3.10), we see that r l2 , d E s RN is O-linear, hence belongs to Page p2q 2 pEq and is a D E -cocycle. Since E 1 is a Lie 8-algebroid, its restriction to Ä 2 E 1 is zero. Lemma 4.3.8 allows to change l2 to an 2-ary bracket ℓ 2 :" l2 `τ2 with τ 2 " 0 on Ä 2 E 1 . Hence, ℓ 2 defines a graded almost Lie algebroid bracket, whose restriction to E 1 is still ℓ 1 2 . (b) Since ℓ 2 is an extension of ℓ 1 2 , its Jacobiator Jac P Page p2q 2 pEq of the 2-ary bracket ℓ 2 preserves E 1 . Also, its restriction ι E1 Jac P Page p2q 2 pE 1 q is the Jacobiator of ℓ 1 2 , and the latter is the D E 1 -coboundary of ℓ 1 3 in view of the higher Jacobi identity of E 1 . Since Jac P Page p2q 2 pEq is a D E -cocycle, Lemma 4.3.8 assures that Jac is the image through D E of some element ℓ 3 P Page p2q 1 pEq which preserves E 1 and whose restriction to

Ä 3 E 1 is ℓ 1 3 .
The proof continues by recursion: at the n-th step, we use Lemma 4.3.8 to construct an n-ary bracket for E that extends the n-ary bracket of E 1 . By construction, the inclusion map ι : E 1 ãÑ E is a morphism for the n-ary brackets for all n ě 1.

Proof of universality

Before proving the universal character of the construction, we need to do some preparations.

Let us prove the following lemma, Lemma 4.3.11. Let Ψ, Ξ : S ' K pE 1 q Ñ S ' K pEq be O-linear Lie 8-algebroid morphisms. Let n P N 0 . If Ξ piq " Ψ piq for every 0 ď i ď n, there exists 1. a Lie 8-morphism of algebroids J 1 : S K pE 1 q Ñ S K pEq 2. and a homotopy pJ t , H t q r0,1s joining Ψ and J 1 , such that 1. the components of polynomial-degree less or equal to n of H t vanish, 2. J piq 1 " Ξ piq for every 0 ď i ď n `1.

Proof. Let us consider pΞ ´Ψq pn`1q :

Ä ' E 1 ÝÑ Ä ' E.
By assumption, pΞ ´Ψq piq " 0 for all i ď n, so that in view of Lemma 2.3.27

1. pΞ ´Ψq pn`1q is a Ψ p0q -co-derivation.

2. Proposition 4.3.5 means that the map the restriction of the map pΞ ´Ψq pn`1q to Ä n`2 E 1 corresponds to a closed element of degree 0 in Page pn`1q pE 1 , Eq equipped with differentials ℓ 1 , ℓ 1 1 .

Proposition 4.3.3 implies that there exists O-linear map H n`1 : Ä n`2 pE 1 q ÝÑ E, a degree ´1 and of polynomial-degree n `1, i.e. an element in Page pn`1q pE 1 , Eq, such that,

pΞ ´Ψq pn`1q " Q p0q E ˝Hn`1 `Hn`1 ˝Qp0q E 1 . (4.14)
We denote its extension to a Ψ p0q -co-derivation of degree ´1 by H pn`1q . We now consider the following differential equation for t P r0, 1s: Hence:

dJ t dt " Q E ˝Ht `Ht ˝QE 1 ,
# J piq t " Ψ piq for i " 0, . . . , n , J pn`1q t " Φ pn`1q `tpΞ ´Ψq pn`1q .
Therefore, applying t " 1 to the previous relation, one finds

# J piq t " Ψ piq for i " 0, . . . , n , J pn`1q 1 " Ψ pn`1q `pΞ ´Ψq pn`1q " Ξ pn`1q
This completes the proof.

Construction of the Lie 8-morphism

Proof (of Theorem 4.2.4). Let us prove item 1. We construct the Taylor coefficients of the Lie 8algebroid Φ by recursion.

The Taylor coefficient of polynomial-degree 0 is obtained out of classical properties of projective resolutions of O-modules. Given any complex pE 1 , ρ 1 , ℓ 1 1 , π 1 q which terminates in A through π, for every free resolution pE, ρ, ℓ 1 , πq of A, there exists a chain map Φ p0q : pE 1 , ℓ 1 1 q Ñ pE, ℓ 1 q as in Equation (4.2), and any two such chain maps are homotopic. We still denote by Φ p0q its extension to an polynomial-degree 0 co-morphism

Ä ' E 1 Ñ Ä ' E.
To construct the second Taylor coefficient, let us consider the map: A. If either one of the homogeneous elements x P E 1 or y P E 1 is not of degree ´1, a straightforward computation gives:

S 2 K pE 1 q Ñ E px, yq Þ Ñ Φ p0q ˝ℓ1 2 px,
ℓ 1 ˝´Φ p0q ˝ℓ1 2 px, yq ´ℓ2 ´Φp0q pxq, Φ p0q pyq ¯¯" Φ p0q ˝ℓ1 1 ˝ℓ1 2 px, yq `ℓ2 ´Φp0q ˝ℓ1 1 pxq, Φ p0q pyq p´1q |x| ℓ 2 ´Φp0q pxq, Φ p0q ˝ℓ1 1 pyq " ´Φp0q ˝ℓ1 2 ´ℓ2 ´Φp0q , Φ p0q ¯¯˝ℓ 1 1 px d yq.
B. If both x, y P E 1 are of degree ´1:

π ´Φp0q ℓ 1 2 px, yq ´ℓ2 pΦ p0q x, Φ p0q yq ¯" π 1 ˝ℓ1 2 px, yq ´π ˝ℓ2 ´Φp0q x, Φ p0q y " rπ 1 pxq, π 1 pyqs ´"π ´Φp0q x ¯, π ´Φp0q x ¯ı " rπ 1 pxq, π 1 pyqs ´rπ 1 pxq, π 1 pyqs " 0.
By Proposition 4.3.3 item 2), there exists Φ p1q P Hom O ´Ä2 E 1 , E ¯, of degree 0, so that Φ p0q ˝ℓ1 2 px, yq `Φp1q ˝ℓ1 1 px d yq " ℓ 1 ˝Φp1q px, yq `ℓ2 pΦ p0q pxq, Φ p0q pyqq for all x, y P E 1 . (4.17) Φ p0q is a chain map and Relation (4.17) can be rewritten in terms of Q E and Q E 1 as follows

# Q p0q E ˝Φp0q " Φ p0q ˝Qp0q E 1 Q p0q E ˝Φp1q ´Φp1q ˝Qp0q E 1 " Φ p0q ˝Qp1q E 1 ´Qp1q E ˝Φp0q (4.18)
The construction of the morphism Φ announced in Theorem 4.2.4 is then done by recursion. The recursion assumption is that we have already defined a O-multilinear co-morphism Φ : S ' K pE 1 q Ñ S ' K pEq with pΦ ˝QE 1 ´QE ˝Φq pkq " 0 for all 0 ď k ď n.

The co-morphism Φ : S ' K pE 1 q Ñ S ' K pEq with Taylor coefficients Φ p0q and Φ p1q satisfies the recursion assumption for n " 1.

Assume now that we have a co-morphism Φ that satisfies this assumption for some n P N, and consider the map T Φ :" Φ ˝QE 1 ´QE ˝Φ. Lemma 2.3.26 implies that T pn`1q Φ is a O-multilinear Φ p0q -co-derivation, and that it corresponds to a D-closed element 1 in Page pn`1q pE 1 , Eq. Since it has no 1 The following remark is crucial. Under the assumptions of Lemma 2.3.26, pΦ˝Q E 1 ´QE ˝Φq pn`1q corresponds to a Dclosed element of degree `1 in the bi-complex Page pn`1q pE 1 , Eq through the chain isomorphism described in Proposition 4.3.5. Here, E, E 1 are equipped with the differentials ℓ1, ℓ 1 1 which are the restriction of the components

Q p0q E , Q p0q E 1 .
component on the last column for degree reason, Proposition 4.3.3 implies that T pn`1q Φ is a coboundary: That is to say that there is a Φ p0q -co-derivation Θ P Page pn`1q pE 1 , Eq (of polynomial-degree n `1 and degree 0) which can be seen as a map Θ :

Ä n`2 pE 1 q Ñ E such that:

T pn`1q Φ " Q p0q E ˝Θ ´Θ ˝Qp0q E 1 .
Consider now the co-morphism Φ whose Taylor coefficients are those of Φ in polynomial-degree 0, . . . , n and Φ pn`1q `Θ in polynomial-degree n `1:

Φpiq :" $ & % Φ piq if 0 ď i ď n, Φ pn`1q `Θ if i " n `1 (4.19)
This is easily seen to satisfy the recursion relation for n `1. This concludes the recursion. The Taylor coefficients obtained by recursion define a Lie 8-algebroid Φ : S ' K pE 1 q ÝÑ S ' K pEq which is compatible by construction with the hooks π, π 1 .

By continuing this procedure, we construct a Lie 8-morphism from S ' K pE 1 q to S ' K pEq. This proves the first item of Theorem 4.2.4.

Construction of a homotopy that joins two such morphisms

Let us prove the second item in Theorem 4.2.4. Notice that in the proof of the existence of the Lie 8-morphism between S ' K pE 1 q and S ' K pEq obtained in the first item, we made many choices, since we have chosen a coboundary at each step of the recursion.

Let Φ, Ψ be two such Lie 8-morphisms between S ' K pE 1 q and S ' K pEq. The polynomial-degree 0 component of the co-morphisms Φ and Ψ restricted to E 1 are chain maps:

¨¨¨/ / E 1 ´2 / / h Φ p0q Ψ p0q E 1 ´1Φ p0q Ψ p0q h π 1 A ¨¨¨/ / E ´2 / / E ´1 π = = = =
which are homotopy equivalent in the usual sense because pE, ℓ 1 q is a projective resolution of A: said differently, there exists a degree ´1 O-linear map h :

E 1 Ñ E such that Ψ p0q ´Φp0q " ℓ 1 ˝h `h ˝ℓ1 1 on E 1 . (4.20)
Let us consider the following differential equation: " Ψ p0q . By construction, J 1 is homotopic to Φ via the pair pJ t , H t q over r0, 1s, and its polynomial-degree 0 Taylor coefficient coincides with the Taylor coefficient of Ψ.

$ & % dJt dt " Q E ˝Ht pJ t q `Ht pJ t q ˝QE 1 ,
From there, the construction goes by recursion using Lemma 4.3.11. Indeed, this lemma allows constructing recursively a sequence of Lie 8-algebroids morphism pΨ n q ně0 and homotopies pJ n,t , H n,t q (with t P rn, n `1s) between Ψ n and Ψ n`1 such that: H piq n,t is zero for t ě n and i ‰ n `1. By Lemma 4.3.11, all these homotopies are compatible with the hooks. These homotopies are glued in a homotopy pJ t , H t q r0,`8r such that for every n P N 0 , the components of polynomial-degree n of the Lie 8-algebroids morphism J

Examples of universal Lie 8-algebroids of Lie-Rinehart algebras

New constructions from old ones

In this section, we explain how to construct universal Lie 8-algebroids of some Lie-Rinehart algebra which is derived from a second one through one of natural constructions as in Section 3.2.1 (localization, germification, restriction), when a universal Lie 8-algebroid of the latter is already known.

Localization

Localization is an useful algebraic tool, specially in algebraic geometry. When O is an algebra of functions, it corresponds to study local properties of a space, or germs of functions.

Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra over O. Let S Ă O be a multiplicative-closed subset containing no zero divisor. We recall from item 2, Example 3.2.1 that the localization S ´1A -A b O S ´1O of A at S comes equipped with a natural structure of Lie-Rinehart algebra over the localization algebra S ´1O. Recall that for φ : E ÝÑ T a homomorphism of O-modules, there is a well-defined homomorphism of O-modules,

φ b id : E b O S ´1O ÝÑ T b O S ´1O, φ b id px b f s q :" φpxq b f s that can
ρ 1 : S ´1E ´1 ÝÑ DerpS ´1Oq x s Þ ÝÑ ρ 1 `x s ˘: S ´1O ÝÑ S ´1O f u Þ ÝÑ 1 s ¨´ρpxqrfsu´fρpxqrus u 2
for

x P E, f P O, ps, uq P S ˆS;

3. ℓ 1 k " S ´1ℓ k , for all k P Nzt2u;

4. The binary bracket is more complicated because of the anchor map: we set

ℓ 1 2 ˆ1 s x, 1 u y ˙" 1 su ℓ 2 px, yq ´ρpxqrus su 2 y `ρpyqrss s 2 u x
for x, y P E, ps, uq P S 2 (with the understanding that ρ " 0 on E ´i with i ě 2); 5. π 1 " S ´1π.

One can check that these operations above are well-defined and for all z P S ´1E ´1 the map ρ 1 pzq is indeed a derivation on S ´1O. Proof. The object ppE 1 ' , ℓ 1 ' , ρ 1 qq described above is also a Lie 8-algebroid terminating in S ´1A through π 1 . It is universal because localization preserves exact sequences [And].

Restriction

When O Y is the ring of functions of an affine variety Y (see Section 5.1), to every subvariety X Ă Y corresponds its zero locus, which is an ideal I X Ă O Y . A Lie 8-algebroid or a Lie-Rinehart algebra over O Y may not restrict to a Lie-Rinehart algebra over O X : it only does so when one can quotient all brackets by I X , which geometrically means that the anchor map takes values in vector fields tangent to X. We can then "restrict", i.

Germification

Let W Ď C N be an affine variety and O W its coordinates ring (see Section 5.1). For a P W , consider O W,a the local ring at a. Note that O W,a » pO W q ma [START_REF] Hartshorne | Algebraic Geometry[END_REF], where m a " tf P O W | f paq " 0u and pO W q ma is the localization w.r.t the complement of m a , Proposition 4.4.1 implies the following statement:

Sections vanishing on a codimension 1 subvariety

Let pA, r¨, ¨s , ρ A q be an arbitrary Lie-Rinehart algebra over O. For any ideal I Ă O, IA is also a Lie-Rinehart algebra (see Example 3.2.1). When O are functions on a variety M , I are functions vanishing on a subvariety X and A is a O-module of sections over M , IA corresponds geometrically to sections vanishing along X. It is not an easy task. In codimension 1, i.e. when I is generated by one element, the construction can be done by hand. Proposition 4.4.5. Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra over a commutative algebra O. Let pE, ℓ k " t¨¨¨u kě1 , ρq be a Lie 8-algebroid that terminates in A through a hook π. For any element χ P O, the O-module A 1 " χA Ď A is closed under the Lie bracket, so the triple pχA, r¨, ¨sA , ρ A q is a Lie-Rinehart algebra over O. A Lie 8-algebroid pE 1 " E, ℓ 1 k " t¨¨¨u 1 kě1 , ρ 1 q hooked in χA through π 1 can be defined as follows:

1. The brackets are given by (a) t¨u 1 1 " t¨u 1 , (b) the 2-ary bracket: tx, yu 1 2 :" χtx, yu 2 `ρpxqrχs y `p´1q |x||y| ρpyqrχs x, (4.22) for all x, y P E ' , with the understanding that ρ "

0 on E ď´2 , (c) t¨¨¨u 1 k " χ k´1 t¨¨¨u k for all k ě 3, 2. ρ E 1 " χρ, 3. π 1 " χπ.
Proof. We leave it to the reader. Proof. If χ is not a zero-divisor in A (i.e. if a Þ Ñ χa is an injective endomorphism of A), then the kernel of π 1 coincides with the kernel of π, i.e. with the image of t¨u 1 " t¨u 1 , so that pE, ℓ 1 , χπq is a resolution of χA.

Algebra extension

Blow-up

Let us investigate the behavior of the universal Lie 8-algebroids under blow-up.

We recall that the blowup of C N `1 at the origin is given by, B 0 pC N `1q " tpx, ℓq P C N `1 ˆPN | x P ℓu together with the map

π : B 0 pC N `1q ÝÑ C N `1 px, ℓq Þ Ñ x.
For O " Crz 0 , . . . , z N s the coordinate ring of C N `1, the blow-up of C N `1 at the origin is covered by affine charts: in the i-th affine chart U i , the coordinate ring is

O U i " Crz 0 {z i , . . . , z i , . . . , z N {z i s.
By Remark 3.2.1, and Proposition 4.4.7 for any Lie-Rinehart algebra A whose anchor map takes values in vector fields vanishing at 0 P C N `1, we obtain a Lie 8-algebroid of O U i b O A that we call blow-up of at 0 in the chart U i . Proposition 4.4.7 then says that the blow-up at 0 of the universal Lie 8-algebroid of A, in each chart, is a Lie 8-algebroid that terminates in the blow-up of A (as defined in remark 3.2.1). It may not be the universal one, see Example 4.4.10.

EXAMPLES OF UNIVERSAL LIE 8-ALGEBROIDS OF LIE-RINEHART ALGEBRAS 73

Example 4.4.10 (Universal Lie-8-algebroids and blow-up: a counter example). Consider the polynomial function in N `1 variables φ " ř N i"0 z 3 i . Let us consider the singular foliation F φ Ă XpC N q as in Example 4.4.6. Its generators are ∆ ij :" together with the anchor map ρ which assigns e i to r ∆ iN , for i " 1 . . . N ´1. On the other hand, the blow-up of the universal Lie 8-algebroid of F φ is not homotopy equivalent to a Lie algebroid. Let us show this point. In this case, the Lie 8-algebroid is given in Section 4.4.6: notice that E ´i ‰ 0 for i " 1, . . . , N and that ℓ 1 | 0 " 0. The pull-back Lie 8-algebroid p Ẽ, p lk q kě1 , ρ, πq verifies by construction that Ẽ´i ‰ 0 for i " 1, . . . , N and that l1 | x " 0 for every x in the inverse image of zero, and such a complex can not be homotopy equivalent to a complex of length 1. In other words, O U i b O ¨is not an exact functor in this case (which is a classical fact in algebraic geometry).

z 2 i B Bz j ´z2 j B Bz i , for 0 ď i ă j ď N . Let us consider its blow-up in the chart U N . Geometrically speaking, Ă F φ " O U N b O F φ is the O U N -module generated by the blown-up vector fields r ∆ ij " z N ´z2 i B Bz j ´z2 j B Bz i ¯, j ‰ N , and r ∆ iN " z N ´zN z 2 i B Bz N ´B Bz i ´z2 i ř N ´1 j"0 z j B Bz j ¯of
This example tells us that the blow-up of the universal Lie 8-algebroid of an affine variety W may not be the universal Lie 8-algebroid of its blow-up, (even locally).

Universal Lie 8-algebroids of some singular foliations

Singular foliation is defined in Section 3.2. The coming example uses the notion "multi-derivation", it is useful to recall the definition and write down some basic operations on them. We refer the reader to Section 3.1 of [START_REF] Laurent-Gengoux | Poisson structures. Grundlehren der mathematischen Wissenschaften A series of comprehensive studies in mathematics[END_REF] for more details on this notion. Definition 4.4.11. A skew symmetric k-multilinear map P P Hom K pO k , Oq is a said to a k-multiderivation of O if P is a derivation in each of its argument, i.e., for every i P t1, . . . , ku and for all f, g P O we have When O happen to be the coordinate ring of some affine variety W , k-multi-derivations of O are called k-multi-vector fields on W , and denote by X k pW q the module of k-multi-vector fields on W . In general, it is denoted by X k pOq.

P rf 1 , . . . ,
Remark 4.4.12. By skew-symmetry argument, it is enough that the relation (4.24) holds for the first slot, i.e. for i " 1.

Lemma 4.4.13. The graded module X ' pOq :" À kě0 X k pOq comes equipped with graded algebra structure whose product ^is defined as follows, pP ^Rqrf 1 , . . . , f k`l s :" ÿ σPSpk,lq ϵpσqP rf σp1q , . . . , f σpkq sRrf σpk`1q , . . . , f σpk`lq s, for P P X k pOq, R P X l pOq and f 1 , . . . , f k`l P O. By convention X 0 pOq " O. Also, f ^X :" f X for all f P O. Lemma 4.4.14. For every f P O, the contraction by f , ι f : X k pOq Ñ X k´1 pOq, @ k ě 1, which is defined as follows

2 P Þ Ñ ι f pP qrf 1 , . . . , f k´1 s :" P rf, f 1 , . . . , f k´1 s, for all f 1 , . . . , f k´1 P O (4.25) satisfies • ι f ˝ιf " 0,
• for all P P X k pOq and R P X l pOq, ι f pP ^Rq " ι f pP q ^R `p´1q k P ^ιf pRq.

(4.26)

Proof. The first item is true by skew-symmetry of multi-derivations. Now let us prove the second item. For f 1 , . . . , f k`l´1 P O, we have

ι f pP ^Rqrf 1 , . . . , f k`l´1 s " pP ^Rqrf, f 1 , ¨¨¨, f k`l´1 s " ÿ σPSpk´1,lq ϵpσqP rf, f σp1q , . . . , f σpk´1 qsRrf σpkq , . . . , f σpk`l´1q sp ´1q k ÿ σPSpk,l´1q
ϵpσq P rf σp1q , . . . , f σpkq sRrf, f σpk`1q , . . . , f σpk`l´1q q looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon here, the inversion number with f is k " ι f pP q ^R `p´1q k P ^ιf pRq.

We have used the fact that for every σ P Spk, lq, one has that either σp1q " 1 or σpk `1q " 1.

Remark 4.4.15. Notice the following:

• For all f P O and X P DerpOq, one has ι f pXq " Xrf s.

• For any two derivation X, Y P DerpOq we have pX ^Y qrf, gs " Xrf sY rgs ´Y rgsXrf s.

• For a family of derivations X 1 , . . . X k P DerpOq, the k-multi-derivation X 1 ^¨¨¨^X k applied to

f 1 , ¨¨¨, f k P O is equal to the determinant X 1 rf 1 s ¨¨¨X 1 rf k s . . . . . . X k rf 1 s ¨¨¨X k rf k s .
• When O is the algebra of polynomial functions in d variables Krx 1 , . . . , x d s, a k-multi-derivation P admits the coordinate expression

P " ÿ 1ďi 1 㨨¨ăi k ďd P rx i 1 , . . . , x i k s B Bx i 1 ^¨¨¨^B Bx i k .

Lie derivative

For P P X k pOq and X P DerpOq, the Lie derivative L X P P X k pOq of P along X is defined as

pL X P qrf 1 , ¨¨¨, f k s :" X rP rf 1 , ¨¨¨, f k ss ´k ÿ j"1 P rf 1 , . . . , Xrf j s, . . . , f k s. (4.27)
There is another important operation on multi-derivations, the so-called "Schouten bracket" which is a generalization of the commutator of derivations, also Lie derivative of multi-derivation along a vector field. We will not recall how it is defined here, since we do not really make use of it. We refer the reader to, e.g [START_REF] Laurent-Gengoux | Poisson structures. Grundlehren der mathematischen Wissenschaften A series of comprehensive studies in mathematics[END_REF] for this notion.

Vector fields annihilating a Koszul function φ

This universal Lie 8-algebroid was already described in Section 3.7 of [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF], where the brackets were simply checked to satisfy the higher Jacobi identities -with many computations left to the reader. Here, we give a theoretical explanation of the construction presented in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF].

Let O be the algebra of all polynomials on V :"

C d . A function φ P O is said to be a Koszul polynomial, if the Koszul complex . . . ιφ Ý Ñ X 3 pV q ιφ Ý Ñ X 2 pV q ιφ Ý Ñ XpV q ιφ Ý Ñ O ÝÑ 0 (4.28)
is exact in all degree, except in degree 0. By virtue of a theorem of Koszul [START_REF] Eisenbud | Commutative algebra. With a view toward algebraic geometry[END_REF], see [START_REF] Hideyuki | Commutative ring theory[END_REF] Theorem 16.5 piq, φ is Koszul if ´Bφ Bx 1 , ¨¨¨, Bφ Bx d ¯is a regular sequence. From now on, we choose φ a Koszul function, and consider the Lie-Rinehart algebra (which is a singular foliation) F φ :" tX P XpV q : Xrφs " 0u " Kerpι φ q : XpV q ιφ Ý Ñ O.

(4.29)

The Koszul complex (4.28) truncated of its degree 0 term gives a free resolution pE, d, ρq of F φ , with E ´i :" X i`1 pV q, d :" ι φ , and ρ :" ´ιφ .

Remark 4.4.16. Exactness of the Koszul complex implies in particular that F φ is generated by the vector fields:

" Bφ Bx i B Bx j ´Bφ Bx j Bφ Bx i , | 1 ď i ă j ď d * . (4.30)
In [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF], this resolution is equipped with a Lie 8-algebroid structure, whose brackets we now recall.

Proposition 4.4.17. A universal Lie 8-algebroid of F φ Ă XpV q is given on the free resolution `E´' " X '`1 pV q, d " ι φ , ρ " ´ιφ ˘by defining the following n-ary brackets:

tB I 1 , ¨¨¨, B In u n :" ÿ i 1 PI 1 ,...,inPIn ϵpi 1 , . . . , i n qφ i 1 ¨¨¨in B I i 1 1 '¨¨¨'I in n ;
(4.31)

and the anchor map given for all i, j P t1, . . . , nu by

ρ ˆB Bx i ^B Bx j ˙:" Bφ Bx j B Bx i ´Bφ Bx i B Bx j . (4.32)
Above, for every multi-index J " tj 1 , . . . , j n u Ď t1, . . . , du of length n, B J stands for the n-vector field B

Bx j 1

^¨¨¨^B Bx jn and φ j 1 ¨¨¨jn :" To understand this structure, let us first define a sequence of degree `1 graded symmetric polyderivations on X ' pV q (by convention, i-vector fields are of degree ´i `1) by:

B n φ Bx j 1 ¨¨¨Bx jn . Also, I 1 ' ¨¨¨' I n is a multi-index
tB i 1 , . . . , B i k u 1 k :" B k φ Bx i 1 ¨¨¨Bx i k . (4.33)
We extend them to a graded poly-derivation of X ' pV q.

Lemma 4.4.18. The poly-derivations pt¨¨¨u 1 k q kě1 are O-multilinear and equip X ' pV q with a (graded symmetric) Poisson 8-algebra structure. Also, t¨u 1

1 " ι φ .

Proof. For degree reason, tF, X 1 , . . . , X k´1 u 1 k " 0 for all X 1 , . . . , X k´1 P X ' pV q and all F P X 0 pV q " O. This implies the required O-multilinearity. It is clear that the higher Jacobi identities hold since brackets of generators tδ i 1 , ¨¨¨, δ in u 1 are elements in O, and all brackets are zero when applied an element in O.

Proof (of Proposition 4.4.17). The brackets introduced in Proposition 4.4.17 are modifications of the Poisson 8-algebra described in Lemma 4.4.18. By construction, t¨¨¨u 1 n " t¨¨¨u n when all arguments are generators of the form B I for some I Ă t1 . . . , nu of cardinal ě 2. By O-multilinearity, this implies t¨¨¨u 1 n " t¨¨¨u n when n ě 3, or when n " 2 and no argument is a bivector-field, or when n " 1 and the argument is not a bivector field. As a consequence, all higher Jacobi identities hold when applied to n-vector fields with n ě 3.

Let us see what happens when one of the arguments is a bivector field, i.e. in the case where we deal with at least an element of degree ´1. Let us assume that there is one such element, i.e.

Q 1 " B i ^Bj , Q 2 " B I 2 , . . . , Q n " B In with |I j | ě 3, j " 2, . . . , n.
Then, in view of the higher Jacobi identity for the Poisson 8-brackets pt¨¨¨u 1 k q kě1 gives:

0 " ÿ 2ďkďn´2 ÿ σPS k,n´k ϵpσq ! ␣ Q σp1q , . . . , Q σpkq ( 1 k , Q σpk`1q , . . . , Q σpnq ) 1 n´k`1 (4.34) `ÿ σPS n´1,1 ,σpnq‰1 ϵpσq ! ␣ Q σp1q , . . . , Q σpn´1q ( 1 n´1 , Q σpnq ) 1 2 (4.35) `ÿ σPS 1,n´1 ,σp1q‰1 ϵpσq ! ␣ Q σp1q ( 1 1 , Q σp2q . . . , Q σpnq ) 1 n (4.36) `p´1q ř n k"2 |B I k | ␣ tQ 2 , . . . , Q n u 1 n´1 , Q 1 ( 1 2 (4.37) `␣tQ 1 u 1 1 , Q σp2q . . . , Q σpnq ( 1 n . (4.38)
In lines (4.34)-(4.35)-(4.36) above, we have t¨¨¨u 1 " t¨¨¨u for all the terms involved. This is not the case for (4.37)-(4.38). Indeed: Since the image of its anchor map are vector fields tangent to W , it is clear that the universal Lie 8-algebroid of Proposition 4.4.17 restricts to W . To prove Proposition 4.4.19, it suffices to check that the restriction i W XpV q to W of the Koszul complex is still exact, except in degree 0.

! tB I 2 , ¨¨¨, B In u 1 n´1 , B i ^Bj )1 2 " ␣ tB I 2 ,
Lemma 4.4.20. The restriction to the zero locus W of φ of the Koszul complex (4.28), namely the complex, . . .

ιφ Ý Ñ i W X 3 pV q ιφ Ý Ñ i W X 2 pV q ιφ Ý Ñ i W F is a free resolution of i W F in the category of O W -modules.
Proof. Let k ě 2 be an integer. For a given P P i W X k pV q, the relation ι φ P " 0 means that for any P P X k pV q extending P , there exists U P X k´1 pV q such that ι φ P " φU . By exactness of the Koszul complex (4.28), one has, U " ι φ Q for some Q P X k pV q. Hence, P ´φ Q is an extension of the bivector field P such that ι φ p P ´φ Qq " 0. Using one more time exactness of the Koszul complex (4.28), we construct R P X k`1 pV q such that P " φQ `ιφ R . Thus, P " i W P " ι φ i W R " ι φ R.

Vector fields vanishing on subsets of a vector space

Let O be the algebra of smooth or holomorphic or polynomial or formal functions on K d , and I Ă O be an ideal. Then IDerpOq, i.e. vector fields of the form:

ř d i"1 f i B Bx i , with f 1 , . . . , f d P I, is a Lie-Rinehart algebra (It is also a singular foliation).
Remark 4.4.21. Geometrically, when I corresponds to functions vanishing on a sub-variety N Ă K n , IDerpOq must be interpreted as vector fields vanishing along N .

EXAMPLES OF UNIVERSAL LIE 8-ALGEBROIDS OF LIE-RINEHART ALGEBRAS 78

Let us describe a Lie 8-algebroid that terminates at IDerpOq, then discuss when it is universal. Let pφ i q iPI be generators of I. Consider the free graded algebra K " Orpµ i q iPI s generated by variables pµ i q iPI of degree ´1. The degree ´1 derivation B :"

ř iPI φ i B Bµ i squares to zero. The O-module K ´j of elements degree j in K ' is made of all sums ř i 1 ,...,i j PI f i 1 ...i j µ i 1 ¨¨¨µ i j with f i 1 ...i j P O. Consider the complex of free O-modules ¨¨¨B b O id ÝÑ K ´2 b O DerpOq Bb O id ÝÑ K ´1 b O DerpOq (4.39)
Proposition 4.4.22. The complex (4.39) comes equipped with a Lie 8-algebroid structure that terminates in IDerpOq through the anchor map given by µ i

B Bx j Þ Ñ φ i B
Bx j for all i P I, and j P 1, . . . , d.

Proof. First, one defines a O-linear Poisson-8-algebra structure on the free algebra generated by pµ i q iPI (in degree ´1) and ´B Bx j ¯d j"1 (in degree 0) and 1 by:

" µ i , B Bx j 1 , . . . , B Bx jr * 1 r`1 :" B r φ i Bx i 1 . . . Bx ir (4.40)
all other brackets of generators being equal to 0. Since the brackets of generators take values in O, and since an n-ary bracket where an element of O appears is zero, this is easily seen to be a Poisson 8-structure. The general formula is

␣ µ I 1 b O B xa 1 , . . . , µ In b O B xa n ( 1 n :" ÿ j " 1, . . . , n ij P Ij ϵ B n´1 φ i j Bx a 1 ¨¨¨x Bx a j ¨¨¨Bx an µ I 1 ¨¨¨µ i j I j ¨¨¨µ Inb O Bx a j , ( 4.41)
where µ J " µ j 1 . . . µ js for every list J " tj 1 , . . . , j s u, where ϵ is the Koszul sign, and where for a list J containing j, J j stands for the list J from which the element j is crossed out, as in Equation (4.31).

The O-module generated by µ i 1 ¨¨¨µ i k b O B xa , i.e. the complex (4.39) is easily seen to be stable under the brackets t¨¨¨u 1 k for all k ě 1, so that we can define on K b O DerpOq a sequence of brackets pℓ k " t¨¨¨u k q kě1 by letting them coincide with the previous brackets on the generators, i.e. t¨¨¨u n is given by Equation (4.41) for all n ě 1. The brackets are then extended by derivation, O-linearity or Leibniz identity with respect to the given anchor map, depending on the degree. In particular,

t¨¨¨u 1 k " t¨¨¨u k for k ě 2. For k " 1, t¨¨¨u 1 1 " t¨¨¨u 1 on ' iě2 K ´i b O DerpOq. For k " 2, we still have t¨¨¨u 1 2 " t¨¨¨u 2 on ' i,jě2 K ´i d K ´j b O DerpOq.
Let us verify that all required axioms are satisfied. For n " 2, Equation (4.41) specializes to:

ℓ 2 pµ i b O B xa , µ j b O B x b q " Bφ j Bx a µ i b O B x b ´Bφ i Bx b µ j b O B xa
which proves that the anchor map is a morphism when compared with the relation:

rφ i B xa , φ j B x b s " Bφ j Bx a φ i B x b ´Bφ i Bx b φ j B xa .
The higher Jacobi identities are checked on generators as follows:

1. When there are no degree ´1 generators, it follows from the higher Jacobi identities of the Poisson 8-structure (4.40) and the O-multilinearity of all Lie 8-algebroid brackets involved.

CHAPTER 5

Universal Lie 8-algebroids of affine varieties

In this chapter, we apply the results of Chapter 4 to answer some elementary but open questions that have to do with algebraic geometry, such as the interaction between the singularities of an affine variety and its Lie algebra of vector fields. Notice that the Theorem 2.1 of Chapter 4.2 only says that it is possible to associate a Lie 8algebroid structure to an affine variety by considering the Lie-Rinehart algebra made of its vector fields. But the construction can be extremely complicated, see e.g. Section 5.3. We only have an existence theorem. This leads to the natural question:

Question. How is the geometry of an affine variety related to its universal Lie 8-algebroid?

For instance, in view of the construction of Section 3.10. It is also relevant to ask about the effect of blow-ups on this construction. We will see in Example 4.4.10 that blow-ups may change a universal Lie 8-algebroid to a Lie algebroid. But this example does not tell us really how the higher brackets disappear under the effect of blow-ups. One of the important question we may ask is about the description of "the big theorem" of Hironaka [START_REF] János | Lectures on resolution of singularities / János Kollár[END_REF] in terms of the universal Lie 8-algebroids obtained at each step while resolving singularities. This question remains open, but there are several other problems about which we are able to make some progresses. Those are quite modest, but, at least, we want to have the question clarified. Application to "blowup-up" will be discusses in Section 7.2.

Background on affine varieties and some constructions

We recall definitions and some main properties of the notion of affine variety in order to fix notations. Our main references for this chapter are [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Eisenbud | Commutative algebra. With a view toward algebraic geometry[END_REF][START_REF] Lakshmibai | The Grassmannian Variety: Geometric and Representation-Theoretic Aspects[END_REF].

In this chapter we will sometimes see C d as the d-dimensional affine space which is commonly denoted by A d C , forget about its vector space structure, but here we will not make any notational distinctions. The latter is equipped with the Zariski topology that is, the topology whose closed subsets are the zero set of some ideal I Ď Crx 1 , . . . , x d s ": O, i.e. which are of the form ta " pa 1 , . . . , a d q P C d | f paq " 0, @f P Iu. One can check that these subsets indeed define a topology on C d

I W " tf P O | f pxq " 0, @ x P W u.
In general, we have I W ‰ I because the vanishing ideal of the affine variety W " tx 2 " 0u is the ideal I W " xxy ‰ xx 2 y. Notice that I W can be defined for any arbitrary subset

W Ă C d . It is easy to check that (a) for S Ď T Ď O, one has ZpSq Ě ZpT q, (b) for U Ď V Ď C d , one has I U Ě I V .
2. The ideal I W is larger than I. Hilbert's Nullstellensatz theorem (e.g. see Theorem 1.6 of [START_REF] Eisenbud | Commutative algebra. With a view toward algebraic geometry[END_REF]) claims that I W " ? I :" tf P O | f N P I, for some N P Nu.

3. We have, W " ZpI W q: if x P W , then by definition of I W , one has f pxq " 0 for all f P I W . Whence, W Ď ZpI W q. Conversely, it is clear that I Ď I W . This fact proves the other inclusion.

4. Also, by Noetheriality of the polynomial ring O, the ideal I W Ă O is generated by a finite number of generators. In the sequel, we shall define an affine variety W as the zero locus of an ideal generated by a finite set of polynomials φ 1 , . . . , φ r P O.

5. The Zariski closure V of a subset V Ď C d is equal to ZpI V q: by definition of I V , one has

V Ď ZpI V q. Conversely, V Ď V " ZpIq for some ideal I Ď O. By item 1.(b), I V Ď I V .
In particular, I Ď I V . By item 1.(a), this implies ZpI V q Ď ZpIq " V .

Definition 5.1.3. Let W Ď C d be an affine variety.

1. A function F : W Ñ C is said to be a polynomial if F pxq " f pxq, @x P W , for some element

f P O. The set CrW s of polynomial functions on W is, under the restriction map f P O Þ Ñ f | W , isomorphic the quotient O{I W ": O W , called the coordinate ring of W .
2. Elements of the Lie algebra of C-linear derivations, DerpO W q ": XpW q, of O W are called vector fields on W .

Remark 5.1.4. Notice that 1. the coordinate ring O W of an affine variety besides being a ring is also a vector space over C, hence it is a C-algebra. This algebra is generated by the images x1 , . . . , xd in O W through the projection map of the coordinate functions x 1 , . . . , x d P O.

For each a P W , the kernel of the evaluation map ev a : O W Ñ C, F Þ Ñ F paq, is the maximal ideal m a :" kerpev a q made of all polynomial functions on W that vanish at a.

2.

W " C d is an affine variety with I W " t0u, and O W " O.

3. W " tau Ď C d is an affine variety with I W " px 1 ´x1 paq, . . . , x d ´xd paqq the maximal ideal of a, and O W " C.

4. For an affine variety W Ă C d with corresponding ideal I W , we have

DerpO W q » tX P XpC d q | XrI W s Ă I W u I W XpC d q .
Lemma 5.1.5. For every affine variety W Ď C d , the ring O W is Noetherian.

Proof. Let I be an ideal of O W . Denote by p : O Ñ O{I W be the quotient map. Then p ´1pI q is also an ideal of O. By Noetheriality of O, p ´1pI q is finitely generated. In particular, I " ppp ´1pI qq is finitely generated. This shows that O W is Noetherian.

Germs

Here we mention the notion of local rings. We refer the reader to [Cha14, Hid89, Eis95] for more details.

Let W Ď C N be an affine variety and O W its coordinates ring. We recall for U Ď W an open subset, a function f : U ÝÑ C is said to be regular at a P U if there exists g, h P O W with hpaq ‰ 0 such that f " g h in a neighborhood of a, namely there exists an open set

V Ă U that contains a such that f | V " g h | V .
A function germ at a point a P W is an equivalence class pf q a of pairs pU, f q with a P U Ă W an open subset containing a, and f : U ÝÑ C is regular at a, under the relation equivalence: pU, f q " pV, gq if f | W " g| W on an open subset W Ď U XV . The set of equivalence classes of the above equivalence relation inherits naturally an associative C-algebra, that is called germs of regular functions at a and is denoted by O W,a . Also, a function germ pf q a at a P W has a well-defined value at a, given by the image of any representative pU, f q at a, namely pf q a paq :" f paq. Since the map

O W,a ÝÑ pO W q m W,a , pU, f q Þ Ñ f | U "
g h with g, h P O W and h does not vanish on U , is a bijection. One has, O W,a » pO W q m W,a [START_REF] Hartshorne | Algebraic Geometry[END_REF]. Here m W,a " tf P O W | f paq " 0u and pO W q m W,a is the localization w.r.t the complement of m W,a . It is important to notice that O W,a is a local ring. We denote the unique maximal ideal of O W,a again by m W,a . Also, It is worth it to notice that O W,a isomorphic to the quotient O a {I a of the local ring O a of C d at a by the ideal I a which is spanned by the ideal I W in O a .

The Zariski tangent space

Let a P W Ď C d a point of an affine variety W . There are several equivalent descriptions of the tangent space of W variety at the point a. Here we define it as pointwise derivations of the local ring O W,a , see [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF]Šafarevič Igor Rosticlavovič[END_REF] or Appendix B.2 of [START_REF] Laurent-Gengoux | Poisson structures. Grundlehren der mathematischen Wissenschaften A series of comprehensive studies in mathematics[END_REF], for more details on this topic.

A pointwise derivation of O W at a is a C-linear map

δ a : O W,a Ñ C
satisfying the following Leibniz identity, δ a pF Gq " δ a pF qGpaq `F paqδ a pGq. In particular, if a regular function f is constant in a neighborhood of a then its germ pf q a at a satisfies δ a ppf q a q " 0, since δ a p1 ¨1q " δ a p1q ¨1 `1 ¨δa p1q " 2δ a p1q δ a p1q " 0.

It is not hard to check that the set of all pointwise derivations of O W at a is a C-vector space.

Remark 5.1.6. Assume now that W " C d . Denote by pe 1 , . . . , e d q the canonical basis of C d . For every i P t1, . . . , du

pe i q a : O W,a Ñ C, pf q a Þ Ñ lim tÑ0 f pa `te i q ´f paq t ": Bf Bx i paq,
is a well-defined pointwise derivation of O at a. One can show that (see e. δ a ppx i q a q pe i q a , (5.1)

i.e.,

δ a pf q a " d ÿ i"1 Bf Bx i paq δ a px i q a .
(5.2) Hence, pointwise derivations pe i q a , i " 1, . . . , d form a basis for the vector space of pointwise derivation of O at a. Definition 5.1.7. The Zariski tangent space T a W of W Ď C d at a P W is the vector space of all pointwise derivations of O W at a.

Proposition 5.1.8. [R94] For a P W , one has T a W » ´mW,a {m 2 W,a ¯˚.

The vector space m W,a {m 2 W,a is called the cotangent space to W at a.

Remark 5.1.9. Notice that, 1. by Remark 5.1.6, we have

T a C d » C d .
2. the tangent space T a W of W Ď C d at a P W can be seen as pointwise derivations δ a of O at a of the form (5.1) such that δ a ppf q a q " 0 for all f P I W . From this point of view, one has

T a W » # pv 1 , . . . , v d q P C d ˇˇˇˇd ÿ i"1 v i Bf Bx i paq " 0, @f P I W + .

Singularities of an affine variety W

In this section we recall some definitions and some facts on singularities on affine varieties and fix some notations. We refer the reader to [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF]Šafarevič Igor Rosticlavovič[END_REF] for the full theory.

There are various equivalent ways to define the dimension of an affine variety W , we refer the reader to Page 4 of [START_REF] Hartshorne | Algebraic Geometry[END_REF] also to the Chapter 11 of [START_REF] Joe | Algebraic geometry: a first course/Joe Harris[END_REF] for more details. The dimension dim W of W is defined to be the maximal length d of the chains W 0 Ă W 1 Ă ¨¨¨Ă W d of distinct nonempty irreducible sub-varieties of W . Notice that a chain of irreducible sub-varieties corresponds to a chain of prime ideals in O W , by Noetheriality it must be of finite length.

Definition 5.1.10. A point a P W is said to be regular if dim T a W " dim W. Otherwise, we say that a is singular. The set of regular points of W is denoted by W reg , and the singular ones by W sing .

We say that W is regular if W sing " H. Proposition 5.1.11. [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF]Šafarevič Igor Rosticlavovič[END_REF]We have the following

1. For every a P W , dim T a W ě dim W 2. There is an open dense open subset of W such that the map a Þ Ñ dim T a W is constant. In particular,
• regular points of W form an open dense subset of W ,

• and singular points a (closed) proper sub-variety of W .

Remark 5.1.12. In particular, a P W is a singular point of W if only if dim T a W ą dim W . That is,

W sing " ta P W | dim T a W ą dim W u.

Local coordinates at a point

Let a P W Ď C d . We recall that (see e.g [R94]) that a family of elements t 1 , . The following proposition explains how an affine variety looks around a regular point (see [START_REF] Hauser | Blowups and Resolution[END_REF] Proposition 3.5, also [START_REF] Jong | Local Analytic Geometry: Basic Theory and Applications[END_REF]).

Proposition 5.1.15. Let W Ď C d be affine variety of codimension k, (i.e., k " d ´dim W ). Then, W is regular at a point a P W if and only if there exist local coordinates y 1 , . . . , y d of C d at a such that W is locally of the form y 1 " ¨¨¨" y k " 0.

Three main constructions

Consider an affine variety W which is given by an ideal I W Ă O, with O the algebra of polynomials in d-variables, and O W " O{I W the algebra of functions on W . There are three natural Lie-Rinehart algebras associated to W :

1. The O W -module XpW q of vector fields on W (i.e. derivations of O W ) is a Lie-Rinehart algebra over O W ; its anchor map is the identity.

2. The O-module X W pC d q of vector fields on C d tangent to W (i.e. derivations of O preserving I W ) is a Lie-Rinehart algebra with respect to the O-module structure; its anchor map is the inclusion X W pC d q ãÑ XpC d q.

3. The O-module I W XpC d q of vector fields on C d vanishing at every point of W (i.e. I W -valued derivations of O) is a Lie-Rinehart algebra with respect to the O-module structure; its anchor map is again the inclusion I W XpC d q ãÑ XpC d q.

These three Lie-Rinehart algebras are related:

1. There is an inclusion

I W XpC d q Ă X W pC d q
2. the restriction of X W pC d q to W coincides with XpW q. Let us justifies this. Every vector field on W extends to C d : to see that, let δ P XpW q, we have δpx i `IW q " f i `IW for some

f i P Crx 1 , . . . , x d s, i " 1, . . . , d.
We define the vector field r δ :"

d ÿ i"1 f i B Bx i
on C d . The vector field r δ restricts to δ on W , since for every f P Crx 1 , . . . , x n s, r δpf q `IW " δpf `IW q.

In particular, r δpI W q Ă I W .

Note that the Lie-Rinehart algebras X W pC d q, I W XpC d q Ă XpC d q are finitely generated as O-modules, since O is Noetherian (Proposition B.2.7). Whence, these Lie-Rinehart algebras are singular foliations on the complex manifold C d in the sense of Example 3.2.4.

Remark 5.1.16. What happens if we take a look at the evaluation map at some point a P C d ? Any vector field X "

d ÿ i"1 Xrx i s B Bx i P X W pC d q tangent to W induces a pointwise derivation of O W at a P W
as follows:

1. We extend X by localization at the maximal ideal m a to a derivation pXq P DerpO W,a q.

2. We define X| a ppf q a q :"

d ÿ i"1
Xrx i spaqpe i q a ppf q a q P T a W .

X| a is well-defined, since XrI W s Ă I W for all f P I W , in particular X| a ppf q a q " 0 for all f P I W .

In fact, we do not need to localize X to define X| a :" pXrx 1 spaq, . . . , Xrx

d spaqq P T a W ãÑ C d .
But, by developing the minors µ i paq's along their last columns, it takes the form,

µ i paq " ˘Bφ 1 Bx j paqC 1 ˘¨¨¨˘B φ r´1 Bx j paqC r´1 .
Notice that the determinants C 1 , . . . , C r´1 are the same for each H j . Thus,

d ÿ j"1 v j µ i paq " r´1 ÿ s"1 ˘˜d ÿ j"1 v j Bφ s Bx j paq ļooooooooomooooooooon "0 C s " 0.
Item 4. is obtained as follows: the local ring at a is by definition the localization O a of Crx 1 . . . , x d s with respect to the multiplicative set of all polynomials that do not vanish at a. By Proposition 5.1.15 a P W is a regular point if and only if there exists "local coordinates" y 1 , . . . , y d P O a such that W is locally of the form y 1 " ¨¨¨" y k " 0, i.e. the localization of I W is generated by these variables. Hence, the tangent space at m is the vector space, spant B By i |m , i ě k `1u. Therefore, for v P T a W the local vector field

X " dim W ÿ i"1 v i B By k`i
maps O a to O a , in particular it maps O to O a and we have XrI W s Ă pI W q ma . Therefore, for every, i P t1, . . . , du there exists a polynomial function g i that does not vanish at a such that g i Y rx i s P Crx 1 , . . . , x d s. Hence, the vector field X " g 1 ¨¨¨gr g 1 paq¨¨¨grpaq X is tangent to W satisfies Xpaq " v and XrI W s Ă I W . This concludes the proof.

Remark 5.1.18. We may not have equality in item 1. in Lemma 5.1.17. To see this, consider the cups,

W " tpx, yq | C 2 | x 3 ´y2 " 0u.
It is clear that the tangent space T 0 W of W at 0 P W is the whole space C 2 . But the vector fields in X W pC 2 q vanish at zero, since it is spanned as a Crx, ys-module by the Hamiltonian 2y B Bx `3x 2 B By and the weighted Euler vector field 2x B Bx `3y B By (see Proposition 5.3.1).

The following lemma shows that the vector fields that are tangent to W are also tangent to every strata of the stratification that consists of by taking the singular locus W sing of the singular locus of W then the singular locus pW sing q sing of the singular locus W sing and so on.... We obtain a sequence of inclusions of the form W Ą pW sing q sing looooomooooon ":W 1 Ą ppW sing q sing q sing looooooooomooooooooon ":W 2 Ą ¨¨¨.

(5.5) Lemma 5.1.19. We have the following inclusions

X W pC d q Ď X W 1 pC d q Ď ¨¨¨Ď X W i pC d q Ď ¨¨¨(5.6)
Proof. Let us prove that if X P XpC d q is such that XrI W s Ă I W then XrI W sing s Ă I W sing , where I W sing is the ideal of functions on the singular part of W . Since I W sing is obtained by considering the minors of order k " d ´dim W of k elements chosen into the generators φ 1 , . . . , φ r . That is, W sing is given the ideal A φ 1 , ¨¨¨φ r , P rφ i 1 , ¨¨¨, φ i k s, P P X k pC d q, for integers 1 ď i 1 ă ¨¨¨ă i k ď r E (5.7)

Let us explain why the vector fields that tangent to W are also tangent to its singular locus. For a vector field X P X W pC d q one has by Formula (4.27) that,

X rP rφ i 1 , ¨¨¨, φ i k ss " pL X P qrφ i 1 , ¨¨¨, φ i k s `k ÿ j"1 P rφ i 1 , . . . , Xrφ i j s, . . . , φ i k s.
(5.8)

Notice that pL X P qrφ i 1 , ¨¨¨, φ i k s P I sing since pL X P q P X k pC d q. On the other hand, for every j there exists polynomial functions f 1 , . . . , f r such that Xrφ i j s " ř r i"1 f l φ l . Since P is a multi-derivation, one has,

P rφ i 1 , . . . , Xrφ i j s, . . . , φ i k s " r ÿ l"1 φ l P rφ i 1 , . . . , f l , . . . , φ i k sr ÿ i"1 f l P rφ i 1 , . . . , φ l , . . . , φ i k s
It is now clear that the RHS of the equation (5.8) is in the ideal I sing . The proof goes by recursion.

Here is a direct consequence of Lemma 5.1.19.

Theorem 5.1.20. Every vector field X P XpW q is tangent to the stratification (5.5), i.e. X P XpW i q for each i ě 1.

The coming example shows that the inclusions (5.6) may be strict. This Example can also be found in the problem list [START_REF] Louis | Singular Foliations : Problems sessions[END_REF] The straight line x " y " 0 is a strata of the previous affine variety W . Any vector field tangent to W is tangent to this straight line. Let us show that it has to vanish at every point of this straight line. If not, its flow at time t would map a point p0, 0, z 0 q to a point p0, 0, z 1 q with z 1 ‰ z 0 . Its differential then induce a linear automorphism of the normal bundle of that straight line that has to preserve the straight lines x " 0, y " 0, x `y " 0. Since a linear endomorphism of C 2 preserving three straight lines has to be a multiple of the identity map, this differential cannot map the straight line x `z0 y to the straight line x `z1 y.

On their universal Lie 8-algebroids

Let W be an affine variety. The following is a direct consequence of the results of Chapter 4.

Proposition 5.1.22. Let W be an affine variety as above.

1. XpW q admits a universal Lie 8-algebroid made of free O W -modules of finite rank.

2. I W XpC d q and X W pC d q admit universal Lie 8-algebroids made of finitely many free O-modules of finite rank.

Proof. Classical theorems of commutative algebras allows equipping the three Lie-Rinehart algebras above with resolutions of a certain type:

1. Since O is a Noetherian regular ring, X W pC d q and I W XpC d q admit free resolutions by finitely generated O-modules. By Hilbert Syzygy Theorem B.2.8, those can be chosen to be of finite length.

Since

: i W U O pX W pC d qq ÝÑ U O W pXpW qq.
Notice that this morphism is in general not a Lie 8-quasi-isomorphism.

Universal Lie 8-algebroid of an affine variety

We give the following definition, Definition 5.2.1. A Lie 8-algebroid of an affine variety W is the homotopy class of Lie-8-algebroid associated to the Lie-Rinehart algebra XpW q over O W .

Example 5.2.2 (Vector fields on hyperelliptic curves). We follow the notations of [START_REF] Billig | Lie algebras of vector fields on smooth affine varieties[END_REF]. Let us consider on C 2 the hyperelliptic curve H given by the equation y 2 " 2hpxq, where h is a monic polynomial of odd degree 2ν [START_REF] Billig | Lie algebras of vector fields on smooth affine varieties[END_REF] for more details). A free resolution is described as follows: E ´1 is the O H -module, generated by two elements that we denote by τ, µ. The anchor is defined then by ρpτ q " X, ρpµq " Y.

Then we choose E ´2 to be the O H -module given by the generator η, and we set E ´i " 0 for i ě 3. The differential map ℓ 1 " d is chosen to be zero, except on degree ´2 where it is the O H -linear map d : E ´2 ÝÑ E ´1 given by dpηq " yτ ´dpxqµ.

Let us now describe a universal Lie 8-algebroid structure: The 2-ary bracket is defined on generators of E ´1 by tτ, µu 2 " h 1 pxq dpxq τ ´y d 1 pxq dpxq µ.

Then we extend this bracket to the whole space E ´1 by O H -linearity, skew-symmetric and Leibniz identity. Notice that tdη, τ u 2 " tdη, µu 2 " 0, thus, one can define the 2-bracket by tη, τ u 2 " tη, µu 2 " 0. We extend all brackets using Leibniz identity. All k-ary brackets are zero for k ě 3. The k-ary bracket ℓ k , k ě 1 and ρ restrict to the exact complex ¨¨¨/ / E ´3 ℓ 1 / / E ´2 ℓ 1 / / Sker a pρq ρ / / DerpO W q .

(5.9)

Let us check that ℓ 2 is well-defined: for all e, e 1 P Sker a pρq, ρpℓ 2 pe, e 1 qqrOs " rρpeq, ρpe 1 qspOq, (since ρ is morphism of brackets)

Ă m a , (by definition of the commutator r¨, ¨s).

The latter Lie 8-algebroid goes to quotient to a Lie 8-algebroid ˆp' iě2 E ´i m a E ´i q ' ker a pρq, l' , ρ ˙(5.10)

over O W {m a , where ker a pρq :" Skerapρq maSkerapρq . Since this quotient is the base field C, we obtain in fact a Lie 8-algebra.

Remark 5.2.3. In particular, if a P W is an isolated singular point, then E ´1 " Sker a pρq. In that case, U O W pXpW qq is a universal of the Lie-Rinehart algebra A a " tδ P XpW q | δrm a s Ă m a u " XpW q. By Section 4.4.1 again, (5.10) is a Lie 8-algebra on Tor O W pXpW q, Cq.

Lie 8-algebroids on minimal resolutions

The germ at a of the Lie-Rinehart algebra of vector fields on W is easily checked to coincide with the Lie-Rinehart algebra of derivations of O W,a .

Here is an immediate consequence of Proposition 4.4.4. Proposition 5.2.4. Let W be an affine variety. For every a P W , the germ at a of the universal Lie 8-algebroid of W is the universal Lie 8-algebroid of DerpO W,a q.

To describe this structure, let us start with the following Lemma. Lemma 5.2.5. Let a be a point of an affine variety W . The universal Lie 8-algebroid of DerpO W,a q can be constructed on a resolution ppE a ´iq iě1 , ℓ 1 , πq, with E a ´i free O W,a -modules of finite rank for all i ě 1, which is minimal in the sense that ℓ 1 pE ´i´1 q Ă m a E ´i for all i ě 1.

Proof. Since Noetherian property is stable by localization, the ring O W,a is a Noetherian local ring. Proposition 8.2 in [May] assures that O W,a b O W DerpO W q admits a free minimal resolution by free finitely generated O W,a -modules. Since O W,a is a local ring with maximal idea m a , we can assume that this resolution is minimal. In view of Theorem 4.2.1, there exists a Lie 8-algebroid structure over this resolution, and the latter is an universal of DerpO W,a q. By Theorem 4.2.1, a resolution of DerpO W,a q as in Lemma 5.2.5 comes equipped with a universal Lie 8-algebroid structure for DerpO W,a q. The quotient with respect to m a is a Lie 8-algebra of the isolated singular point a with trivial 1-ary bracket. Using Corollary 4.2.13 and its subsequent discussion, we can prove the next statement. Proposition 5.2.6. For any universal Lie 8-algebroid structure on a resolution of DerpO W,a q as in Lemma 5.2.5, the quotient with respect to the ideal m a is a representative of the Lie 8-algebra of the 5.3. SOME EXAMPLES OF UNIVERSAL LIE-ALGEBROIDS OVER AN AFFINE VARIETY 94 isolated singular point a, with trivial 1-ary bracket, on a graded vector space canonically isomorphic to Tor O W pX W , Cq (C being a O W -module through evaluation at a).

In particular, its 2-ary bracket is a graded Lie bracket on Tor O W pX W , Cq which does not depend on any choice made in the construction, and its 3-ary bracket is a Chevalley-Eilenberg cocycle whose class is also canonical.

Some examples of universal Lie-algebroids over an affine variety

Vector fields tangent to W : a codimension one example

The zero-set in W Ă V " C d of a weight homogeneous polynomial function φ P CrX 1 , ¨¨¨, X d s admitting only an isolated singularity at the origin is one of the simplest possible example of a nonsmooth affine variety W Ă V " C d . We give in this section a description of U O pXpW qq. Although our description is not complete, it will show how complex the universal Lie 8-algebroids may be, even for simple objects.

First, let us describe XpW q " DerpO W q. We denote by ω 1 , . . . , ω d the weights of the variables x 1 , . . . , x d and by |φ| the weighted degree of φ. Recall that, by definition, O W :" CrX 1 ,¨¨¨,X d s xφy .

Proposition 5.3.1. As a O W -module, XpW q :" DerpO W q is generated by the restrictions to W of the following vector fields in C d :

1. the weighted Euler vector field Ý Ñ E :"

ř d i"1 ω i x i B x i
2. the dpd ´1q{2 vector fields given by:

X ij :" Bφ Bx i B x j ´Bφ Bx j B x i with 1 ď i ă j ď d.
We start with a lemma. Recall that a homogenous function φ with an isolated singularity at 0 is a Koszul function (see Example 4.4.6), so that the Koszul complex, i.e. the complex of poly-vector fields on V " C d , equipped with ι φ :

¨¨¨ι φ ÝÑ X 2 pC d q ιφ ÝÑ X 1 pC d q ιφ ÝÑ O
has no cohomology except in degree 0. We denote it by pX ' , ι φ q.

Lemma 5.3.2. If P P X i`1 pC d q, Q P X i pC d q satisfy ι φ pP q " φQ, then there exists R P X i`2 pC d q such that P "

1 |φ| Ý Ñ E ^Q `ιφ pRq
Proof. This follows from the easily checked fact that ι φ pQq " 0, so that

ι φ ˆ1 |φ| Ý Ñ E ^Q˙" φQ.
This implies that

ι φ ˆP ´1 |φ| Ý Ñ E ^Q˙" 0
and the existence of R now follows from the exactness of the Koszul complex.

SOME EXAMPLES OF UNIVERSAL LIE-ALGEBROIDS OVER AN AFFINE VARIETY 95

Proof (of Proposition 5.3.1). Any X P XpW q is the restriction to W of a vector field X in V tangent to W , i.e. that satisfies Xrφs " f φ for some f P O. By Lemma 5.3.2 the vector field X can be written as X " f |φ| Ý Ñ E `ιφ pP q, for some bivector field P P X 2 pC d q. The restriction to W of the first (resp the second) term is tangent to W and is of the first (resp. second) type described in Proposition 5.3.1. This completes the proof.

We now intend to construct a free resolution of XpW q in the category of O W . Let us denote by ^Ý Ñ E : X i pV q Ñ X i`1 pV q the map ω Þ Ñ ω ^Ý Ñ E . The map ^Ý Ñ E is a chain map from the restriction i W X ' to W of the Koszul complex, shifted by one to itself. More precisely, ^Ý Ñ E in the diagram below is a chain map:

¨¨¨ι φ / / i W X 2 pV q ιφ / / ^Ý Ñ E i W XpV q iφ / / ^Ý Ñ E O W ^Ý Ñ E ¨¨¨ι φ / / i W X 3 pV q ιφ / / i W X 2 pV q ιφ / / XpW q Lemma 5.3.3. The chain map ¨^Ý Ñ E is a quasi-isomorphism.
Proof. It is quite clear for i ě 1 since the complex pE 1 r1s, ι φ q has no cohomology. Let us check bijectivity in the last column. Surjectivity follows from Proposition 5.3.1. To check injectivity consider a function f P O such that

f Ý Ñ E " ι φ pπq `φQ, (5.11)
for some bivector field π P X 2 pV q and vector field Q P XpV q. Upon taking ι φ to both sides, Equation (5.11) implies, f |φ|φ " φι φ pQq.

Hence, we obtain f " 1 |φ| ι φ pQq.

The lemma 5.3.3 implies that the mapping cone construction provides a free resolution of the O W -module of vector fields of XpW q on W namely:

˜E´i " i W X i´1 pV q ' i W X i`1 pV q, i ě 1; D " ˜´ι φ 0 ´^Ý Ñ E ι φ ¸, π ¸.
(5.12)

In degree ´1 we read E ´1 " O W ' i W X 2 pV q and π is defined on the generators of E ´1 as πp1 ' 0q :"

Ý Ñ E and πp0 ' B x i ^Bx j q :" Bφ Bx i B x j ´Bφ Bx j B x i , for all, 1 ď i ă j ď d.
Let us now describe some of the k-ary brackets:

Proposition 5.3.4. The Lie-Rinehart algebra XpW q of vector fields on W admits a universal Lie 8-algebroid whose 1. underlying complex is (5.12) (which is a free resolution of XpW q), 2. 1-ary bracket is given by the resolution (5.12).

3. anchor map ρ :" π.

SOME EXAMPLES OF UNIVERSAL LIE-ALGEBROIDS OVER AN AFFINE VARIETY 96

4. In the case where the generator 1 ' 0 appears together with a bivector field on W , one can define the 2-ary bracket on elements of degree ´1 which makes the anchor map a morphism as follows, t1 ' 0, 0 ' B x i ^Bx j u 2 :" p|φ| ´2q0 ' B x i ^Bx j .

5. The k-ary brackets can be chosen to be the same as in the structure given by Proposition 4.4.17 on i W X i`1 pV q for i ě 1.

6. the 3-ary bracket can be chosen to be zero when evaluated at 1 ' 0 and with two other elements of i W X 2 pV q.

Corollary 5.3.5. XpW q does not come from a Lie algebroid of rank 1 `dpd`1q 2 .

Remark 5.3.6. In general, it is hard to compute the generators of XpW q. But, there is a case where we know all generators: it is when W is a complete intersection with isolated singularity at zero such that I W is generated by weight homogeneous polynomials φ 1 , . . . , φ r . In that case, X W pC d q is generated by I W XpC d q, the Euler vector field, and the Hamiltonian vector fields (see [START_REF] Hauser | Affine varieties and lie algebras of vector fields[END_REF][START_REF] Siebert | Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0[END_REF]).

The computation of the Lie 8-algebroid remains complicated.

Conclusion:

As a particular case of Section 4, we notice that a Lie 8-algebroid can be associated to any affine variety. Explicit computations are difficult. Some applications to Mohsen's resolution of singularities will be given in Section 7.2 There are still many open questions on the geometric meaning of this Lie 8-algebroid structure. Although, we have not answered many questions , we state concepts, lemmas, and counterexamples that we hope to be able to use in the future.

CHAPTER 6

Universal Q-manifolds of a singular foliation

The aim of this chapter is to lay the ground for the subsequent results. More precisely, to explain how Theorem 4.2.1 extends the results on singular foliations of Sylvain Lavau's PhD [START_REF] Lavau | Lie 8-algebroids and singular foliations[END_REF] followed by a referred version by C. Laurent-Gengoux, S. Lavau and T. Strobl in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]. The results of Section Chapter 4 extend the latter for arbitrary Lie-Rinehart algebras and also to the infinite case, and it still holds even when we do not have a geometric resolution. We introduce the notion of longitudinal vector fields on a N Q-manifold and prove a new result on their cohomology. This chapter is taken from the textbook [START_REF] Laurent-Gengoux | Geometry of singular foliations: a draft of an introduction[END_REF] in which I am co-author. We refer the reader to [Vor10, BP13, LMP20, LLS20] for more details on this topic.

Throughout of this chapter M is a smooth, real analytic or complex manifold and K P tR, Cu. We denote by O the sheaf of functions on M .

Q-manifolds

Let us first define N-graded manifolds.

Graded manifolds

In words, as the name suggests graded manifolds are for graded vector spaces what manifolds are for vector spaces, in the sense that roughly speaking manifolds look locally like R n and graded manifolds are locally like R n ˆV for some graded vector space V . In this chapter we introduce the notion of graded manifolds, their vector fields, their morphisms, etc. Definition 6.1.1. A (positively) graded manifold over the base manifold M is a sheaf for some graded vector space E " ' 8

E : U Þ Ñ EpUq
i"1 E ´i. Sections of the sheaf E are called functions on E. It is convenient to denote a graded manifold as a pair pM, Eq where E is implicit. Remark 6.1.2. A function ξ P E j is a formal sum ξ " `8 ÿ i"0 ξ piq (6.1) with ξ piq P E an element of polynomial-degree i and degree j. For degree reasons, the sum must be finite.

Local coordinates of a graded manifold

Recall that for U Ă M an open set, one has pE i q U " ÝÑ U ˆKrkpE i q for every i ě 1. Hence, the graded coordinates on the graded manifold pM, Eq is the data made of:

In degree 0: a system of coordinates px 1 , . . . , x n q of M on U In degree i ě 1: a local trivialization pξ 1 i , . . . , ξ

rkpE i q i q of E i on U.
That is, a system of graded coordinates of pM, Eq on U is px 1 , . . . , x n , ξ 1 1 , . . . , ξ rkpE 1 q 1 , . . . , ξ 1 i , . . . , ξ

rkpE i q i , . . . q.
Elements of EpUq are "polynomials" in tpξ j i q j"1,...,rkpE ˚iq , i ě 1u with coefficients in OpUq.

Example 6.1.3. The sheaf of differential forms pM, E " ΩpM qq on a manifold M is a graded manifold since for every point m P M , it takes the form O U b K ^'T mM where U is an open neighborhood of m. Exterior forms can be seen as sections on the graded vector bundle E ´1 " T M .

Example 6.1.4. Let k be a positive integer. A finite dimensional vector space E and its dual E ˚can be seen as graded vector bundles of respective degree ´k and k over a point. E is a graded manifold over M " tptu, with functions isomorphic to ^E˚f or k odd and SpE ˚q for k even.

Definition 6.1.5. A morphism of graded manifolds between the two graded manifolds pM, Eq and pM 1 , E 1 q with respective base manifolds M and M 1 is a pair made of a smooth or real analytic or holomorphic map ϕ : M ÝÑ M 1 called the base map and a sheaf morphism over it, i.e. a family of graded algebra morphisms:

E 1 pU 1 q Ñ Epϕ ´1pU 1 qq,
compatible with the restriction maps, such that Φpf αq " ϕ ˚pf qΦpαq. (6.2) for all f P O 1 U 1 and α P E 1 pU 1 q.

A homotopy between two morphisms of graded manifolds Φ, Ψ : pM, Eq ÝÑ pN, E 1 q is a morphism of graded manifold pM, Eq ˆpr0, 1s, Ωpr0, 1sqq ÝÑ pM 1 , E 1 q whose restrictions to t " 0 resp. t " 1 coincide with Φ and Ψ respectively.

Vector fields on graded manifolds

Vector fields on manifolds are derivations of its algebra of functions. For a graded manifold, the analog of functions are the sheaf of sections ΓpS ' pE ˚qq. Since it is not commutative but graded commutative, one has to consider graded derivations. A graded derivation of degree k of E is the data, for every

U Ă M of a linear map Q : E ' pUq ÝÑ E '`k pUq,
compatible with all restriction maps, that increases the degree by `k and satisfies:

QrF Gs " QrF sG `p´1q ki F QrGs for every F P E i pUq, G P EpUq. Since we think geometrically, we say "vector fields of degree k" instead of graded derivation. Definition 6.1.6. Let pM, Eq be a graded manifold. For U Ă M and k P Z let X k pEqpUq :" Der k pEpUqq be the EpUq-module of derivation of degree k on EpUq. The correspondence U Þ ÝÑ X ' pEqpUq is a sheaf of E-module. Its sections are called vector fields on E.

Let us list some important facts on vector fields on E:

1. the E-module X ' pEq :" ' kPZ X k pEq of vector fields on E is naturally graded. The E-module X ' pEq of vector fields on E is a graded Lie subalgebra of the graded Lie algebra Hom K pE, Eq whose graded Lie bracket is the graded commutator. Precisely, the graded Lie bracket rP, Qs " P ˝Q ´p´1q kl Q ˝P (6.3) of two vector fields P, Q of degree k, l respectively is a vector field of degree k `l. It is easily checked that the bracket (6.3) fulfills (a) rP, Qs " ´p´1q jk rQ, P s, (graded skew-symmetry) (b) p´1q jl rP, rQ, Rss `p´1q jk rQ, rR, P ss `p´1q kl rR, rP, Qss " 0, (graded Jacobi identity) (c) rP, f Qs " P rf sQ `f rP, Qs, (Leibniz identity) for f P O and P, Q, R are vector fields of degree j, k and l respectively.

2. Their description in local coordinates: notice that any homogeneous element e P ΓpE ´kq corresponds to a vertical1 vector field ι e P X ´kpEq (i.e. it is O-linear) of degree ´k defined by contraction with e ι e pξq :" xξ, ey, ξ P ΓpE ˚q (6.4) and we extend by O-linear derivation, where x¨, ¨y is the dual pairing between E ˚and E. Notice that ι e is by construction of polynomial-degree ´1.

Let pU, x 1 , . . . , x n q be a coordinate chart of M and pξ j i q j"1,...,rkpE i q with i ě 1 be a homogeneous local trivialization of E ˚i, it should be understood that ξ j i is the j-th elements of the local frame in ΓpE ˚iq. Let pe j i q j"1,...,rkpE ´iq , i ě 1 be the dual basis of pξ j i q j"1,...,rkpE ˚iq , i ě 1, then for every pair i, j, ι e j i " B Bξ j i is the partial derivative with respect to ξ j i P ΓpE ˚iq. By choosing a T M -connection on E, it is easy to check that for any k P Z the family

ˆξj 1 i 1 d ¨¨¨d ξ j l i l B Bx j ˙l ě 0 i 1 ¨¨¨i l " k j 1 , . . . , j l j " 1, . . . , n Y ˜ξj 1 i 1 d ¨¨¨d ξ j l i l B Bξ j i ¸l ě 0 i 1 ¨¨¨i l ´i " k j 1 , .
. . , j l i ě 1, j " 1, . . . , rkpE ´iq form a basis for X k pEqpUq up to permutations of the ξ j 1 i 1 d ¨¨¨d ξ j l i l 's. Here we adopt the convention i 0 " j 0 " 0 and ξ 0 " 1 P ΓpS 0 pE ˚qq » O. Whence, any vector field Q P X k pEqpU q admits coordinates decomposition as follows

Q " ÿ l ě 0 i 1 ¨¨¨i l " k j 1 , . . . , j l j " 1, . . . , n 1 l! j Q j 1 ¨¨¨j l i 1 ¨¨¨i l ξ j 1 i 1 d ¨¨¨d ξ j l i l B Bx j `ÿ i ě 1, l ě 0 i 1 ¨¨¨i l ´i " k j 1 , . . . , j l j " 1, . . . , rkpE ´iq 1 l! ij Q j 1 ¨¨¨j l i 1 ¨¨¨i l ξ j 1 i 1 d ¨¨¨d ξ j l i l B Bξ j i .
for some functions Q j 1 ¨¨¨j l i 1 ¨¨¨i l P O. These functions can be chosen in a unique manner to satisfy, e.g. ij Q j σp1q ¨¨¨j σplq i σp1q ¨¨¨i σplq " ϵpσqQ j 1 ¨¨¨j l i 1 ¨¨¨i l for any permutation σ of t1, . . . , lu.

For example, if Q is of degree `1, then it can be written in these notations as

Q " ÿ 1 ď u ď rkpE ´1 q j " 1, . . . , n j Q u 1 ξ u 1 B Bx j `ÿ i ě 1, " l ě 0 i 1 ¨¨¨i l ´i " 1 j 1 , . . . , j l j " 1, . . . , n 1 l! ij Q j 1 ¨¨¨j l i 1 ¨¨¨i l ξ j 1 i 1 d ¨¨¨d ξ j l i l B Bξ j i .
This following lemma says that vector fields of polynomial-degree ´1 are all the types (6.4).

Lemma 6.1.7. Let pM, Eq be a graded manifold. For i ě 1, a vector field P P X ´ipEq of polynomialdegree ´1 and of degree ´i, is of the form ι e for some section e P ΓpE ´iq.

Proof. Note that a vector field P P X ´ipEq of polynomial-degree ´1 is vertical, i.e. P pf q " 0, since functions of M are of polynomial-degree zero. Hence, in local coordinates pU, x 1 , . . . , x n q M and pξ j i q j"1,...,rkpE ˚iq with be a homogeneous local trivialization of E ˚i and pe j i q j"1,...,rkpE ´iq be the dual basis of pξ j i q j"1,...,rkpE ˚iq : a polynomial-degree ´1 vector field P P X ´ipEq of degree ´i is forced to be of the form

P | U " rkpE ´iq ÿ j"1 f j pxq B Bξ j i (6.5)
with f j P C 8 pU q. Now, choose a local section e U P Γ U pE ´iq of the form, e U :"

ÿ jě1
f j e j i . (6.6)

It is then clear that ι e U " P | U . Since local sections of E ´i given on different coordinates chart domains U a and U b coincide on U a X U b and then lift to a global section of E ´i. Thus, ι e " P for some e P ΓpE ´iq.

N Q-manifolds

Definition 6.1.8. A dg-manifold or N Q-manifold is a positively graded manifold pM, Eq endowed with a degree `1 homological vector field Q on E, i.e., Q P X 1 pEq is such that Q 2 " 0.

They shall be denoted as a triple pM, E, Qq.

Example 6.1.9. Given a finite dimension Lie algebra pg, r¨, ¨sq of dimension d. We assume that g is concentrated in degree ´1. It is clear that pM " tptu, E " ^'g ˚q is a graded manifold over M " tptu. This graded manifold carries a dg-manifold structure. Precisely, we define the corresponding homological vector field as follows: fix a basis pe i q i"1,...,n of g and let these global coordinate functions pξ i q i"1,...,n on g be its dual. We have re i , e j s "

n ÿ l"1 λ l ij e l
for some coefficients λ l ij P K. One can check that the degree `1 vector field

Q " 1 2 n ÿ i,j,l"1 λ l ij ξ i ^ξj B Bξ l
corresponds to the Chevalley-Eilenberg differential pd CE , ^'g ˚q. Therefore, Q 2 " 0. Notice that Q 2 " 0 is equivalent to the Jacobi identity for r¨, ¨s.

Example 6.1.10. Given a differential graded vector bundle ppE ´iq iě1 , dq over M . There is a natural dg manifold given by its sheaf of sections pM, E " ΓpSpE ˚qq. In particular, the deferential map d : E ÝÑ E is dualized as a degree `1 map S 1 pE ˚q ÝÑ S 1 pE ˚q that we extend to a C 8 pM q-linear derivation on E squared to zero.

Example 6.1.11. Let E " T r1sM be the shifted bundle of M . It induces a graded manifold structure pM, E " ΩpM qq over M . This graded manifold carries a dg-manifold structure Q that corresponds to the de Rham differential on ΩpM q. In terms of coordinates, the homological vector field Q reads

n ÿ i"1 dx i B Bx i .
Let us introduce some vocabularies that will need to use. Definition 6.1.12. Let pM, E 1 , Q 1 q and pM, E, Qq be two N Q-manifolds.

1. A linear map Φ : E ÝÑ E 1 is said to be of polynomial-degree/degree j P Z provided that, for all function α P E of polynomial-degree/degree i, Φpαq is of polynomial-degree/degree i `j. Any map Φ : E ÝÑ E 1 of degree i decomposes w.r.t the polynomial-degree as follows:

Φ " ÿ rPZ Φ prq with Φ prq : E ÝÑ E 1 a map of polynomial-degree r.

Remark 6.1.13. When Φ : E ÝÑ E 1 is a graded morphism of algebras, necessarily one has Φ prq " 0 for all r ă 0. Furthermore, for all n, r P N and all ξ 1 , . . . , ξ k P ΓpV q one has:

Φ prq pξ 1 d ¨¨¨d ξ n q " ÿ i 1 `¨¨¨`in"r
Φ pi 1 q pξ 1 q d ¨¨¨d Φ pinq pξ n q. (6.7)

Obviously, in this case Φ is determined uniquely by the image of ΓpV q. Definition 6.1.14. Let pM, E, Qq and pM, E 1 , Q 1 q be two N Q-manifolds over M with sheaves of functions E and E 1 respectively. A morphism of N Q-manifold over M from pM, E 1 , Q 1 q to pM, E, Qq is a morphism of graded manifolds Φ : E ÝÑ E 1 (of degree 0) over the identity map which intertwines Q and Q 1 , i.e., Φ ˝Q " Q 1 ˝Φ. (6.8) Remark 6.1.15. It is important to notice that

• morphisms of N Q-manifolds over M are by definition O-linear, since they are defined over the identity map

• the component Φ prq of polynomial-degree r ě 0 of any O-linear map Φ : E ÝÑ E 1 maps ΓpE ˚q to ΓpS r`1 pE 1˚q q. By O-linearity, it gives rise to a section ϕ r P ΓpS r`1 pE 1˚q b Eq. Therefore, one has Φ prq pξq " xϕ r , ξy (6.9) for all ξ P ΓpE ˚q. It follows that Φ is entirely determined by the collection `ϕr P ΓpS k`1 pE 1˚q b Eq ˘rě0 when Φ is an algebra morphism or a Ξ-derivation for some map Ξ : E ÝÑ E 1 . In such case, for r ě 0, ϕ r P ΓpS r`1 pE 1˚q b Eq is then called the r-th Taylor coefficient of Φ. We also call the 0-th Taylor coefficient ϕ 0 : E 1 Ñ E the linear part of Φ. The latter is a chain map

¨¨¨/ / E 1 ´3 ϕ 0 d 1p3q / / E 1 ´2 ϕ 0 d 1p2q / / E 1 ´1 ϕ 0 ρ 1 / / T M id ¨¨¨/ / E ´3 d p3q / / E ´2 d p2q / / E ´1 ρ / / T M (6.10)

N Q-manifolds -Lie 8-algebroids

We have seen in Section 2.2 that Lie 8-algebroids (possibly infinite dimension) over O are one-to-one with co-differentials of the graded symmetric algebra, which are compatible with the action of O. This correspondence provides a simple characterization of Lie 8-algebroids over an arbitrary commutative unital algebra O. In the finite dimensional case, we can work without co-differentials by using T.

Voronov's higher derived brackets construction [START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF]. Roughly speaking, it is shown in [START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF] that (finite dimensional) Lie 8-algebroids as in Remark 2.1.13 are the same as N Q-manifolds. However, it is important to note the latter correspondence fails in infinite dimension case, since the identification Γ pS ' pE ˚qq » Γ pS ' pEq ˚q fails. We will not explain entirely the construction here, but we refer the reader to [START_REF] Voronov | Higher derived brackets for arbitrary derivations[END_REF][START_REF] Voronov | Higher derived brackets and homotopy algebras[END_REF][START_REF] Voronov | Q-manifolds and higher analogs of Lie algebroids[END_REF] also to [START_REF] Bonavolontà | On the category of Lie n-algebroids[END_REF][START_REF] Jotz | Modules and representations up to homotopy of Lie n-algebroids[END_REF] for more details on the construction.

The following statement is similar to our Proposition 2.2.4 the difference lies in the fact that ours remains valid in infinite dimension. Proposition 6.1.16 (T. Voronov). Let pM, Eq be a graded manifold of finite dimension in each degree.

There is a one-to-one correspondence between:

1. (finitely generated) negatively graded Lie 8-algebroids pE, pℓ k q kě1 , ρq over M , and 2. homological vector fields Q P X `1pEq.

The relation stated in Theorem 6.1.16 can be enlightened in terms of the k-ary brackets pℓ k q as follows:

r¨¨¨rrQ, ι e 1 s , ι e 2 s , . . . , ι e k s p´1q " ι ℓ k pe 1 ,...,e k q , for all e 1 , . . . , e k P ΓpEq (6.11) ρpeqrf s " rQ, ι e s p0q pf q, for all f P O, e P ΓpE ´1q (6.12)

In particular

Universal Q-manifolds

This section can be understood as a consequence of the main results of the first part of the thesis. We recover the result on universal Q-manifolds of a singular foliation [LLS20, Lav17], whose existence was proved under the condition that geometric resolutions exist. We recall that the existence of such a resolution is not guaranteed, unlike the case of resolution by free modules. We refer the reader to Appendix B and B.3 for more details on resolutions of modules and geometric resolutions of singular foliations.

Theorem Moreover, the homotopy equivalence between them is unique up to homotopy.

Proof. Apply Corollary 4.2.5.

The complex defined by ad Q p0q

Let F be a singular foliation on M that admits a universal algebroid pE, Qq. For a fixed k P N 0 Y t´1u consider the bicomplex defined on Hom O. The total differential is given by the formula,

B ´Ψpkq ¯" B h ´Ψpkq ¯´p´1q |Ψ pkq | B v ´Ψpkq ¯for all Ψ pkq P Hom ' O ´ΓpE ˚q, Γ ´Sk`1 K pE ˚q¯¯.
The following diagram pictures the idea of the bicomplex. The total degree is given by the antidiagonals lines.

. . . . . . . . .

Ò Ò Ò Ñ Hom O ˆΓpE ˚2q, Γ ´Sk`1 K pE ˚q¯k `3˙B h Ñ Hom O ˆΓpE ˚1q, Γ ´Sk`1 K pE ˚q¯k `3˙¨˝ρ Ñ Hom O ´F˚, Γ ´Sk`1 K pE ˚qk`3 ¯¯Ñ 0 B v Ò B v Ò D v Ò Ñ Hom O ˆΓpE ˚2q, Γ ´Sk`1 K pE ˚q¯k `2˙B h Ñ Hom O ˆΓpE ˚1q, Γ ´Sk`1 K pE ˚q¯k `2˙¨˝ρ Ñ Hom O ´F˚, Γ ´Sk`1 K pE ˚qk`2 ¯¯Ñ 0 B v Ò B v Ò B v Ò Ñ Hom O ˆΓpE ˚2q, Γ ´Sk`1 K pE ˚q¯k `1˙B h Ñ Hom O ˆΓpE ˚1q, Γ ´Sk`1 K pE ˚q¯k `1˙¨˝ρ Ñ Hom O ˆF˚, Γ ´Sk`1 K pE ˚q¯k `1˙Ñ 0 Ò Ò Ò 0 0 0
(6.16) There is no cohomology for the total complex which is governed by B " r ¨, Q p0q s since the lines are exact (see Proposition B.1.16). When k " ´1 the bicomplex (6.16) is just the line complex:

¨¨¨B h ÝÑ Hom O `ΓpE ˚2q, O ˘Bh ÝÑ Hom O `ΓpE ˚1q, O ˘Bh ÝÑ Hom O pF ˚, Oq ÝÑ 0 (6.17)
which is also exact.

A result on longitudinal vector fields and examples

In this section, we study the cohomology of longitudinal vector fields, which will help in proving the main results stated in the beginning of Chapter 8.2.

Let F be a singular foliation over M . Definition 6.2.4. Let E be a splitted graded manifold over M with sheaf of function E. A vector field L P XpEq is said to be a longitudinal vector field for F if there exists vector fields X 1 , . . . , X k P F and functions Θ 1 , . . . ,

Θ k P E such that Lpf q " k ÿ i"1 X i rf sΘ i , @f P O. (6.18)
2. Any connection on ΓpE ˚q induces a vector field of degree zero r ∇ X P XpEq by setting for f P O, r ∇ X pf q :" Xrf s. Once a connection is chosen, we have for all k P Z

X k pEq » à jě1 E k`j b O ΓpE ´j q'E k b O XpM q » ' jě1
ΓpSpE ˚qk`j bE ´j q'ΓpSpE ˚qk bT M q. (6.21) Thus, one can realize a vector field P P X k pEq as a sequence P " pp 0 , p 1 , . . .q, where p 0 P ΓpSpE ˚qk b T M q and p i P ΓpSpE ˚qk`i b E ´iq for i ě 1 are called components of P . In the diagram (6.23), P " pp 0 , p 1 , . . .q is represented as an element of the anti-diagonal and p i is on column i. We say that P is of depth n P N if p i " 0 for all i ă n. In particular, vector fields of depth greater or equal to 1 are vertical. Under the decomposition (6.21), the differential map ad Q takes the form

D " D h `ÿ sě0 D vs (6.22)
with D 2 " 0. Here D h " idbd or idbρ, and D vs : ΓpSpE ˚qk bE ´iq Ñ ΓpSpE ˚qk`s`1 b E ´i´s q for i ě 0, s ě 0, we shall denote E 0 :" T M . We denote the latter complex by pX, Dq. They can be represented as anti-diagonal lines in the following commutative diagram, whose lines are complexes of O-modules . . . . . . . . .

¨¨¨/ / ΓpSpE ˚qk`2 b E ´2q O O idbd / / ΓpSpE ˚qk`2 b E ´1q O O idbρ / / ΓpSpE ˚qk`2 b T M q O O ¨¨¨/ / ΓpSpE ˚qk`1 b E ´2q _ _ . . . Qbid `¨¨Ö O idbd / / ΓpSpE ˚qk`1 b E ´1q ^.. . Qbid `¨¨Ö O idbρ / / ΓpSpE ˚qk`1 b T M q f f . . . Qbid `¨¨Ö O ¨¨¨/ / ΓpSpE ˚qk b E ´2q _ _ . . . Qbid `¨¨Ö O idbd / / ΓpSpE ˚qk b E ´1q e e g g . . . Qbid `¨¨Ö O idbρ / / ΓpSpE ˚qk b T M q d d g g . . . Qbid `¨¨Ö O . . . O O . . . O O . . . O O column 2 column 1 column 0 (6.23)
Under this correspondence, we understand longitudinal vector fields as the following.

Lemma 6.2.9. A graded vector field P " pp 0 , p 1 , . . .q P X is longitudinal if

p 0 P E b O F.
The following theorem is crucial for Chapter 8.

Theorem 6.2.10. Let pE, Qq be a universal Q-manifold of F.

Longitudinal vector fields form an acyclic complex.

More precisely, any longitudinal vector field on E which is a ad Q -cocycle is the image through ad Q of some vertical vector field on E.

2. More generally, if a vector field on E of depth n is a ad Q -cocycle, then it is the image through ad Q of some vector field on E of depth n `1.

Proof. Since pE, Qq is an universal Q-manifold of F, lines in (6.23) are exact. It is now a diagram chasing phenomena. Let P " pp 0 , p 1 , . . . , q P X be a longitudinal element which is a D-cocycle.

By longitudinality there exists an element b 1 P ΓpSpE ˚q b E ´1q such that pid b ρqpb 1 q " p 0 . Set P 1 " p0, b 1 , 0, . . .q, that is we extend b 1 by zero on ΓpSpE ˚q b E ď´2 q and ΓpSpE ˚q b T M q. It is clear that P ´DpP 1 q " p0, p 1 1 , p 1 2 , . . .q is also a D-cocycle. In particular, we have D h pp 1 1 q " 0 by exactness there exists b 2 P ΓpSpE ˚q b E ´2q such that D h pb 2 q " p 1 1 . As before put P 2 " p0, 0, b 2 , 0, . . .q. Similarly, P ´DpP 1 q ´DpP 2 q " p0, 0, p 2 2 , p 2 3 , . . .q is a D-cocycle. By recursion, we end up to construct P 1 , P 2 , . . . that satisfy P ´DpP 1 q ´DpP 2 q `¨¨¨" 0, that is, there exists an element B " p0, b 1 , b 2 , . . .q P X such that DpBq " P . This proves item 1.

To prove item 2 it suffices to cross out in the diagram (6.23) the columns number 0, . . . , n ´1, which does not break exactness. The proof now follows as for item 1.

In particular, we deduce from Theorem 6.2.10 the following exact subcomplex. Corollary 6.2.11. Let pE, Qq be a universal Q-manifold of F. The subcomplex V Q of pXpEq, ad Q q made of vertical vector fields P P XpEq that satisfy P ˝Qpf q " 0 for all f P O is exact.

Proof. Let P P XpEq be a vertical vector field which is an ad Q -cocycle (note that we have automatically P ˝Qpf q " 0 for all f P O). By Theorem 6.2.10 there exists a vertical vector field r P P XpEq such that rQ, r P s " P . Moreover, r P P V Q , since for all f P O, 0 " rQ, r P spf q " p´1q | r P | r P ˝Qpf q.

The following remark will be used especially for the proof of Theorem 8.3.1 of Chapter 8.

Remark 6.2.12. For a cocycle P P V Q of degree 0 one has P p´1q " 0 (for degree reason). By Corollary 6.2.11, P is the image by ad Q of an element r P P V Q i.e. such that rQ, r P s " P . Also, one can choose r P p´1q " 0: we have rQ, r P s p´1q " rQ p0q , r P p´1q s " P p´1q " 0.

By exactness of ad Q p0q (see Section 6.2.1), we have P p´1q " rQ p0q , ϑs for some O-linear map ϑ P HompΓpE ˚q, ΓpS 0 pE ˚qqq of degree ´2. Put s P :" r P ´rQ, ϑs, where ϑ is extended to a derivation of polynomial-degree ´1. Clearly, rQ, s P s " P and s P p´1q " r P p´1q ´rQ, ϑs p´1q " r P p´1q ´rQ p0q , ϑs " 0.

Therefore, P " ad Q p s P q with s P p´1q " 0.

Here is direct consequence of Theorem 6.2.10 and Proposition B.1.7.

Corollary 6.2.13. Let F a singular foliation on M . Let pE, Qq be a universal Q-manifold of F. Then,

H ' pXpEq, ad Q q » H ' ˆXpEq LpEq , ad Q ˙. (6.24)
where ad Q is induced by ad Q on the quotient space.

Since all vertical vector fields are longitudinal, there is an isomorphism of O-modules

XpEq LpEq » ΓpS ' pE ˚qq b O ˆXpM q F ˙. (6.25)
Let us describe the differential map ad Q using this isomorphism: let pU, x 1 , . . . , x n q be local coordinates on M and take local trivialisations ξ 1 , ξ 2 , . . . of graded sections in ΓpE ˚q, also let ξ 1 , ξ 2 , . . . the corresponding dual sections in ΓpEq. Notice that locally, any P P XpEq LpEq is represented by a vector field of the form ÿ

j, i 1 ,...,i k f j i 1 ,...,i k pxqξ i 1 d ¨¨¨d ξ i k b B Bx j , f j i 1 ,...,i k P C 8 pUq
since the vector fields in B Bθ i are vertical vector fields. That is to say, P can be represented by an element of the form P " Θ b X with Θ P Γ U pS ' pE ˚qq and X P XpUq, using the decomposition (6.25). Since

rQ, P s " QrΘsX `p´1q |Θ| Θ ¨rQ, Xs " QrΘsX `p´1q |Θ| Θ d ÿ k, |ξ k |"1 ξ k « ÿ j Q j k pxq B Bx j , X ff `vertical vector fields " QrΘsX `p´1q |Θ| ÿ k, |ξ k |"1 Θ d ξ k rρpξ k q, Xs `vertical vector fields we have ad Q pP q " QrΘs b X `p´1q |Θ| ÿ k, |ξ k |"1 Θ d ξ k b rρpξ k q, Xs, (6.26)
where X stands for the class of a vector field X P XpM q in XpM q F . Therefore, locally

ad Q " Q b id `ÿ k, |ξ k |"1 id d ξ k b ∇ Bott ξ k
where ∇ Bott : ΓpE ´1qˆX pM q F ÝÑ XpM q F , is the Bott connection associated to F, i.e. ∇ Bott e X :" rρpeq, Xs.

Remark 6.2.14. Assume that M " R d and F is a singular foliation that admits two leaves: t0u and

R d zt0u. Then XpEq LpEq » ΓpS ' pE ˚qq| 0 b R ˆRd ' I 0 XpM q F ˙.
The differential may be complicated to describe.

Example 6.2.15. Let F " I 0 XpR d q and pE, Qq one of its universal Lie algebroid such that E ´1 "» glpR d q is the transformation Lie algebroid for the action of glpR d q on R d . For this singular foliation,

XpR d q F » R d . Thus, we obtain XpEq LpEq » ΓpS ' pE ˚qq| 0 b C 8 pR d q R d .
and the Bott connection is simply the Chevalley-Eilenberg differential for the action of glpR d q on R d . In particular, it is easy to see that

H 1 pXpEqq " H 1 ˆXpEq LpEq ˙" H 1 CE pglpR d q, R d q " 0
In other words, the universal Lie 8-algebroid is, in that case, rigid, i.e. any formal deformation

Q `ϵQ 1 `ϵ2 Q 2 `¨¨¨of its derivation Q is formally trivial [LGL22a].
Here is a second example where the universal Lie 8-algebroid is rigid.

Example 6.2.16. Let F be the singular foliation given by the action of Lie algebra g " slp2, Rq on R 2 through the vector fields: Hence, as a graded vector space:

e " x B
XpEq LpEq » p^'sl 2 ' Rr2sq b R pR 2 ' Rq.
The differential induced by Q is easily checked to be the Chevalley-Eilenberg differential for the natural slp2, Rq-action on these modules. In particular,

H 1 pXpEqq " H 1 ˆXpEq LpEq ˙" H 1 ˆslp2, Rq, XpR 2 q F ˙" 0,
since the Lie algebra slp2, Rq is semisimple. This implies that the universal Lie 8-algebroid pE, Qq is rigid in the sense of [START_REF] Laurent | On deformations of singular foliations[END_REF].

Conclusion:

We recalled the duality "Lie 8-algebroid ÐÑ Q-manifolds of finite rank in all degree" so that the universal Lie 8-algebroid becomes a universal Q-manifold. We recover [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] the notion of universal Q-manifold pE, Qq of a singular foliation. We prove that ad Q has a no longitudinal cohomology.

a Which squares to zero on vector fields on E.

CHAPTER 7

Isotropy Lie algebras of a singular foliation

In this chapter, we look at an extension of the Androulidakis and Skandalis isotropy Lie algebra [START_REF] Androulidakis | Smoothness of holonomy covers for singular foliations and essential isotropy[END_REF] of a singular foliation at a point.

Let us recall some definitions.

Definition 7.0.1. Let F be a singular foliation on a manifold M . Let I m " tf P C 8 pM q | f pmq " 0u.

1. The tangent space of F at a point m P M is the vector space T m F :" tX| m | X P Fu whose elements are evaluations of the vector fields of F at m. In other words, T m F is the image of the evaluation map ev m :

F Ñ T m M, X Þ Ñ X| m .
2. The set

! m P M ˇˇF ImF ÝÑ T m F is bijective
) is open and dense in M [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF]. It is the set of regular points of pM, Fq and is denoted by M reg .

For any point m P M , the O-module Fpmq :" tX P F | Xpmq " 0u " kerpev m q is a Lie subalgebra of F. The evaluation map goes to quotient and induces the following exact sequence,

0 / / Fpmq I m F / / F I m F / / T m F / / 0. Since I m F Ď
O m b O A in a neighborhood of m. Since r " dimp A I m 1
A q for all points m 1 P U, again Nakayama Lemma B.2.11 implies pe 1 , . . . , e r q are minimal generators on a neighborhood of m 1 . Any representative pe 1 , . . . , e r q of pe 1 , . . . , e r q are local sections of A in a neighborhood U of m that span A on U so that A| U is a free module. This applies that A is a C 8 pUq-module, which is projective. By Serre-Swan theorem, A " ΓpAq for some vector bundle A.

Specialization of a Lie 8-algebroid at a point

• Let pM, E, Qq " pE ' , ℓ ' , ρq be a Lie 8-algebroid over a manifold M with anchor ρ. For every point m P M , the k-ary brackets restrict to the graded vector space evpE, mq :"

˜à iě2 E ´i|m ¸' kerpρ m q
and equipped the latter with a Lie 8-algebra structure that we denote by Istropy m pE, Qq. For every k ě 1, the restriction goes as follows:

tx 1 , . . . , x k u k :" ℓ k ps 1 , . . . , s k q |m (7.2)
for all x 1 , . . . , x k P evpE, mq and s 1 , . . . , s k P ΓpEq sections of E such that s i pmq " x i with i " 1, . . . , k. These brackets are well-defined. It is clear that for k ‰ 2, since ℓ k is linear over functions. But it is not immediate that the 2-ary bracket is well-defined as well. Let us check that.

On one hand, the new brackets t¨¨¨u k have values in evpE, mq for degree reason, except may be for the 2-ary bracket when applied to elements of degree ´1 (i.e. elements of the kernel of ρ m ) but in that case it is in the kernel of ρ m since ρ m ptx 1 , x 2 u 2 q " ρ m pℓ 2 ps 1 , s 2 q |m q " ρpℓ 2 ps 1 , s 2 qq |m " rρps 1 q, ρps 2 qs |m " 0

In the last line we have used the fact that the Lie bracket of two vector fields that vanish at m is a vector field that vanishes again at m.

On the other hand, the 2-ary bracket t¨, ¨u2 is also well-defined when applied to elements of degree less or equal to ´2, we need to verify when we take the bracket with at least an element of degree ´1. Let pe i 1 , . . . e i rkpE ´iq q be a local trivialization of E ´i on a neighborhood U of the point m P M . For x 1 P kerpρ m q and x 2 P E ´i|m write

x 1 " rkpE ´1q ÿ k"1 λ k e 1 k pmq, x 2 " rkpE ´iq ÿ k"1 µ k e i k pmq
for some scalars pλ i q in K. The scalars pλ k q, pµ k q extend to functions pf k q, pg k q on U. Therefore, we have

tx 1 , x 2 u 2 " ℓ 2 ps 1 , s 2 q |m with s 1 " rkpE ´1q ÿ k"1 f k e 1 k , s 2 " rkpE ´iq ÿ k"1 g k e i k .
If r s 2 is another extension of x 2 , then ps 2 ´r s 2 qpmq " 0 and this is equivalent to pg k ´r g k qpmq " 0 for k " 1, . . . , rkpE ´iq. It follows that

ℓ 2 ps 1 , s 2 ´r s 2 q |m " rkpE ´iq ÿ k"1 ℓ 2 `s1 , pf k ´r g k qe i k ˘|m " rkpE ´iq ÿ k"1
( ( ( ( ( ( ( pf k ´r g k qpmqℓ 2 `s1 , e i k ˘|m `(( ( ( ( ( ( ( ρps 1 q |m rf k ´r g k se i k " 0.

Hence, evpE, mq : ¨¨¨t

¨u1 "ℓ 1 |m ÝÑ E ´3| m t¨u 1 "ℓ 1 |m ÝÑ E ´2| m t¨u 1 "ℓ 1 |m
ÝÑ kerpρ m q comes equipped with a Lie 8-algebra whose brackets are pt¨¨¨u k q kě1 .

• Any Lie 8-morphism of algebroids Φ : pM, E 1 , Q 1 q Ñ pM, E, Qq induces a Lie 8-algebra morphism Φ |m : S ' pV 1 |m q Ñ S ' pV |m q since it is O-linear.

• We define the graded vector space à iě1 H ´ipE ' , mq (7.3) as the cohomology group of the complex

¨¨¨ℓ 1| m / / E ´3|m ℓ 1| m / / E ´2|m ℓ 1| m / / kerpρ m q / / 0.
One can check that when pM, E, Qq is universal for a singular foliation F, the graded space (7.3) does not depend on the underlying geometric resolution of F and is denoted HpF, mq.

This construction can be extended to any Lie 8-algebroid over O for any maximal ideal I. Let pE, ℓ ' , ρq be a Lie 8-alegrboid over O. Define

E ´i| I " $ ' & ' % te P E ´1 | ρpeqrOs Ď Iu for i " 1 E ´i IE ´i for i ě 1 (7.4)
The construction is purely formal. Also, the cohomology of the complex

¨¨¨ℓ 1 | I / / E ´3| I ℓ 1 | I / / E ´2| I ℓ 1 | I / / E ´1| I
denoted by H ' pE, Iq does not depend on the choice of a free resolution of the Lie-Rinehart algebra A, we denote it by H ' pE, Iq.

The isotropy Lie 8-algebra of a singular foliation

We assume that pM, E, Qq is universal for F. Note that the Lie 8-algberoid obtained by specialising at some point m P M does not induce directly a Lie 8-algberoid on the graded space HpF, mq but the 2-ary bracket t¨, ¨u2 goes to quotient directly on elements of degree ´1 i.e. to H ´1pF, mq, because

td p2q m px 1 q, x 2 u 2 " d p2q m ptx 1 , x 2 u 2 q
for all x 1 P E ´2|m and x 2 P kerpρ m q. That endows H ´1pF, mq with Lie algebra structure.

Proposition 7.1.1. The Androulidakis and Skandalis isotropy Lie algebra g m " Fpmq ImF of the singular foliation F at a point m P M , is isomorphic to H ´1pF, mq equipped with the induced Lie algebra structure.

Proof. For m P M , we construct a Lie algebra isomorphism ζ : kerpρmq impd p2q m q Ñ g m as follows: For an element u P kerpρ m q, let r u be an extension of u to a local section on E ´1. By construction, one has ρpr uq P Fpmq. Let r ρ m be the surjective linear map defined by r ρ m : kerpρ m q ÝÑ g m , u Þ ÝÑ rρpr uqs.

Since any other extension r u for u differs from the first one by a section in I m ΓpE ´1q, the map r ρ m is well-defined. Surjectivity is due to the fact that every vector field of F vanishing at m P M is of the form ρpeq with e a (local) section of E ´1 whose value at m belongs to kerpρ m q. In addition, it is not hard to see that r ρ m is a morphism of brackets.

It remains to show that kerpr ρ m q " impd p2q m q: let u P kerpr ρ m q Ă kerpρ m q and r u be a local section of E ´1 that extends u. By definition of u, the class of ρpr uq is zero in g m , therefore, there exists some functions f i P I m and X i P F, i " 1, . . . , k, local generators such that ρpr uq "

k ÿ i"1 f i X i . This implies that, ρpr u ´k ÿ i"1 f i e i q " 0.
where for i " 1, . . . , k, e i is a (local) section of E ´1 whose image through ρ is X i . Since pE ' , d ' , ρq is a geometric resolution, there exists a (local) section q P ΓpE ´2q such that

r u " k ÿ i"1
f i e i `dp2q q (7.5) By evaluating Equation (7.5) at m, we find out that u P impd p2q m q. Conversely, for v P E ´2|m , choose a (local) section q of E ´2 through v. Therefore, d p2q q P ker ρ, is a (local) extension of d

p2q m v P impd p2q m q. The image of d p2q m v through r
ρ m is obviously zero. This proves that kerpr ρ m q " impd p2q m q.

However, if the underlying complex of pM, E, Qq is minimal at m then, for every i ě 2, the vector space H ´ipF, mq is canonically isomorphic to E ´i|m . Also, H ´1pF, mq is canonically isomorphic to kerpρ m q. Definition 7.1.2. Let pM, E, Qq be a universal Lie 8-algebroid of a singular foliation F whose underlying complex is minimal at m. Then, HpF, mq carries a Lie 8-algebra structure given by Istropy m pE, Qq called the isotropy Lie 8-algebra of the singular foliation F at m.

One can show that this definition is independent of any choices made in the construction.

Remark 7.1.3. By Proposition 7.1.1, the isotropy Lie algebra of the singular foliation F at a point m P M in the sense of Androulidakis and Skandalis, is isomorphic to the degree minus one component H ´1pF, mq » kerpρ m q of the isotropy Lie 8-algebra of F at m. Remark 7.1.4. For an arbitrary Lie-Rinehart algebra A and a maximal ideal I, it is still true that pH ' pA, Iq, d " 0q is quasi-isomorphic to pE| I , ℓ 1 | I q and it is still true that the bracket of the universal Lie 8-algrbroid over O constructed in Section 4.2 restricts to a 8-algebra structure on pE| I , ℓ 1 | I q. By Homotopy transfer, this Lie 8-algebra can be transferred to pH ' pA, Iq, d " 0q. It is not clear whether Proposition 7.1.1 holds true or not. » g m as Lie algebras.

The subset of regular points of F in M satisfies

M reg " tm P M | rkpd p2q m q " dimpker ρ m qu.
It is open and dense in M .

3. The restriction of the foliation F to M reg is the set of sections of a subbundle of TM, i.e., is a regular foliation.

If pE, d, ρq is of finite length, then the regular leaves have the same dimension.

Proof. Item 1. is proved in Proposition 7.1.1 (see also Proposition 4.14 [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]). For items 2. and 3. we refer the reader to Proposition 1.5 of [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF].

A blow-up procedure for a singular foliation

We present in this section an interpretation of a blow-up procedure invented by Mohsen in [START_REF] Mohsen | Blow-up groupoid of singular foliations[END_REF].

Grassmann bundle

For E a finite dimensional vector space, we denote by Gr ´ℓpEq the set of all codimension ℓ P N vector subspaces in E. Let us recall a few facts on Gr ´ℓpEq see [KES00, Joe92, LB15], our main reference is [R94]:

1. It is a metric space for the distance dpV, V 1 q " ∥P V ´PV 1 ∥, where P V stands for the orthogonal projection of E onto V Ă E with respect to an arbitrary metric on E.

2. The topology does not depend on the metric and makes Gr ´ℓpEq a manifold.

3. It is moreover a compact manifold and an projective variety 1 .

Affine coordinates charts:

One of the ways of defining the standard affine coordinates on the Grassmanian Gr ´ℓpEq is as follows (Example 1.24 of [R94]): Fix a basis e 1 , . . . , e d"dim E for E. Any element V P Gr ´ℓpEq, i.e. a vector subspace V Ă E of codimension ℓ, can be viewed as a d ˆpd ´ℓq matrix M V " pC 1 , . . . , C d´ℓ q whose columns are formed by linearly independent column vectors obtained by choosing a basis for V . The homogeneous coordinates of V in Gr ´ℓpEq are the components of the n ˆpd ´ℓq matrix M V . Any other choice of basis for V gives another maximal rank matrix M 1 V and an invertible pd´ℓqˆpd´ℓq-matrix P P GLpd´ℓ, Kq such that M V " M 1 V ˝P . In particular, since M V has full rank, there exists a family of integers 1 ď i 1 ă ¨¨¨ă i d´ℓ ď d, such that M V is equivalent to a matrix M 1 V whose submatrix made of the rows i 1 , ¨¨¨, i d´ℓ is the identity matrix. For example, if the first d ´ℓ rows of M V are linearly independent, then the matrix is equivalent to the matrix ˜Id´ℓ A where A " pa ij q is a ℓ ˆpd ´ℓq-matrix. In that case, V admits a basis of the form

v j :" e j `ℓ ÿ k"1
a kj e k , j " 1, . . . , d ´ℓ.

(7.7)

V is completely determined by A.

One define an atlas on Gr ´lpEq as follows: Consider the map

ψ 1,...,d : M l,d´l pKq ÝÑ M d,d´l pKq A Þ ÝÑ ˜Id´ℓ A ¸.
1 For the notion of projective variety and notations, we refer the reader e.g. to the book [R94].

For a permutation σ P S d , we define the map ψ σp1q,...,σpdq that associates every A P M l,d´l pKq the matrix in M d,d´l pKq that permutes the lines of ˜Id´ℓ A ¸w.r.t σ, that is to say ψ σp1q,...,σpdq pAq :" P pσq ˝ψ1,...,d pAq where P pσq is the permutation matrix associated to σ.

Statement: For every ordered integers 1 ď i 1 ă ¨¨¨ă i d´ℓ ď d, the coordinates chart on Gr ´ℓpEq is the open set U i 1 ,¨¨¨,i d´ℓ Ă Gr ´ℓpEq consisting of all sub-vector space of E such that for every basis the submatrix which is made of the rows i 1 , ¨¨¨, i d´ℓ is invertible, and the coordinate map is ψ σp1q,...,σpdq with σ is a permutation sending 1, . . . , d ´ℓ on i 1 , ¨¨¨, i d´ℓ .

Grassmann bundle: For E ´1 Ñ M a vector bundle of rank d, the disjoint union:

Gr ´ℓpE ´1q :" ž mPM Gr ´ℓpE ´1| m q
comes equipped with a natural manifold structure and Π : Gr ´ℓpE ´1q ÝÑ M is a fibration. It is called the Grassmann pd ´ℓq-plane bundle. To fix some notations, the fiber at

m P M is Π ´1pmq " tpV, mq | V P Gr ´ℓpE ´1| m qu .
For every open subset U Ă M on which E ´1 is trivial, Π ´1pU q » Gr ´ℓpR d q ˆU.

A blow-up procedure

Settings:

In what follows, we are given a foliated manifold pM, Fq with M connected. We assume that a geometric resolution of finite length exists. Under these assumptions, all the regular leaves have the same dimension. Let M reg the set of the regular points of pM, Fq. Denote by ℓ the common dimension of the regular leaves.

Remark 7.2.1. For most of the present discussion, the whole geometric resolution is not needed: is it sufficient to assume that there exists vector bundles E ´1, E ´2 and that all regular leaves have the same dimension.

Let pE ' , ℓ ' , ρq be a universal Lie 8-algebroid of F. Consider the underlying geometric resolution pE, d, ρq :

¨¨¨ℓ 1 "d p4q ÝÑ E ´3 ℓ 1 "d p3q ÝÑ E ´2 ℓ 1 "d p2q ÝÑ E ´1 ρ ÝÑ T M. (7.8)
Notice that for every point m P M reg , dimpd p2q m q " dim kerpρ m q " rkpE ´1q ´ℓ.

Therefore, there exists a natural section of Π on M reg defined by:

σ : M reg ÝÑ Gr ´ℓpE ´1q, m Þ ÝÑ impd p2q m q (7.9)
We consider Ă M :" σpM reg q the closure of the graph of σ in Gr ´ℓpE ´1q, and denote by π the restriction of Π to Ă M .

Recall from Section 7.1 that evpE, mq :

¨¨¨ÝÑ E ´3| m d p3q m ÝÑ E ´2| m d p2q m
ÝÑ kerpρ m q comes equipped with a Lie 8-algebra pt¨¨¨u k q kě1 whose unary bracket is d

π ´1pmq " " V Ă E ´1|m ˇˇˇD pm n q P M N reg , such that, impd p2q mn q ÝÑ nÑ`8 V as m n ÝÑ nÑ`8 m * is non-empty.
2. For all m P M , and V P π ´1pmq one has,

(a) impd p2q m q Ď V Ď kerpρ m q. (b) The 2-ary bracket t¨, ¨u2 on ker ρ m restricts to V and the image of V in kerpρmq impd p2q m q » g m , is a
Lie subalgebra of codimension ℓ ´dimpL m q, where L m is the leaf through m.

3. For all m P M reg , π ´1pmq " kerpρ m q " impd p2q m q is reduced to a point in Gr ´ℓpR rkpE ´1q q.

Ă

M does not depend on the choice of a geometric resolution.

5. The projection π : Ă M Ñ M is proper and onto.

Proof. Let us prove item 1 for m P M . By compactness of the Grassmanian manifold Gr ´ℓpE ´1|m q » Gr ´ℓpR rkpE ´1q q, out of any sequence in M N reg that converges to m, we can extract a sequence n Þ Ñ κ ´impd p2q mn q ¯P Gr ´ℓpE ´1|m q that has a limit, where κ is a local trivialization of the vector bundle E ´1 which identifies the fibers E ´1|m and E ´1|m n . This proves item 1.

Let us show item 2.paq: Let V P π ´1pmq and pm n q P M N reg such that m n ÝÑ nÑ`8 m and impd p2q mn q ÝÑ nÑ`8

V . Let v P impd p2q m q. We have v " d p2q m u for some u P E ´2|m . Choose a (local) section r u of E ´2 through u. It implies that d p2q mn r upm n q ÝÑ nÑ`8 d p2q m u, hence d p2q m u P V . Thus, impd p2q
m q Ď V . For any element v P V , there exists a sequence v n P kerpρ mn q " impd p2q mn q, n P N that converges to v. In particular, ρ mn pv n q " 0 for all n. Hence, by continuity, one has v P kerpρ m q.

To prove item pbq, choose a local frame e 1 , . . . , e r , . . . e ℓ`r of E ´1 such that e 1 pmq, . . . , e r pmq is an orthogonal basis of V for an arbitrary Hermitian structure on E ´1. For i, j P t1, . . . , l `ru, let pc k ij q P OpUq be a family of functions over U such that

ℓ 2 pe i , e j q " l`r ÿ k"1 c k ij e k .
In particular, for every ti, ju Ă t1, . . . , l `ru, te i pmq, e j pmqu 2 "

l`r ÿ k"1 c k ij pmqe k pmq. Let v 1 , v 2 P V with v 1 " r ÿ i"1
α i e i pmq and v 2 "

r ÿ i"j
β i e i pmq. There exists sequences

v n 1 " r`l ÿ i"1 α i n e i pmq ÝÑ nÑ`8 v 1 and v n 2 " r`l ÿ i"1 β i n e i pmq ÝÑ nÑ`8 v 2
with v n 1 , v n 2 P ker ρ xn , for all n P N. In particular, the sequences pα i n q, pβ i n q P K N with i P t1, . . . , r `lu

satisfy α i n ÝÑ nÑ`8 α i , β i n ÝÑ nÑ`8
β i (α i " β i " 0 for i ě r `1). For every n P N we have,

r`l ÿ i,j,k"1 α i n β j n c k ij pm n qe k pm n q " tv n 1 , v n 2 u 2 P ker ρ xn " impd p2q mn q. (7.11) Since r`l ÿ i,j,k"1 α i n β j n c k ij pm n qe k pm n q ÝÑ nÑ`8 r`l ÿ i,j,k"1 α i β j c k ij pmqe k pmq " tv 1 , v 2 u 2 , one has, tv 1 , v 2 u 2 P V .
Item 3. is a consequence of 2.paq, since m P M reg if and only if kerpρ m q " impd p2q m q (by item 1. of Lemma 7.1.5). Item 4. follows from the existence of homotopy equivalence between any two geometric resolutions. Item 5. follows from the fact that the projection Π is with compact fibers. This concludes the proof.

The corollary below is a direct consequence of item 2.(b) of Theorem 7.2.2.

Corollary 7.2.3. There is a natural inclusion

Ă M ãÑ ž mPM
Gr ℓ´dimpLmq pg m q.

(7.12)

Proof. Let m P M . Since elements V P π ´1pmq satisfy impd p2q m q Ď V Ď kerpρ m q, they correspond injectively to a (unique) sub-vector space of codimension ℓ ´dim L m in g m . In particular, this implies π ´1pmq ãÑ Gr ℓ´dimpLmq pg m q. Lemma 7.2.4. For every m P M , the set K m " tV P Gr ´ℓpE ´1|m q | V Ď ker ρ m u Ă Gr ´ℓpE ´1|m q is locally an affine variety.

Proof. Let e 1 , . . . , e d be a local frame of E ´1 on an open subset U Ă M . One has,

ρpe i q " dim M ÿ k"1 f k i B Bx k P F, (7.13)
for some local functions f k i P C 8 pUq. Without any lost of generality, consider for example the standard coordinates chart U 1,...,d´ℓ for the grassmanian Gr ´ℓpE ´1|m q. Let V P U 1,...,d´ℓ and let pa ij q be the homogeneous coordinates of V . Define the sections

r v j :" e j `ℓ ÿ k"1
a kj e k , j " 1, . . . , d ´ℓ (7.14) By construction, the r v j 's, evaluated at m, form a basis for V . We have,

ρpr v j q " dim M ÿ k"1 ˜f k j `ℓ ÿ s"1 a sj f k s ¸B Bx k .
V Ď ker ρ m if and only if

f k j pmq `ℓ ÿ s"1 a sj f k s pmq " 0, j " 1, . . . , d ´ℓ k " 1, . . . , dim M. (7.15)
Therefore, K m is defined by the polynomial Equations (7.15).

Theorem 7.2.5. If F is a polynomial singular foliation on

M P ␣ C N , R N ( , then Ă M is a locally affine variety.
Proof. We can choose a polynomial geometric resolution. Denote by x " px 1 , . . . , x N q the local coordinates on W Ď M . We are using notations of Lemma 7.2.4. In Equation (7.13), the functions

f k i are polynomial in px 1 , . . . , x N q. By item 1, Theorem 7.2.2, every element V of Ă M " Ť xPM π ´1pxq is obtained as a limit impd p2q xn q ÝÑ nÑ`8 V P π ´1pxq
with px n q P M N reg , such that, x n ÝÑ nÑ`8

x. Fix n P N 0 . In the notations of Lemma 7.2.4, take V " impd p2q xn q. Thus, in the coordinate chart U 1,...,d´ℓ (see Section 7.2.1), the coordinates pa n ij q of impd p2q xn q satisfy the polynomial equations

f k j px n q `ℓ ÿ s"1
a n sj f k s px n q " 0, j " 1, . . . , d ´ℓ k " 1, . . . , N.

(7.16)

One has,

f k j px n q `ℓ ÿ s"1 a n sj f k s px n q " 0 ÝÑ nÑ`8 f k j pxq `ℓ ÿ s"1 a sj f k s pxq " 0,
where for all s, j, a n sj ÝÑ nÑ`8 a sj . Therefore, Ă M is given in local coordinates W ˆU1,...,d´ℓ by elements that satisfy 1. Equation (7.15) 2. and that are limit of elements of (7.15) in nearby regular points, elements which are unique.

Hence, it is, on this affine variety, the irreducible components of (7.15) that projects onto M .

The pull-back of F to Ă M : Let X P F. There exists a section υ of the vector bundle p : E ´1 Ñ M such that ρpυq " X. Consider the linear vector field r X P XpE ´1q defined as follows r Xrp ˚f s :" p ˚pρpυqrf sq, @ f P C 8 pM q,

x r Xrαs, ey :" ρpυqrxα, eys ´xα, ℓ 2 pυ, eqy, @ α P ΓpE ˚q, e P ΓpE ´1q.

Notice that r X depends on the choice of the almost Lie algebroid bracket ℓ 2 . The following items hold (see [START_REF] Laurent-Gengoux | Geometry of singular foliations: a draft of an introduction[END_REF], Lemma 2.1.19. p. 61).

The flow

ϕ r X t : E ´1 Ñ E ´1 of r
X when it is defined, is a vector bundle isomorphism over ϕ X t .

2. The diagram below commutes,

E ´1 ϕ Ă X / / ρ E ´1 ρ T M dϕ X t / / T M (7.17)
where ϕ X t is the flow of X.

In particular, ϕ r X t preserve the grassmanian Gr ´ℓpE ´1q.

Proposition 7.2.6. For every X P F, choose υ P ΓpE ´1q such that ρpυq " X. The vector field r X induces a vector field Ă Ă X on Gr ´ℓpE ´1q such that

1. Ă Ă X is tangent to Ă M . 2. Ă Ă X projects onto X.
Proof. Since for every V with px n q Ă M reg . For every n P N 0 , one has

x P M , ϕ r X t | x is an isomorphism of E ´1| x , therefore ϕ r X t | x preserves Π ´1pxq " Gr ´ℓpE ´1| x q. We define Ă Ă X for pV, xq P Gr ´ℓpE ´1q by Ă Ă XpV q :" d dt | t"0 ´ϕ r X t | x ¯|V P T V Gr ´ℓpE ´1| x q. ( 7
ϕ r X t | xn ´imd p2q xn ¯" imd p2q ϕ X t pxnq , since ρ ˝ϕ r X t " dϕ X t ˝ρ. Thus, ϕ r X t | x pV q " lim nÑ`8 ϕ r X t | xn ´imd p2q xn " lim nÑ`8 ´imd p2q ϕ X t pxnq ¯P π ´1 `ϕX t pxq ˘.
By consequence, for V P Ă M , we have Ă Ă XpV q P T V Ă M , by Equation (7.18).

Corollary 7.2.7. Ă Ă X does not depend on the choice of the almost bracket ℓ 2 .

Proof. This is an obvious consequence of item 3. in Proposition 7.2.6, since π : Ă M ÝÑ M is invertible on an open (dense) subset.

Denote by r

F the module generated by the Ă Ă X's, with X P F. For F polynomial, r F is a singular foliation on the locally affine variety Ă M , because r X is still polynomial.

Corollary 7.2.8. If F is a polynomial singular foliation, then it is lifted to a projective singular foliation r F on the locally affine variety Ă M .

Generalization. The same construction would apply if, instead of considering impd p2q q Ď E ´1, we consider impd pi`1q q Ď E ´i or impρq Ď T M for i ě 1. Denote by Ă M i the transformation that we would obtain by such a construction. A lift r F i of F can be defined exactly in the same manner by considering a lift r X of X P F on E ´i (or T M ) associated to ℓ 2 : ΓpE ´1q ˆΓpE ´iq Ñ ΓpE ´iq or to rX, ¨s.

More precisely, consider for i ě 0, the Grassmann bundle Gr ´ℓi pE ´iq, with E 0 :" T M , where ℓ i :" rkpd i`1 m q (with the understanding that d p1q " ρ for i " 0) at regular points. These bundles are manifolds that project to M through a proper map. Over M reg , there exists a unique V P Gr ´ℓi pE ´iq such that d piq m pV q " 0. Theses define maps

σ i : M reg ãÝÑ Gr ´ℓi pE ´iq.
We define Ă M i :" σ i pM reg q. This is not a manifold in general. However, it is a locally affine variety if F is a polynomial singular foliation on R N or, C N as in Theorem 7.2.5. Moreover, 1. independent of the choice of a geometric resolution, as in Theorem 7.2.2,

the projection Ă

M i Ñ M is proper and onto, as in Theorem 7.2.2, 3. F lifts to a singular foliation on Ă M i , and it is one-to-one to M reg , as in Proposition 7.2.6.

For i " 0 the same conclusion holds for

σ 0 : M ÝÑ Gr ℓ 0 pT M q m Þ ÝÑ T m F.
Let us give some examples.

Example 7.2.9. Notice that Gr ´ℓpC ℓ q " tptu, so that if d piq is into on an open dense subset, the construction degenerates, in this case Gr ´ℓpC ℓ q ˆM » M

1. If F is a projective singular foliation, then Ă M 1 » M : because ΓpE ´1q » F and E ´1 ρ
Ñ T M is injective on the open dense subset M reg , i.e. ℓ 0 " rkpE ´1q. Hence Gr ´ℓ0 pE ´1q » M .

If F admits open dense orbits, Ă

M 0 » M , since Gr ´dim M pT M q » M .

3. Let W Ď M " C d be an affine variety generated by (independent functions) φ, ψ P Crx 1 , . . . , x d s, i.e. I W " xφ, ψy. Let F W be the singular foliation of (polynomial) vector fields vanishing on W . At regular points

x P M reg , T x F W " d, hence 7.2. A BLOW-UP PROCEDURE FOR A SINGULAR FOLIATION 123 (a) Ă M 0 » M . (b) Also, • E ´1 " pC d ' C d q ˆM, ρ : e i ' 0 Þ Ñ φ B Bx i and 0 ' e i Þ Ñ ´ψ B Bx i , for all i " 1, . . . , d. • E ´2 " C d ˆM, and d p2q : e i Þ Ñ ψ K e i ' φ K e i with, K " gcdpφ, ψq impd p2q x q " ker ρ x " B ψpxq Kpxq u ' φpxq Kpxq u, with u P C d F
For a convergent sequence py n q in M zW , pker ρ yn q converges if and only if r ψpynq Kpynq : φpynq Kpynq s converges in P 1 pCq. In that case, Ă M 1 is the closure of the graph tpy, rψpyq : φpyqsq, y P M zW u, which is the blow up of

C d along W . • Ă M 2 » M since d p2q
x is injective at regular points, so that Gr ´dpE ´2q » M .

4. Let W be the affine variety defined by φ P Crx 1 , . . . , x d s. Consider the singular foliation F φ " tX P XpC d q | Xrφs " 0u. For every y P C d , pT y F φ q K " x∇ y φy. For convergent sequence y n ÝÑ nÑ`8 y P W . The sequence impρ yn q converges if and only if ∇ yn φ converges in Gr ´pd´1q pC d q, that is " Bφ Bx 1 py n q : ¨¨¨: Bφ Bx d py n q ı converges in the projective space P d´1 pCq. Therefore, Ă M 0 is the closure of the graph of the map, y Þ Ñ py,

" Bφ Bx 1 pyq : ¨¨¨: Bφ Bx d pyq ı q which is the blow up of C d along the singular locus of W .
Let us conclude this chapter by an open question. Can we desingularize a singular affine variety W Ď C d by applying the constructions above to the singular foliation F " XpW q of vector fields tangent to W ? We should then understand Ă W 0 as the Nash modification of W [START_REF] Lê | Geometry of Tangents on Singular Spaces and Chern Classes[END_REF]. The meaning of Ă W 1 is unclear. We would like to relate the Ă W i 's and the r F i together and go further in the universal Lie 8-algebroid, e.g. to understand the role of the 3-ary bracket in this procedure.

Conclusion:

In this chapter, we recall the notion of isotropy Lie (8)-algebra of a singular foliation (and of a Lie-Rinehart algebra).

Then, we use the geometric resolution (i.e. the resolution on which the universal Lie 8algebroid is built) to recover several notions of resolution of singularities: one being due to Nash and a second one to Mohsen.

CHAPTER 8

Symmetries of singular foliations through Lie 8-algebroids

This chapter is one of the main application of results in Chapter 8.3.1 and Section 6.2. These results are taken from my article [START_REF] Louis | On symmetries of singular foliations[END_REF].

We introduce the notion of weak symmetry actions of a lie algebra g on a singular foliation F and study the interaction of those on the universal Lie 8-algebroids of F. Also, in the subsequent chapter, we apply these results to the problem of extending a strict Lie algebra action on a sub-affine variety on the ambient space. Convention 8.0.1. Throughout this chapter, M stands for a smooth or complex manifold, or an affine variety over C. We denote the sheaf of smooth or complex, or regular functions on M by O and the sheaf of vector fields on M by XpM q, and Xrf s stands for a vector field X P XpM q applied to f P O. Also, K stands for R or C.

Definitions and examples

Definition 8.1.1. Let F Ă XpM q be a singular foliation over M .

• A diffeomorphism ϕ : M ÝÑ M is said to be a symmetry of F, if ϕ ˚pFq " F.

• A vector field X P XpM q is said to be an infinitesimal symmetry of F, if rX, Fs Ă F. The Lie algebra of infinitesimal symmetries of F is denoted by spFq.

In particular, F Ă spFq, since rF, Fs Ă F.

Proposition 8.1.2. [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF][START_REF] Garmendia | On the inner automorphisms of a singular foliation[END_REF] Let M be a smooth or complex manifold. The flow of an infinitesimal symmetry of F, if it exists, is a symmetry of F.

As we will see later, one of the consequences of our future results is that any symmetry X P spFq of a singular foliation F admits a lift to a degree zero vector field on any universal N Q-manifold over F that commutes with the homological vector field Q. This allows us to have an alternative proof and interpretation of Proposition 8.1.2 (see Section 8.2).

Let pg, r¨, ¨sg q be a Lie algebra over K " R or C, depending on the context. From now on and in the sequel g is concentrated in degree ´1. Definition 8.1.3. A weak symmetry action of the Lie algebra g on a singular foliation F on M is a K-linear map ϱ : g ÝÑ XpM q that satisfies:

• @ x P g, rϱpxq, Fs Ď F,

• @ x, y P g, ϱprx, ys g q ´rϱpxq, ϱpyqs P F.

When x Þ ÝÑ ϱpxq is a Lie algebra morphism, we speak of strict symmetry action of g on F. There is an equivalence relation on the set of weak symmetry actions which is defined as follows: two weak symmetry actions, ϱ, ϱ 1 : g ÝÑ XpM q are said to be equivalent if there exists a linear map φ : g ÝÑ F such that ϱ ´ϱ1 " φ. Remark 8.1.4. It is important to notice that when F is a regular foliation and M {F is a manifold, any weak symmetry action of a Lie algebra g on F induces a strict action of g over M {F. Definition 8.1.3 is a way of extending this idea to all singular foliations.

Here is a list of some examples.

Example 8.1.5. Let π : M ÝÑ N be a submersion. Since any vector field on N comes from a πprojectable vector field on M , therefore any Lie algebra morphism g ÝÑ XpN q can be lifted to a weak symmetry action g ÝÑ XpM q on the regular foliation Γpker dπq, and any two such lifts are equivalent.

Furthermore, any weak action of a Lie algebra g on a singular foliation F on N can be lifted to a class of weak symmetry actions on the pull-back foliation π ´1pFq, (see Definition 1.9 in [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF]).

Example 8.1.6. Let F be a singular foliation on M . For any point m P M , consider g m " Fpmq I m F the isotropy Lie algebra of F at m (see Definition 7.0.2). Let us denote its Lie bracket by r¨, ¨sgm .

1. Consider ϱ : g m Ñ Fpmq Ă XpM q a section of the projection map,

I m F / / Fpmq / / / / g m ϱ r r (8.1)
Then, rϱpxq, I m Fs Ă I m F and ϱprx, ys gm q ´rϱpxq, ϱpyqs P I m F. Hence, the map ϱ : g m Ñ XpM q is a weak symmetry action of the singular foliation I m F. A different section ϱ 1 of the projection map yields an equivalent weak symmetry action of g m on I m F. An obstruction class for having a strict symmetry action equivalent to ϱ will be given later in Section 9.

2. In particular, for k ě 1, let us denote by g k m the isotropy Lie algebra of the singular foliation I k m F at m. Any section ϱ k : g k m ÝÑ XpM q of the projection map

I k`1 m F / / I k F / / / / g k m ϱ k s s (8.2)
is a weak symmetry action of the Lie algebra g k m on the singular foliation I k`1 m F.

Example 8.1.7. Let pA, r¨, ¨sA , ρq be an almost Lie algebroid on a smooth manifold M , and let F " ρpΓpAqq. Assume there exists p P M such that A| p is a Lie algebra and ρ| p " 0. The question of finding a map A| p ÝÑ ΓpAq such that the composition

A| p ÝÑ ΓpAq ρ ÝÑ XpM q (8.3)
be a Lie algebra morphism, i.e., to ask whether the singular foliation F comes from the transformation Lie algebroid of A| p , can be formulated as a weak symmetry action of A| p on the singular foliation I p F as follows: Consider a linear map

A| p ÝÑ ΓpAq (8.4) a Þ ÝÑ r a
such that for all a P A| p and r a a section of A such that r a| p " a. It is easily checked that the map in (8.4) is not a Lie algebra morphism, but it satisfies Ć ra, bs A|p ´"r a, r b ı A P I p ΓpAq and

" Ą A| p , I p ΓpAq ı Ă I p ΓpAq.
Therefore, the map a Þ ÝÑ ρpr aq is a weak symmetry action of A| p on I p F. Notice that the isotropy Lie algebra g 1 p "

IpF I 2 p F of the singular foliation I p F, is Abelian. In Chapter 9, we show that in this case, the obstruction of having a Lie algebra morphism in (8.3) is a cocycle of Chevalley-Eilenberg of A| p valued in g 1 p .

The next example comes from [START_REF] Laurent | The holonomy of a singular leaf[END_REF], and follows the same patterns as in Examples 8.1.5 and 8.1.6. It is based on the notion of Ehresmann connection. Let us first recall quickly this concept for the sake of completeness and clarity. There are several equivalent manners of viewing Ehresmann connections (see [START_REF] Ivan | Natural operations in differential geometry[END_REF] Section 9, Page 76), the most relevant approach in this context is the following: An Ehresmann connection on a smooth fiber bundle 1 π : E Ñ M is a vector subbundle H of T E, such that T E " H ' V , where V :" tξ P T E | π ˚pξq " 0u is called the "vertical bundle" whose fiber at e P E is V e " T e pE πpeq q. The subbundle H is called the "horizontal bundle". Given a connection as defined above, for every e P E, the linear map π ˚: T e E Ñ T πpeq M restricts to an isomorphism, H e Ñ T πpeq M , whose inverse T πpeq M Ñ H e is called the horizontal lift.

Example 8.1.8. Let pM, Fq be a singular foliation on a smooth manifold M and L Ă M a leaf. Let rL, M s be a neighborhood of L in M equipped with some projection 2 π : rL, M s Ñ L. According to [START_REF] Laurent | The holonomy of a singular leaf[END_REF], upon replacing rL, M s be a smaller neighborhood of L if necessary, there exists an Ehresmann connection (that is a vector sub-bundle H Ă T rL, M s with H ' kerpπ ˚q " T rL, M s) which satisfies that ΓpHq Ă F. Such an Ehresmann connection is called an Ehresmann F-connection and induces a C 8 pLq-linear section ϱ H : XpLq Ñ F proj of the surjection F proj Ñ XpLq, where F proj stands for vector fields of F π-projectable on elements of XpLq. The section ϱ H is a weak symmetry action of XpLq on the transverse foliation T :" Γpker π ˚q X F. When the Ehresmann connection H is flat, ϱ H is bracket-preserving, and defines a strict symmetry of XpLq on the transverse foliation T .

1 this generalizes connections to arbitrary fiber bundle π : E Ñ M , here the total space E is itself a smooth manifold, and has its own tangent bundle T E Ñ E.

2 such projection comes with the Tubular neighborhood theorem [dS01].

Example 8.1.9. Consider, for a fixed k, the singular foliation F k :" I k 0 XpR d q generated by all vector fields vanishing to order k at the origin. The action of the Lie algebra glpR d q on R d which is given by,

glpRq ÝÑ XpR d q, pa ij q 1ďi,jďd Þ ÝÑ ÿ 1ďi,jďd a ij x i B Bx j is a strict symmetry of F k .
Example 8.1.10. Let φ :" pφ 1 , . . . , φ r q be a r-tuple of homogeneous polynomial functions in d variables over K. Consider the singular foliation F φ (see [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF] Section 3.2.1) which is generated by all polynomial vector fields X P XpK d q that satisfy Xrφ i s " 0 for all i P t1, . . . , ru. The action

K Ñ XpK d q, λ Þ Ñ λ Ý Ñ E , is a strict symmetry of F φ .
Here Ý Ñ E stands for the Euler vector field.

Example 8.1.11. Let W Ă C d be an affine variety and I W Ă Crx 1 , . . . , x d s its corresponding ideal. Let us denote by XpW q :" DerpCrx 1 , . . . , x d s{I W q the Lie algebra of vector field of W . Any vector field on W can be extended (not unique) to a vector field on C d (see Section 5.1.1). Let F W :" I W XpC d q the singular foliation made of vector fields vanishing on W . Since every vector field on W can be extended to a vector field on C d tangent to W . Any Lie algebra morphism ϱ : g ÝÑ XpW q extends to a linear map r ϱ : g ÝÑ XpC d q that makes this diagram commutes

XpC d q g r ϱ = = ϱ / / XpW q
This extension r ϱ is a weak symmetry action of g on F W over the ambient space C d . Two different extensions yield equivalent symmetry actions.

A Lie 8-morphism lifting a weak symmetry of a foliation

We recall that O is the sheaf of smooth/complex functions on a smooth/complex manifold M , or the algebra of regular functions on an affine variety over C. We refer the reader to the Chapters 6 and 4 for the notion of (universal) Lie 8-algebroid of a singular foliation. We denote them by pE, Qq and their functions by E.

For the sake of clarity, let put this chapter in context, and fix some notations.

Let pg, r¨, ¨sg q a Lie algebra and pE, Qq a Lie 8-algebroid over M . In the sequel, the Lie algebra g is concentrated in degree ´1. The differential graded Lie algebra pXpEq, r¨, ¨s , ad Q q of vector fields on E is shifted by 1, i.e. a derivation of degree k in X k pEq is of degree k ´1 as an element of the shifted space X k pEqr1s. The graded symmetric Lie bracket on XpEqr1s is of degree `1 and given on homogeneous elements u, v P XpEqr1s as tu, vu :" p´1q |v| ru, vs.

In the sequel, we write pXpEqr1s, r¨, ¨s , ad Q q instead of pXpEqr1s, t¨, ¨u, ad Q q.

Let pS ' K g, Q g q respectively pS ' K pXpEqr1sq, Qq be the corresponding formulations in terms of coderivations of the differential graded Lie algebras pg, r¨, ¨sg q and pXpEqr1s, r¨, ¨s , ad Q q. Precisely, Q g is the co-derivation defined by putting for every homogeneous monomial x 1 ^¨¨¨^x k P S k K g,

Q g px 1 ^¨¨¨^x k q :" ÿ 1ďiăjďk p´1q i`j´1 rx i , x j s g ^x1 ^¨¨¨p x i ¨¨¨p x j ¨¨¨^x k , ( 8.5)
and Q " Qp0q `Q p1q is the co-derivation of degree `1 where the only non-zero Taylor coefficients are, Qp0q :

S 1 K pXpEqr1sq ad Q ÝÑ XpEqr1s and Qp1q : S 2 K pXpEqr1sq t¨,¨u ÝÑ XpEqr1s.
The following is a particular case of Example 2.3.5, see also [START_REF] Lada | Strongly homotopy Lie algebras[END_REF].

Definition 8.2.1. A Lie 8-morphism 3 Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q is a graded coalgebra morphism Φ :

pS ' K g, Q g q ÝÑ pS ' K pXpEqr1sq , Qq of degree zero which satisfies, Φ ˝Qg " Q ˝Φ. (8.6)
In order words, it is the datum of degree zero linear maps

´Φ k : S k`1 K g ÝÑ X ´kpEqr1s ¯kě0 that satisfies ÿ 1ďiăjďn`2 p´1q i`j´1 Φn prx i , x j s g , x 1 , . . . , p x ij , . . . , x n`2 q " rQ, Φn`1 px 1 , . . . , x n`2 qs ÿ i `j " n i ď j σ P S i`1,j`1
ϵpσqr Φi px σp1q , . . . , x σpi`1q q, Φj px σpi`2q , . . . , x σpn`2q qs (8.7) where p

x ij means that we take x i , x j out of the list. When there is no risk of confusion, we write Φ for Φ.

Remark 8.2.2. It is important to notice that:

1. Definition 8.2.1 and Definition 6.1.14 are compatible when M " tptu. Therefore, morphisms in both sense match.

In [MZ12]

, Definition 8.2.1 corresponds to the definition of actions of a Lie 8-algebras of finite dimension on Lie 8-algebroids of finite rank. Here we only have a Lie algebra. In contrast to theirs, we do not assume that g is finite dimensional.

Remark 8.2.3. It follows from the axioms (8.7) that for all x, y P g, rQ, Φ 0 pxqs " 0 and Φ 0 prx, ys g q ´rΦ 0 pxq, Φ 0 pyqs " rQ, Φ 1 px, yqs. (8.8)

In particular, if the homological vector field Q vanishes at some point m P M , then the map x Þ ÝÑ pP P XpEq, P |m Þ Ñ rΦ 0 pxq, P s |m q endows the vector space XpEq |m » pSpE ˚q b Eq |m with a g-module structure. Moreover, the restriction of the map Φ 1 : ^2 g ÝÑ X ´1pEq |m at m is a 2-cocycle of Chevalley-Eilenberg.

Remark 8.2.4. Let pE, Qq be a Lie 8-algebroid and F its basic singular foliation. Any Lie 8morphism Φ : pg, r¨, ¨sg q ÝÑ pX ' pEqr1s, r¨, ¨s , ad Q q gives a weak symmetry action of g on F. If Q |m " 0 for some point m P M , the g-action defined in Remark 8.2.3, is independent of the equivalence class of the weak symmetry action.

The following lemma explains what the 0-Taylor coefficient of a Lie 8-morphism as in Definition 8.2.1 induces on the linear part of pE, Qq. More details will be given in Proposition 8.2.10 and Remark 9.1.1. Lemma 8.2.5. The 0-th Taylor coefficient Φ 0 : g ÝÑ X 0 pEq induces 1. a linear map ϱ : g ÝÑ XpM q, x Þ ÝÑ pϱpxqrf s :" Φ 0 pxqrf s, f P Oq and 2. a linear map x P g Þ ÝÑ ∇ x P Der 0 pEq, i.e. for each x P g, ∇ x : E ÝÑ E is a degree zero map that satisfies ∇ x pf eq " f ∇ x peq `ϱpxqrf se, for f P O, e P ΓpEq.

such that xΦ 0 pxq p0q pαq, ey " ϱpxqrxα, eys ´xα, ∇ x peqy, for all α P ΓpE ˚q, e P ΓpEq. (8.9) Φ 0 pxq p0q stands for the polynomial-degree zero of Φ 0 pxq.

Proof. Using Lemma 6.1.7, we have for every x P g, and e P ΓpEq, rΦ 0 pxq, ι e s p´1q " ι ∇xe , for some K-bilinear map ∇ x : ΓpE ´'q ÝÑ ΓpE ´'q that depends linearly on x P g and that satisfies ∇ x pf eq " f ∇ x peq `ϱpxqrf se, for f P O, e P ΓpEq. (8.10)

To see (8.10), compute rΦ 0 pxq, ι f e s p´1q : ι ∇xpf eq " rΦ 0 pxq, ι pf eq s p´1q

" Φ 0 pxqrf sι e `f rΦ 0 pxq, ι e s p´1q

" ι ϱpxqrf se`∇xe .

In particular, one has for all α P ΓpE ˚q, e P ΓpEq, xΦ 0 pxq p0q pαq, ey " Φ 0 pxq p0q rxα, eys ´rΦ 0 pxq p0q , ι e s p´1q pαq " ϱpxqrxα, eys ´xα, ∇ x peqy.

Homotopies

The following is a particular case of Definition 2.3.14 of Section 2.3.2. We rewrite it in this special context for the sake of clarity.

Definition 8.2.6. Let Φ, Ψ : pS ' K g, Q g q ù `S' K pXpEqr1sq, Q˘b e Lie 8-morphisms. We say Φ, Ψ are homotopic over the identity of M if the following conditions hold:

1. there a piecewise rational continuous path t P ra, bs Þ Ñ Ξ t : pS ' K g, Q g q ù `S' K pXpEqr1sq, Q˘m ade of Lie 8-morphisms that coincide with Φ and Ψ at t " a and b, respectively, 2. and a piecewise rational path t P ra, bs Þ Ñ H t of Ξ t -co-derivations of degree ´1 such that dΞ t dt " Q ˝Ht `Ht ˝Qg . (8.11) Remark 8.2.7. Homotopy equivalence in the sense of the Definition 8.2.6 is an equivalence relation, and it is compatible with composition of Lie 8-morphisms. Also, we "glue" infinitely many equivalences, as in Lemma 2.3.22.

Convention 8.2.8. In the sequel, Q g and Q will be in implicit.

Remark 8.2.9. Definition 8.2.6 is slightly more general than the equivalence relation [START_REF] Rajan | L 8 -algebra actions[END_REF]. In [START_REF] Rajan | L 8 -algebra actions[END_REF], it is explained that Lie 8-oid morphisms are Maurer-Cartan elements in some Lie 8-algebroid

g'E of certain form, and they define equivalence as gauge-equivalence of the Maurer-Cartan elements. This gauge equivalence corresponds to homotopies as above for which all functions are smooth. Also, we do not require nilpotence unlike in Definition 5.1 of [START_REF] Rajan | L 8 -algebra actions[END_REF]. Last, we do not assume g to be of finite dimension.

Proposition 8.2.10. Let g be a Lie algebra and pE, Qq a Lie 8-algebroid over M .

1. Any Lie 8-morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q induces a weak symmetry action of g on the basic singular foliation F " ρpΓpE ´1qq of pE, Qq.

2. Homotopic Lie 8-morphisms Φ, Ψ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q induce equivalent weak symmetry actions ϱ a , ϱ b of g on the basic singular foliation F.

Proof. Item 1. is a consequence of Remark 8.2.3. Indeed, take ϱ : g ÝÑ XpM q as in Lemma 8.2.5 (1.) We claim that ϱ is a weak symmetry action of g on F: Let x, y P g, and e P ΓpE ´1q and f P O.

• rΦ 0 pxq, Qs " 0 entails,

A Φ 0 pxq p0q " Q p1q pf q ı , e E " A Q p1q ´Φ0 pxq p0q rf s ¯, e E
ϱpxqrxQrf s, eys ´xQrf s, ∇ x peqy " ρpeqrϱpxqs, (by Lemma 8.2.5 (2.))

ϱpxqrρpeqsrf s ´ρp∇ x peqqrf s " ρpeqrϱpxqs By consequence, rϱpxq, ρpeqs " ρp∇ x peqq P F. Therefore, rϱpxq, Fs Ď F.

• By Lemma 6.1.7, there exists a skew-symmetric linear map η : ^2 g ÝÑ ΓpE ´1q such that Φ 1 px, yq p´1q " ι ηpx,yq . Therefore, the polynomial-degree zero of Equation (8.8) evaluated at an arbitrary function f P O yields: Φ 0 prx, ys g q p0q pf q ´rΦ 0 pxq, Φ 0 pyqs p0q pf q " rQ, Φ 1 px, yqs p0q pf q ùñ Φ 0 prx, ys g qpf q ´"Φ 0 pxq p0q , Φ 0 pyq p0q ı pf q " " Q p1q , Φ 1 px, yq p´1q ı pf q ùñ ϱprx, ys g qrf s ´rϱpxq, ϱpyqsrf s "

" Q p1q , ι ηpx,yq ı pf q
" ρpηpx, yqqrf s.

Since f is arbitrary, this proves item 1. Using Proposition 2.3.17, Φ " Ψ implies for x P g that Ψpxq ´Φpxq " Q ˝Hpxq ` H ˝Qg pxq " rQ, Hpxqs (8.12)

with H : g ÝÑ X ´1pEq a linear map. Let β : g ÝÑ ΓpE ´1q be a linear map such that Hpxq p´1q " ι βpxq . Taking the polynomial-degree zero of both sides in Equation (8.12) and evaluating at f P O we obtain that pϱ a pxq ´ϱb pxqq rf s "

" Q p1q , Hpxq p´1q ı " " Q p1q , ι βpxq ı rf s " ρpβpxqqrf s.
Since f is arbitrary, this proves item 2.

Proposition 8.2.10 tells us that Lie 8-morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q induces weak symmetry action on the base manifold M . The aim of the next section is to look at the opposite direction. It responds to the following question: Do any weak symmetry action of a Lie algebra on a singular foliation comes from a Lie 8-morphism? If so, can we extend uniquely?

Now we define what we call "lift" of a weak symmetry action of a Lie algebra g on a singular foliation F to a Lie 8-algebroid pE, Qq over F. Definition 8.2.11. Let F be a singular foliation over M and pE, Qq a Lie 8-algebroid over F. Consider a weak symmetry action ϱ : g ÝÑ XpM q of g on F.

• We say that a Lie 8-morphism of differential graded Lie algebras Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q lifts the weak symmetry action ϱ to pE, Qq if for all x P g, f P O, Φ 0 pxqpf q " ϱpxqrf s.

• When Φ exists, we say then Φ is a lift of ϱ on pE, Qq.

Main statements

We now state the main theorem of this chapter. Proposition 8.2.10 showed that a Lie 8-morphism between a Lie algebra g and a Lie 8-algebroid pE, Qq induces a weak action of g on the basic foliation. In this section, we show that any weak symmetry action of a Lie algebra g on a singular foliation F arises this way.

More precisely, Theorem 8.3.1. Let F a be a singular foliation over a smooth manifold (or an affine variety) M and g a Lie algebra. Let ϱ : g ÝÑ XpM q be a weak symmetry action of g on F. The following assertions hold:

1. for any universal Lie 8-algebroid pE, Qq of the singular foliation F, there exists a Lie 8morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q that lifts ϱ to pE, Qq, for some linear map ^3g ÝÑ X ´2pEq. These sets of compatibility conditions continue to higher multilinear maps.

any two such

Corollary 8.3.4. Any symmetry X P XpM q of the singular foliation F can be lifted to a degree zero vector field Z P X 0 pEq that commutes with Q, i.e. such that rZ, Qs " 0.

Proof. To construct Z, it suffices to apply Theorem 8.3.1 for g " R and take Z to be the image of 1 through Φ 0 : R ÝÑ X 0 pEq.

Remark 8.3.5. In particular, Corollary 8.3.4 has the following consequences:

1. for any admissible t, the flow Φ Z t : E ÝÑ E of Z induces an isomorphism of vector bundles E ´1 ÝÑ E ´1. Since rQ, Zs " 0, the following diagram commutes,

ΓpE ´1q ρ pΦ Z t q p0q / / ΓpE ´1q ρ XpM q pφ X t q˚/ / XpM q
where ϕ X t is the flow of X at t.

Consequently

, for any open set U Ă M which is stable under φ X t , there exists an invertible matrix M t X with coefficients in OpU q that satisfies

`ϕX t ˘˚¨X 1 . . . X n ‹ ‹ ' " M t X ¨X1 . . . X n ‹ ‹ ' ,
for some generators X 1 , . . . , X n of F over U . As announced earlier, we recover Proposition 8.1.2, that is, `ϕX t ˘˚pFq " F.

Let pE, Qq and pE 1 , Q 1 q be two universal Lie 8-algebroids of F. A direct consequence of Ricardo Campos's Theorem 4.1 in [START_REF] Campos | Homotopy equivalence of shifted cotangent bundles[END_REF] is that the differential graded Lie algebras pX ' pEqr1s, r¨, ¨s , ad Q q and `X' pEqr1s, r¨, ¨s , ad Q 1 ˘are homotopy equivalent over the identity of M . This leads to the following statement.

Corollary 8.3.6. Let ϱ : g ÝÑ XpM q be a weak symmetry action of a Lie algebra g on F. Then, there exist Lie 8-morphisms, Φ : g ù pX ' pEqr1s, r¨, ¨s , ad Q q and Ψ : g ù `X' pE 1 qr1s, r¨, ¨s , ad Q 1 ˘that lift ϱ, and Φ, Ψ make the following diagram commute up to homotopy

g Φ w w Ψ ( ( pX ' pEqr1s, r¨, ¨s , ad Q q o o " / / `X'

Proof of 8.3.1

This section is devoted to the proof of the main results, i.e. Theorem 8.3.1.

Let F be a singular foliation, and pE, Qq a universal Lie 8-algebroid of F. We start with the following lemma. Lemma 8.3.7. For every weak symmetry Lie algebra action of g on F there exists a linear map, Φ 0 : g Ñ X 0 pEq, such that rQ, Φ 0 pxqs " 0 and Φ 0 pxqrf s " ϱpxqrf s for all x P g, f P O.

Proof. For x P g, let y ϱpxq P X 0 pEq be any arbitrary extension of ϱpxq P spFq to a degree zero vector field on E. Since ϱpxq is a symmetry of F, the degree `1 vector field r y ϱpxq, Qs is also a longitudinal vector field on E, see Example 6.2.6 item 3. In addition, r y ϱpxq, Qs is a ad Q -cocycle. By item 1 of Theorem 6.2.10, there exists a vertical vector field Y pxq P apEq of degree zero such that rQ, Y pxq `y ϱpxqqs " 0.

(8.17)

Let us set for x P g, Φ 0 pxq :" Y pxq `y ϱpxq. By construction, we have, rQ, Φ 0 pxqs " 0 and Φ 0 pxqrf s " ϱpxqrf s for all x P g, f P O.

We will need the following lemma. Lemma 8.3.8. Assume pE, Qq is a universal Lie 8-algebroid over M . Let Φ : pS ' K g, Q g q ÝÑ pS ' K XpEqr1s, Qq be a coalgebra morphism which is a Lie 8-morphism up to polynomial-degree n ě 0, i.e. `Φ ˝Qg ´Q ˝Φ ˘piq " 0 for all integer i P t0, . . . , nu.

Then, Φ can be lengthened to a 8-morphism up to polynomial-degree n `1.

Proof. For convenience, we omit the variables x P S ' K g. Since ϱprx, ys g q ´rϱpxq, ϱpyqs P F for all x, y P g, and since ρ : ΓpE ´1q ÝÑ F surjective, we have ϱprx, ys g q ´rϱpxq, ϱpyqs " ρ pηpx, yqq for some element ηpx, yq P ΓpE ´1q depending linearly on x and y. Now we consider the vertical vector field of degree ´1, ι ηpx,yq P X ´1pU F q which is defined on ΓpE ˚q as: where the first term (resp. the second term) is cancelled by O-linearity of Q p0q (resp. for degree reason). Hence, by Corollary 6.2.11 and Remark 6.2.12, the degree zero vector field Λpx, yq `rQ, ι ηpx,yq s is of the form rQ, Υpx, yqs for some vertical vector field Υpx, yq P X ´1pEq of degree ´1 with Υpx, yq p´1q " 0. For all x, y P g, we define the Taylor coefficient Φ 1 : ^2 g ÝÑ XpEq as Φ 1 px, yq :" Υpx, yq `ιηpx,yq . By construction, we have the following relation, Φ 0 prx, ys g q ´rΦ 0 pxq, Φ 0 pyqs " rQ, Φ 1 px, yqs, @x, y P g (8.20)

ι ηpx,
Consider for x, y, z P g, ϑpx, y, zq " Φ 1 prx, ys g , zq ´rΦ 0 pxq, Φ 1 py, zqs`ö px, y, zq. (8.21)

Here, ö px, y, zq stands for circular permutation of x, y and z with Koszul sign. For degree reason ϑpx, y, zq is O-linear. Moreover, ϑpx, y, zq is a ad Q -cocycle:

rQ, Φ 1 prrx, ys g , zs g qs `ö px, y, zq " ´rΦ 0 prx, ys g q , Φ 0 pzqs `ö px, y, zq " rrΦ 0 pzq, Qs, Φ 1 px, yqs ´rrΦ 1 px, yq, Φ 0 pzqs, Qs`ö px, y, zq " rQ, rΦ 0 pxq, Φ 1 py, zqss`ö px, y, zq.

Here, we have used the fact that rQ, Φ 0 pxqs " 0 for all x P g, and the Jacobi identity for the Lie brackets r¨, ¨sg and r¨, ¨s. By Corollary 6.2.11, there exists a derivation of degree ´2 denoted by Φ 2 px, y, zq P X ´2pEqr1s that satisfies, ϑpx, y, zq " rQ, Φ 2 px, y, zqs. (8.22) So far, in the construction of the Lie 8-morphism, we have shown the existence of a Lie 8morphism Φ : S ' K g ÝÑ S ' K pXpEqr1sq up to polynomial-degree 2 that is p Φ ˝Qg q piq " p Q ˝Φq piq with i " 0, 1, 2. The proof continues by recursion or by applying directly Lemma 8.3.8. This proves the part 1. of the theorem.

Before proving item 3 of Theorem 8.3.1 we will need the following lemma. For convenience, we sometimes omit the variables in g. Lemma 8.3.9. For any two Lie 8-morphisms Γ, Ω : pS ' K g, Q g q ù pS ' K pXpEqr1sq, Qq which coincide up to polynomial-degree n ě 1, i.e. Γ piq " Ω piq , for 0 ď i ď n, their difference in polynomial-degree n `1, namely, Γ pn`1q ´Ωpn`1q : S n`2 K g ÝÑ X ´n´1 pEqr1s is valued in ad Q -coboundary.

Proof. Indeed, a direct computation yields Q ˝pΓ ´Ωq " pΓ ´Ωq ˝Qg ùñ Qp0q ˝pΓ ´Ωq pn`1q ´ppΓ ´Ωq ˝Qg q pn`1q looooooooooomooooooooooon "0 " 0 ùñ rQ, Γ pn`1q ´Ωpn`1q s " 0 ùñ Γ pn`1q ´Ωpn`1q " rQ, H pn`1q s (by item 1 of Theorem 6.2.10) for some linear map H pn`1q : S n`2 K g ÝÑ X ´n´2 pEqr1s.

Let us show item 2 of Theorem 8.3.1. Let Φ, Ψ : g ÝÑ XpEqr1s be two different lifts of the action g ÝÑ XpM q. We denote by Φ, Ψ : S ' K g ÝÑ S ' K pXpEqr1sq the unique comorphisms given respectively by the Taylor coefficients

$ & % Φprq : S r`1 K g Φr ÝÑ X ´rpEqr1s Ψprq : S r`1 K g Ψr Ý Ý Ñ X ´rpEqr1s
, for r ě 0. (8.23)

For any x P g, the degree zero vector field Φ 0 pxq ´Ψ0 pxq P X 0 pEq is vertical. Moreover, we have, rQ, Φ 0 pxq ´Ψ0 pxqs " 0. By Corollary 6.2.11 there exists a vector field H 0 P X ´1pEq of degree ´1, such that Ψ 0 pxq ´Φ0 pxq " rQ, H 0 pxqs Here H t is the extension of H p1q as the unique Θ t -co-derivation where all its arities vanish except the polynomial-degree 1 which is given by H p1q . In polynomial-degree zero, pΘ p0q t q tPr0,1s is constant and has value Θ Hence, Θ p1q t " Φp1q `tp Ψp1q ´Ξp1q 1 q with Ψpiq " Θ piq 1 for i " 0, 1. We then continue this procedure by gluing all these homotopies as in the proof of item 2 of Theorem 4.2.4. We obtain at last a Lie 8-morphism Ω such that Φ " Ω and Ω piq " Ψpiq for i ě 0. That means Ω " Ψ, therefore Φ " Ψ. This proves item 2. of Theorem 8.3.1.

g Ψ 0 ´Φ0 H 0 w w X ´1pEqr1s ad Q / / X
Let us prove item 3 of Theorem 8.3.1. Given two equivalent weak symmetry actions ϱ, ϱ 1 of g on a singular foliation F, i.e. ϱ, ϱ 1 differ by a linear map g ÝÑ XpM q of the form x Þ Ñ ρpβpxqq for some linear map β : g ÝÑ ΓpE ´1q. Let Φ, Φ 1 : g ù pX ' pEqr1s, r¨, ¨s , ad Q q be a lift into a Lie 8-morphism of the action ϱ and ϱ 1 respectively. One has for all x P g and f P O, `Φ0 pxq ´Ψ0 pxq ´rQ, ι φpxq s ˘pf q " ρpφpxqqrf s ´xQpf q, φpxqy " 0.

Since rQ, Φ 0 pxq ´Ψ0 pxq ´rQ, ι φpxq ss " 0, by Corollary 6.2.11 there exists a vertical derivation p Hpxq P X ´1pEq of degree ´1 depending linearly on x P g such that Φ 0 pxq ´Ψ0 pxq " rQ, p Hpxq `ιφpxq s.

Let Hpxq :" p Hpxq `ιφpxq , for x P g. The proof continues the same as for item 2 of Theorem 8.3.1

Particular examples

We recall that for a regular foliation F on a manifold M , the Lie algebroid T F Ă T M , whose sections form F, is a universal Lie 8-algebroid of F. Its corresponding Q-manifold is given by the leafwise De Rham differential on Γp^'T ˚F q.

Example 8.3.10. Let F be a regular foliation on a manifold M . Any weak symmetry action g ÝÑ XpM q, x Þ ÝÑ ϱpxq, of F, can be lifted to Lie 8-morphism Φ : g ù pX ' pEqr1s, r¨, ¨s , ad Q q given explicitly as follows:

x P g Þ ÝÑ Φ 0 pxq " L ϱpxq P X 0 p^'T ˚F q (8.28)

x ^y P ^2g Þ ÝÑ Φ 1 px, yq " ι χpx,yq P X ´1p^'T ˚F q (8.29)

and `Φi : ^i`1 g ÝÑ X ´ip^'T ˚F q ˘" 0, for all i ě 2, where χpx, yq :" ϱprx, ys g q ´rϱpxq, ϱpyqs for

x, y P g. Also, L X stands for the Lie derivative on multi-forms w.r.t X P XpM q, and ι X is the internal product.

Example 8.3.11. Let F be a singular foliation on a manifold M together with a strict symmetry action ϱ : g ÝÑ XpM q such that g Ă F. Hence, C 8 pM qg is a singular foliation which is the image of the transformation Lie algebroid g ˆM . The universality theorem (see [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF][START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF]) provides the existence of a Lie 8-morphism ν : g ÝÑ U F . Let us call its Taylor coefficients ν n : ^n`1 g ÝÑ E ´n´1 , n ě 0. We may take for example the 0-th and 1-th Taylor coefficients of a Lie 8-morphism that lifts ϱ as: Φ 0 pxq :" rQ, ι ν 0 pxq s P X 0 pU F q, for x P g.

Φ 1 px, yq :" rQ, ι ν 1 px,yq s p´1q ´ÿ kě0 rrQ, ι ν 0 pxq s, ι ν 0 pyq s pkq P X ´1pU F q, for x, y P g.

Note that in this case the action ϱ is equivalent to zero, therefore by item 3 of Theorem 8.3.1 the Lie 8-morphism Φ is homotopic to zero.

Lifts of weak symmetry actions and Lie 8-algebroids

In this section, g is a finite dimensional Lie algebra that we see as the trivial vector bundle over M with fiber g.

The following theorem says that any lift of strict symmetry action of g on a singular foliation F induces a Lie 8-algebroids with some special properties and vice versa. See [START_REF] Rajan | L 8 -algebra actions[END_REF], Prop. 3.3, for a proof of the following statement. Proposition 8.4.1. Let pE, Qq be a Lie 8-algebroid over a singular foliation F. Any Lie 8-morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q with g of finite dimension induces a Lie 8-algebroid pE 'g,

Q 1 q with Q 1 :" d CE `Q `ÿ kě1,i 1 ,...,i k "1,...,dimpgq 1 k! ξ i 1 d ¨¨¨d ξ i k Φ k´1 pξ i 1 , . . . , ξ i k q, (8.30)
where d CE is the Chevalley-Eilenberg complex of g, and ξ 1 , . . . , ξ dimpgq P g ˚is the dual basis of some basis ξ 1 , . . . , ξ dimpgq P g and for all k ě 0, Φ k : S k`1 g ÝÑ X ´kpEqr1s is the k-th Taylor coefficients of Φ.

In the dual point of view, (8.30) corresponds to a Lie 8-algebroid over the complex

¨¨¨ℓ 1 ÝÑ E ´3 ℓ 1 ÝÑ E ´2 ℓ 1 ÝÑ g ' E ´1 ρ 1 ÝÑ T M (8.31)
whose brackets satisfy 1. the anchor map ρ 1 sends an element x ' e P g ' E ´1 to ϱpxq `ρpeq P ϱpgq `T F,

the binary bracket satisfies

ℓ 2 pΓpE ´1q, ΓpE ´1qq Ă ΓpE ´1q and ℓ 2 pΓpE ´1q, xq Ă ΓpE ´1q, @ x P g 3. the g-component of the binary bracket on constant sections of g ˆM is the Lie bracket of g.

Conversely, if there exists a Lie 8-algebroid pE 1 , Q 1 q whose underlying complex of vector bundles is of the form (8.31) and that satisfies item 1, 2 and 3, then there is a Lie 8-morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q which is defined on a given basis ξ 1 , . . . , ξ d of g by:

Φ k´1 pξ i 1 , . . . , ξ i k q " pr ˝r¨¨¨rrQ 1 , ι ξ i 1 s, ι ξ i 2 s, . . . , ι ξ i k s Ă XpEqr1s, k P N, (8.32)
where pr stands for the projection map XpE 1 qr1s ÝÑ XpEqr1s.

Proof. We explain the idea of the proof. A direct computation gives the first implication. Conversely, let us denote by Q 1 the homological vector fields of Lie 8-algebroid whose underlying complex of vector bundles is of the form (8.31). The map defined in Equation (8.32) is indeed a lift into a Lie 8-morphism of the weak symmetry action ϱ:

• It is not difficult to check that, for any ξ P g, one has rQ, Φ 0 pξqs " 0.

• The fact that Φ defines a Lie 8-morphism can be found using Voronov trick [START_REF] Voronov | Higher derived brackets for arbitrary derivations[END_REF][START_REF] Voronov | Higher derived brackets and homotopy algebras[END_REF], i.e, doing Jacobi's identity inside the null derivation 0 " pr ˝r¨¨¨rrrQ 1 , Q 1 s, ι ξ i 1 s, ι ξ i 2 s, . . . , ι ξ i k s. (8.33)

A direct computation of Equation (8.33) falls exactly on the requirements of Definition 8.2.1.

Remark 8.4.2. Let us compute Equation (8.33) for a small number of generators (e.g k " 2, 3) in order to show how it works: from the identity

"" " Q 1 , Q 1 ‰ , ι ξ i 1 ı , ι ξ i 2
ı " 0, one obtains by using twice the Jacobi identity the following relation,

" Q 1 , "" Q 1 , ξ i 1 ‰ , ξ i 2 ‰‰ ´"" Q 1 , ξ i 1 ‰ , " Q 1 , ξ i 2 ‰‰ " 0. (8.34)
One should notice that rrQ 1 , ξ i 1 s , ξ i 2 s splits into two parts. One part where the Chevalley-Eilenberg acts to give ""

d CE , ξ i 1 ‰ , ξ i 2 ‰ " ι rξ i 1 ,ξ i 2 sg , while the other part is "" Q 1 ´dCE , ξ i 1 ‰ , ξ i 2 ‰
. Hence, by putting them in Equation (8.34), afterwards projecting on X pS ' pE ˚qq, we get

pr ˝"Q 1 , ι rξ i 1 ,ξ i 2 sg ı `pr ˝rQ 1 , "" Q 1 ´dCE , ξ i 1 ı , ξ i 2 ı s ´pr ˝"" Q 1 , ξ i 1 ‰ , " Q 1 , ξ i 2 ‰‰ " 0.
From here, we deduce that

Φ 0 prξ i 1 , ξ i 2 s g q " rQ, Φ 1 pξ i 1 , ξ i 2 qs `rΦ 0 pξ i 1 q, Φ 0 pξ i 2 qs.
Here is an application of Theorem 8.4.1.

Corollary 8.4.3.

1. Let g be a Lie algebra and G its Lie group.

2. Let pM, Fq a singular foliation together with a weak symmetry action ϱ : g ÝÑ XpM q.

The following assertions hold:

1. On G ˆM the C 8 pG ˆM q-module generated by Let pE, Qq be a universal Lie 8-algebroid structure of pM, Fq. Let Φ k : ^k`1 g ÝÑ X ´kpEq be the Taylor coefficients of a Lie 8-morphism g ù XpEq that lifts ϱ. Then

$ & % p Ð Ý u ,
Q 1 :" d G dR `Q `ÿ kě1,i 1 ,...,i k "1,...,dimpgq 1 k! ξ i 1 d ¨¨¨d ξ i k Φ k´1 pξ i 1 , . . . , ξ i k q, (8.37)
with pξ 1 , . . . , ξ dimpgq q, pξ 1 , . . . , ξ dimpgq q be dual basis of g and g 1.

is a universal Lie 8-algebroid of pXpGq ˆϱ Fq 2. whose coefficients are left invariant for the action of G on G ˆM given by g ¨ph, mq :" pgh, mq.

Conversely, a left invariant Lie 8-algebroid on (8.36) can be interpreted as Taylor coefficients of a lift of ϱ.

A more general statement of Proposition 8.4.1

We end the chapter with a generalization of Proposition 8.4.1.

In the previous section, Proposition 8.4.1 is stated in the finite dimensional context, i.e. it needs g to be finite dimensional and the existence of a geometric resolution for the singular foliation F. In this section we prove that given a weak symmetry action of a Lie algebra g (may be of infinite dimensional) on a Lie-Rinehart algebra F Ă XpM q (we do not require F being locally finitely generated), such Lie 8-algebroid described at the sections level of the complex (8.31) as Proposition 8.4.1 exists.

We state the following theorem in the context of singular foliations, but the same statement and the same proof are valid word-by-word by replacing F by a Lie-Rinehart algebra. Theorem 8.4.4. Let g be a (possibly infinite dimensional) Lie K-algebra and let ϱ : g ÝÑ XpM q be a weak symmetry action of g on a singular foliation F. Let ppK ´iq iě1 , d, ρq be a free resolution of the singular foliation F over O. The complex of trivial vector bundles over M Proof. (of Theorem 8.4.4) The complex of Equation (8.38) being exact everywhere except in degree ´1 we cannot apply directly Theorem 2.1 in [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF] but we can mimic the proof given for Theorem 2.1 in [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF] to construct the higher brackets when there is no obstruction in degree ´1. For convenience, let us denote R ´1 :" Γpgq ' ΓpE ´1q and R ´i :" ΓpE ´iq for i ě 2. Given a natural number k ě 0, we consider the total complex ˆz Page pkq ' pRq, D " rd, ¨sRN ˙of the following bicomplex . . . . . . . . .

¨¨¨d ÝÑ E ´3 d ÝÑ E ´2 d ÝÑ g ' E ´1 ρ 1 ÝÑ T M (8.
Ò Ò Ò ¨¨¨Ñ Hom O ´Äk`1 R | ´k´3 , R ´3¯d Ñ Hom O ´Äk`1 R | ´k´3 , R ´2¯d Ñ Hom O ´Äk`1 R | ´k´3 , dR ´2¯Ñ 0 δ Ò δ Ò δ Ò ¨¨¨Ñ Hom O ´Äk`1 R | ´k´2 , R ´3¯d Ñ Hom O ´Äk`1 R | ´k´2 , R ´2¯d Ñ Hom O ´Äk`1 R | ´k´2 , dR ´2¯Ñ 0 δ Ò δ Ò δ Ò ¨¨¨Ñ Hom O ´Äk`1 R | ´k´1 , R ´3¯d Ñ Hom O ´Äk`1 R | ´k´1 , R ´2¯d Ñ Hom O ´Äk`1 R | ´k´1 , dR ´2¯Ñ 0 Ò Ò Ò 0 0 0 "-3 column" "-2 column" "-1 column"
(8.39) The map δ stands for the vertical differential which is defined for all Φ P Hom O ´Äk`1 R, R ¯by δpΦq pr 1 , . . . , r k`1 q :" Φ ˝d pr 1 d . . . d r k`1 q, @ r 1 , . . . , r k`1 P R, where here d acts as an O-derivation on r 1 d . . . d r k`1 P Ä k R and the horizontal differential given by Φ Þ Ñ d ˝Φ.

Since the line the bicomplex is exact, the total complex ˆz Page pkq ' pRq, D " rd, ¨sRN ˙is also exact.

Construction of the 2-ary bracket: its construction is almost the same as in [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF] we adapt what has been done to our case. We first construct a 2-ary bracket on R ´1 to extend on every degree. For all k ě 1, let us denote by pe p´kq i q iPI k a basis of ΓpE ´kq. The set tX i " ρpe p´1q i q P F | i P I 1 u is a set of generators of F. In particular, there exists elements c k ij P O and satisfying the skew-symmetry condition c k ij " ´ck ji together with

rX i , X j s " ÿ kPI c k ij X k @i, j P I 1 . (8.40)
By definition of weak symmetry one has rϱpξ i q, ρpe p´1q j qs P F and ϱprξ i , ξ j sq g ´rϱpξ i q, ϱpξ j qs P F for all pi, jq P I g ˆI´1 . (8.41)

Here, pξ i q iPIg is a basis for g. Since ppK ´iq iě1 , d, ρq is a free resolution of F, there exists two O-bilinear maps χ : Γpgq ˆΓpE ´1q Ñ ΓpE ´1q, η : Γpgq ˆΓpgq Ñ ΓpE ´1q defined on generators ξ i , e p´1q j by the relations rϱpξ i q, ρpe p´1q j qs " ρpχpξ i , e p´1q j qq and ϱprξ i , ξ j s g q ´rϱpξ i q, ϱpξ j qs " ρpηpξ i , ξ j qq.

We first define a naive 2-ary bracket on ΓpE ´1q as follows:

1. an anchor map by ρ 1 pe p´1q i q " X i , and ρ 1 pξ i q " ϱpξ i q, for all i P I, I g , 2. a degree `1 graded symmetric operation l2 on R ' as follows: ¯, (c) l2 pξ i , ξ j q " rξ i , ξ j s g `ηpξ i , ξ j q, (d) r ℓ 2 is zero on the other generators, (e) we extend l2 to R using O-bilinearity and Leibniz identity with respect to the anchor ρ 1 .

By paq, pbq, pcq, pdq, peq, l2 satisfies the Leibniz identity with respect to the anchor r ρ and paq, pbq, pcq makes the latter a bracket morphism. The map defined for all homogeneous r 1 , r 2 P R ' by rd, l2 s RN pr 1 , r 2 q " d ˝l 2 pr 1 , r 2 q `l 2 pdr 1 , r 2 q `p´1q |r 1 | l2 pr 1 , dr 2 q , (8.42) is a graded symmetric degree `2 operation pR b Rq ' ÝÑ R '`2 , and rd, l2 s RN | R ´1 " 0. It is O-bilinear, i.e. for all f P O, r 1 , r 2 P R rd, l2 s RN pr 1 , f r 2 q ´f rd, l2 spr 1 , r 2 q " 0.

We also have that ρprd, l2 s RN pr 1 , f r 2 qq " ρp l2 pdr 1 , r 2 qq " 0, for all r 1 P R ´2, r 2 P R ´1, since ρ ˝d " 0. ´Ä2 R ´j´1 , R ´j ¯such as Dpτ 2 q " ´rd, l2 s RN . By replacing l2 by l2 `τ2 we get a 2ary bracket ℓ 2 of degree `1 which is compatible with the differential map d and the anchor map r ρ.

Construction of higher brackets: notice that by construction of the 2-ary bracket ℓ 2 one has, Jacpr 1 , r 2 , r 3 q P dR ´2 for all r 1 , r 2 , r 3 P R ´1. In other words, Jac P Hom O p Ä 3 R ´1, dR ´2q. A direct computation shows dJacpr 1 , r 2 , r 3 q " Jacpdr 1 , r 2 , r 3 q `p´1q |r 1 | Jacpr 1 , dr 2 , r 3 q `p´1q |r 1 |`|r 2 | Jacpr 1 , r 2 , dr 3 q for all r 1 , r 2 , r 3 P R. Which means, rJac, ds RN pr 1 , r 2 , r 3 q " 0 for all r 1 , r 2 , r 3 P R. Thus, DpJacq " 0. It follows that, Jac is a D-coboundary, there exists an element ℓ 3 " ř jě2 ℓ j 3 P z Page p2q 1 pRq with ℓ j 3 P Homp Ä 3 R | ´j´1 , R ´j q such that Dpℓ 3 q " ´Jac.

(8.43)

We choose the 3-ary bracket to be ℓ 3 . For degree reason, the remaining terms of the k-ary brackets for k ě 3 have trivial components on the column ´1 of the bicomplex (8.39). From this point, the proof continues exactly as in Section 4.3.2.

Example 8.4.6. We return to Example 8.1.8. In that case, the Lie algebra g " XpLq that acts on the singular foliation T . In that case, we have ϱpgq X T m T " t0u for all m in rL, M s. We can apply directly Theorem 4.2.1, to obtain a Lie 8-algebroid structure on the complex

¨¨¨d ÝÑ ΓpE ´3q d ÝÑ ΓpE ´2q d ÝÑ g ' ΓpE ´1q ρ 1 ÝÑ XprL, M sq. (8.44)
Notice that here the Lie algebra g is infinite dimensional, therefore we are not allowed to use the duality between Lie 8-algebroid and Q-manifold. Therefore, we cannot use the explicit Formula (8.32) to define at lift Φ. However, we have to rely on the existence theorem 8.3.1 to assure the existence of a lift Φ.

Conclusion:

We show that actions of a Lie algebra g on the leaf space M {F i.e. weak symmetry actions, lift to Lie 8-algebra morphisms g ù XpEq on the DGLA of vector fields on an universal Lie 8-algebroid pE, Qq, provided it exists, in a unique up to homotopy manner.

We explain how to use chapter 4 to get rid of all finite ranks/dimensions assumptions, using the universal Lie 8-algebroids of Lie-Rinehart algebras.

2. A direct computation gives 0 " rrQ, Φ 0 pxqs , ι e s p´1q " rQ, rΦ 0 pxq, ι e ss p´1q ´rΦ 0 pxq, rQ, ι e ss p´1q " " Q p0q , rΦ 0 pxq, ι e s p´1q

ı

´"Φ 0 pxq p0q , rQ, ι e s p´1q ı "

" Q p0q , ι ∇xpeq ı ´"Φ 0 pxq p0q , ι ℓ 1 peq ı " ι ℓ 1 ˝∇xpeq ´ι∇x˝ℓ 1 peq
We have used graded Jacobi identity and the dual correspondence between Lie 8-algebroids and N Q-manifolds (see Proposition 6.1.16). We recapitulate in the following commutative diagram:

¨¨¨d / / ΓpE ´2q d / / ∇x ΓpE ´1q ρ / / ∇x F ad ϱpxq ¨¨¨d / / ΓpE ´2q d / / ΓpE ´1q ρ / / F (9.2)
which means, ℓ 1 ˝∇x " ∇ x ˝ℓ1 and ρ ˝∇x " ad ϱpxq ˝ρ.

Here, ℓ 1 stands for the corresponding unary bracket of pE, Qq. Also, for X P XpM q, ad X :" rX, ¨s.

3. For x, y P g, and e P ΓpEq, the relation (8.14) yields rΦ 0 prx, ys g q ´rΦ 0 pxq, Φ 0 pyqs , ι e s p´1q " rrQ, Φ 1 px, yqs , ι e s p´1q ι ∇ rx,ysg peq ´rΦ 0 pxq, rΦ 0 pyq, ι e ss p´1q `rΦ 0 pyq, rΦ 0 pxq, ι e ss p´1q " " ι ∇ rx,ysg peq ´"Φ 0 pxq p0q , ι ∇ypeq ı `"Φ 0 pyq p0q , ι ∇xpeq ı " rrQ, Φ 1 px, yqs , ι e s p´1q .

x Þ Ñ ∇ x is simply x Þ Ñ ℓ 1 2 px, ¨q,
where ℓ 1 2 is the 2-ary bracket between sections of g and E of the Lie 8-algebroid pQ 1 , E ' gq over the complex

¨¨¨ℓ 1 ÝÑ E ´3 ℓ 1 ÝÑ E ´2 ℓ 1 ÝÑ g ' E ´1 ρ 1 ÝÑ T M (9.5)
like in Proposition 8.4.1. Indeed, for α P ΓpE ˚1q and e P ΓpE ´1q, Φ 0 pxq p0q " pr ˝rQ 1 , ι x s p0q

" pr ˝rQ 1p1q , ι x s.

This implies that:

xΦ 0 pxq p0q α, ey " xQ 1p1q α, x d ey " ρ 1 pxqrxα, eys ´ ρ 1 peqrxα, xys ´xα, ℓ 1 2 px, eqy " ϱpxqrxα, eys ´xα, ℓ 1 2 px, eqy.

An obstruction theory

Let us start with some generalities. Assume we are given

• a Lie algebra g,

• a weak symmetry action ϱ : g ÝÑ XpM q of g on a singular foliation F, together with η : ^2 g ÝÑ ΓpE ´1q such that x, y P g ϱprx, ys g q ´rϱpxq, ϱpyqs " ρpηpx, yqq. (9.6)

• an universal Lie 8-algebroid pE, Q E q of F, Theorem 8.3.1 assures ϱ : g Ñ XpM q admits a lift to a Lie 8-morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q. (9.7)

Equivalently, if g is of finite dimension, (9.7) corresponds (by Proposition 8.4.1) to a Lie 8-algebroid pE 1 , Q 1 q over M such that • pE, Q E q is included as a sub-Lie 8-algebroid in a Lie algebroid pE 1 , Qq over M ,

• its underlying complex is, E 1 ´1 :" g ' E ´1, and for any i ě 2, E 1 ´i " E ´i, namely

¨¨¨d ÝÑ E ´3 d ÝÑ E ´2 d ÝÑ g ' E ´1 ρ 1
ÝÑ T M, (9.8)

• we have, ℓ1 2 px ' 0, y ' 0q " rx, ys g ' ηpx, yq and ℓ 1 2 px, ΓpE ´1qq Ă ΓpE ´1q for all x P g.

Remark 9.2.1. It is important to note that the Lie 8-algebroid pE 1 , Q 1 q can be constructed directly out of the weak symmetry action ϱ : g ÝÑ XpM q, even if g is of infinite dimension (see Theorem 8.4.4).

Remark 9.2.2. In Equation (9.8), the complex pE, ℓ 1 q can be chosen to be minimal at a point m P M , i.e., ℓ 1| m " 0, provided that a geometric resolution of F exists. By Proposition 4.14 in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] the isotropy Lie algebra g m " F pmq ImF of the singular foliation F at the point m P M is isomorphic to kerpρ m q. This allows to denote the latter space also by g m .

In what follows, we will use the 8-algebroid which is described on the complex (9.8).

Lemma 9.2.3. Let m P M be a fixed point of the g-action ϱ. Assume that the underlying complex pE, ℓ 1 q is minimal at a point m, i.e. ℓ 1| m " 0. The map ν : g ÝÑ End pg m q , x Þ ÝÑ ℓ 1 2 px , ¨q|m satisfies (a) νprx, ys g q ´rνpxq, νpyqs `ℓ2 p ¨, ηpx, yqq |m " 0, (b) νpzq `ηpx, yq |m ˘´ηprx, ys g , zq |m `ö px, y, zq " 0.

Proof. The map ν is well-defined. The Jacobi identity on elements x, y P g, e P ΓpE ´1q, evaluated at the point m, implies that νprx, ys g qpe |m q ´rνpxq, νpyqspe |m q `ℓ2 pηpx, yq, eq |m " 0.

This proves item (a). Likewise, Jacobi identity on elements x, y, z P g and since ℓ 1| m " 0 give: ℓ 1 2 pℓ 1 2 px, yq, zq |m `ö px, y, zq " 0 ùñ ℓ 1 2 prx, ys g , zq |m `ℓ1 2 pηpx, yq, zq |m `ö px, y, zq " 0, ùñ νpzq `ηpx, yq |m ˘´ηprx, ys g , zq |m `ö px, y, zq " 0.

Here we have used the definition of ℓ 1 2 on degree ´1 elements and Jacobi identity for the bracket r¨, ¨sg . This proves item (b). By Lemma 9.2.3, g m is equipped with a g-module structure when ηpx, yq |m is for all x, y P g valued in the center the Lie algebra g m . Recall that η is defined by Equation (9.6). Notice that if m P M is a fixed point of the g-action ϱ, then this implies in particular that ηpx, yq| m P ker ρ m " g m . Proposition 9.2.4. Let m P M be a fixed point of the g-action ϱ. Assume that • the underlying complex pE, ℓ 1 q of pE, Qq is minimal at m,

• for all x, y P g, ηpx, yq |m is valued in the center 1 Zpg m q of g m . Then, 1. the restriction of the 2-ary bracket ℓ 1 2 : g b Zpg m q ÝÑ Zpg m q endows Zpg m q with a g-module structure which does not depend neither on the choices of weak symmetry action ϱ, a universal Lie 8-algebroid of F, nor of the Lie 8-morphism Φ : g ÝÑ XpEq.

2. the restriction of the map η : ^2 g ÝÑ ΓpE ´1q at m η |m : ^2 g ÝÑ Zpg m q is a 2-cocycle for the Chevalley-Eilenberg complex of g valued in Zpg m q,

the cohomology class of this cocycle does not depend on the representatives of the equivalence class of ϱ,

4. if ϱ is equivalent to a strict symmetry action, then η |m is exact.

Proof. pE, ℓ 1 q being minimal at m, ℓ 1 2 | m satisfies the Jacobi identity. In particular, for every x P gr1s, ℓ 1 2 px, ¨q| m preserves Zpg m q. By item paq of Lemma 9.2.3, the restriction of the 2-ary bracket

ℓ 1 2 : g b Zpg m q ÝÑ Zpg m q
endows Zpg m q with a g-module, since ℓ 2 p¨, ηpx, yqq| m " 0 by assumption. It is easy to see that if we change the action ϱ to ϱ `ρ ˝β for some vector bundle morphism β : g ÝÑ E ´1 such that β| m : g ÝÑ Zpg m q, the new 2-ary bracket between sections of gr1s and E ´1 constructed as in the proof of Theorem 8.4.4 is modified by px, eq Þ Ñ ℓ 1 2 px, eq `ℓ2 pβpxq, eq. The second term of the latter vanishes at m, by definition of β| m . As a result, the action of ℓ 1 2 on Zpg m q does not depend on the choices made in the construction. This proves item 1.

Item 2 follows from item (b) of Lemma 9.2.3 that tells that η |m : ^2 g ÝÑ Zpg m q is a 2-cocycle for the Chevalley-Eilenberg complex of g valued in Zpg m q.

Let ϱ 1 be a weak symmetry action of g on F which is equivalent to ϱ, i.e., there exists a vector bundle morphism β : g ÝÑ E ´1 (with β| m : g ÝÑ Zpg m q) such that ϱ 1 pxq " ϱpxq `ρpβpxqq for all x P g. Let η 1 : ^2 g ÝÑ E ´1 be such that ϱ 1 prx, ys g q ´rϱ 1 pxq, ϱ 1 pyqs " ρpη 1 px, yqq for all x, y P g. Following the constructions in the proof of Theorem 8.4.4, this implies that η 1 px, yq " ηpx, yq `βprx, ys g q ´ℓ1 2 px, βpyqq `ℓ1 2 py, βpxqq ´ℓ2 pβpxq, βpyqqq, for all x, y P g. (9.9) Equation (9.9) implies that η 1 px, yq| m ´ηpx, yq| m " d CE pβ| m qpx, yq, where d CE stands for the Chevalley-Eilenberg differential. As a consequence, η 1 | m and η| m define the same class in the Chevalley-Eilenberg complex of g valued in Zpg m q. This proves item 3 and 4.

Remark 9.2.5. When ℓ 2| m ‰ 0. The weak symmetry action ϱ is equivalent to strict one if the Maurer-Cartan-like equation (9.9) has no solution with η 1 |m " 0.

Let F be a singular foliation. Let us choose a universal Lie 8-algebroid pE, Qq such that pE, ℓ 1 q is minimal at a point m P M . Such a structure always exists (see Proposition B.2.14). By Proposition 4.14 in [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] the isotropy Lie algebra g m of the singular foliation F at the point m P M is isomorphic to kerpρ m q. The following is a direct consequence of Proposition 9.2.4. Corollary 9.2.6. Let m P M be a fixed point of the g-action ϱ. Assume that the isotropy Lie algebra g m of F at m is Abelian. Then, for any weak symmetry action ϱ of a Lie algebra action g on F such that ϱprx, ys g q ´rϱpxq, ϱpyqs P Fpmq for all x, y P g 1. g m is a g-module.

2. The bilinear map, η |m : ^2 g Ñ g m , is a Chevalley-Eilenberg 2-cocycle of g valued in g m .

3. Its class clpηq P H 2 pg, g m q does not depend on the choices made in the construction. 4. Furthermore, clpηq is an obstruction of having a strict symmetry action equivalent to ϱ.

Example 9.2.7. We return to Example 8.1.6 with m P M a leaf of F. Since the isotropy Lie algebra g k m is Abelian for every k ě 2 the following assertions hold by Corollary 9.2.6: 1. For each k ě 1, the vector space g k`1 m is a g k m -module.

2. The obstruction of having a strict symmetry action equivalent to ϱ k is a Chevalley-Eilenberg cocycle valued in g k`1 m .

Here is a particular case of this example.

Example 9.2.8. Let F :" I 3 0 XpR n q be the singular foliation generated by vector fields vanishing to order 3 at the origin. The quotient g :" I 2 0 XpR n q I 3 0 XpR n q is a trivial Lie algebra. There is a weak symmetry action of g on F which assigns to an element in g a representative in I 2 0 XpR n q. In this case, the isotropy Lie algebra of F at zero is Abelian and ℓ 1 2 pg, g 0 q | 0 " 0. Thus, the action of g on g 0 is trivial. One can choose η : ^2 g ÝÑ g 0 such that η ´x2

i B Bx i , x 2 i B
Bx j ¯" 2e ij , with e ij a constant section in a set of generators of degree ´1 whose image by the anchor is

x 3 i B Bx j . Therefore, η | 0 ´x2 i B Bx i , x 2 i B
Bx j ¯‰ 0. This implies that the class of η is not zero at the origin. Therefore, by item 2 of Corollary 9.2.6 the weak symmetry action of g on F is not equivalent to a strict one.

Example 9.2.9. Consider again the Example 8.4.6 and let pE, ℓ ' , ρq be a universal Lie 8-algebroid of T . Recall that a flat Ehresmann connection is a horizontal distribution whose sections are closed under the Lie bracket of vector fields. Let m P M . Assume in Example 8.1.8 that the section ϱ H : XpLq Ñ F proj satisfies ϱ H pra, bsq ´rϱ H paq, ϱ H pbqs " ρpηpa, bqq P T pmq for some bilinear map η : ^2 XpLq Ñ E ´1. By Proposition 9.2.6, the isotropy Lie algebra g T m of T at m is a XpLq-module, and η is a cocycle of Chevalley-Eilenberg. This provides an obstruction for the F-connection to be flat. Also, we have the following consequence of Corollary 9.2.6 for Lie algebra actions on affine varieties, as in Example 8.1.11. Before going to Corollary 9.2.13 let us write definitions and some facts.

Settings: Let W be an affine variety realized as a subvariety of C d , and defined by some ideal I W Ă Crx 1 , . . . , x d s. We denote by XpW q :" DerpO W q the Lie algebra of vector fields on W , where O W is coordinates ring of W . Now we can introduce the following definition.

Definition 10.1.3. A bi-submersion tower over a singular foliation F on M , is a (finite or infinite) sequence of manifolds and maps as follows

T B : ¨¨¨s i`1 , , t i`1 2 2 B i`1 s i * * t i 4 4 B i s i´1 * * t i´1 4 4 ¨¨¨s 1 * * t 1 4 4 B 1 s 0 + + t 0 3 3 B 0 , ( 10.3)
together with a sequence F i of singular foliations on B i , with the convention that B 0 " M and F 0 " F, such that

• for all i ě 1, F i Ă Γpker ds i´1 q X Γpker dt i´1 q,

• for each i ě 1, B i`1

s i * * t i 4 4 B i is a bi-submersion over F i .
A bi-submersion tower over pM, Fq shall be denoted as pB i`1 , s i , t i , F i q iě0 . The bi-submersion tower over F in (10.3) is said to be of length n P N if B j " B n , s j " t j " id and F j " t0u for all j ě n.

Remark 10.1.4. Let us spell out some consequences of the axioms. For i ě 1, two points b, b1 P B i of the same leaf of F i satisfy s i´1 pbq " s i´1 pb 1 q and t i´1 pbq " t i´1 pb 1 q. Also, for all b

P B i , T b F i Ă pker ds i´1 q | b X pker dt i´1 q | b .
Let us explain how such towers can be constructed out of a singular foliation. Let F be a singular foliation on M . Then, 1. By Proposition 2.10 in [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF], there always exists a bi-submersion B 1 s 0 * * t 0 4 4 M over F.

2. The C 8 pB 1 q-module Γpker ds 0 q X Γpker dt 0 q is closed under Lie bracket. When it is locally finitely generated, it is a singular foliation on B 1 . Then, it admits a bi-submersion B 2

s 1 * * t 1 4 4 B 1 .
Therefore, we have obtained the two first terms of a bi-submersion tower.

3. We can then continue this construction provided that Γpker ds 1 q X Γpker dt 1 q is locally finitely generated as a C 8 pB 2 q-module, and that it is so at each step 1 .

Definition 10.1.5. A bi-submersion tower T B " pB i`1 , s i , t i , F i q over pM, Fq is called exact bisubmersion tower over pM, Fq when F i`1 " Γpkerpds i qq X Γpkerpdt i qq for all i ě 0. It is called a path holonomy bi-submersion tower (resp. path holonomy atlas bi-submersion tower) if B i`1

s i * * t i 4 4 B i is
a path holonomy bi-submersion (resp. a path holonomy atlas) for F i for each i ě 0. When a path holonomy bi-submersion tower is exact, we speak of exact path holonomy bi-submersion tower.

The following theorem gives a condition which is equivalent to the existence of a bi-submersion tower over a singular foliation. The proof uses Lemma 10.1.17 which is stated in the next section.

Theorem 10.1.6. Let F be a singular foliation on M . The following items are equivalent:

1. F admits a geometric resolution.

2. There exists an exact path holonomy bi-submersion tower over pM, Fq.

Convention 10.1.7. For a submersion ϕ : M ÝÑ N and a smooth map ψ : M ÝÑ N , we denote by ψ Γpker dϕq the space of ψ-projectable vector fields in Γpker dϕq Ă XpM q.

Proof. p1q ñ p2q : Assume that F admits a geometric resolution pE, d, ρq. In particular, pE ´1, ρq is an anchored bundle over F. We need to show by recursion on i ě 0 that Γpker ds i q X Γpker dt i q is locally finitely generated because ker d pi`1q or ker ρ is locally finitely generated. We actually repeat at each step i ě 0, the general fact that the pull-back complex of vector bundle by the submersion

φ " t 0 ˝t1 ˝¨¨¨˝t i : B i`1 ÝÑ M ¨¨¨/ / φ ˚E´i´3 φ ˚dpi`3q / / φ ˚E´i´2 φ ˚dpi`2q / / φ ˚E´i´1 / / T B i`1 (10.4)
remains exact at the sections level (at degree 2 ď ´1), since C 8 pB i`1 q is a flat C 8 pM q-module. In addition, for i ě 1, the complex (10.4) defines a geometric resolution of F i`1 .

Let pB 1 , s 0 , t 0 q be a path holonomy bi-submersion over pM, Fq. Consider the map R : Γpt 0 E ´1q ÝÑ t 0 Γpker ds 0 q Ă XpB 1 q (10.5)

t 0 e Þ ÝÑ Ý Ñ e
defined as in Proposition 10.1.16. By Lemma 10.1.17 (1), the map R in (10.5) comes from a vector bundle morphism t 0 E ´1 ÝÑ ker ds 0 and is surjective on an open subset V 1 Ă B 1 , by item 2 of Lemma 10.1.17. In particular the map R in (10.5) restricts to a surjective map kerpt 0 ρq ÝÑ ker `dt 0 | Γpker ds 0 q ˘" Γpker ds 0 q X Γpker dt 0 q Ă XpB 1 q (10.6)

By exactness in degree ´1, ker ρ " d p2q pΓpE ´2qq. Therefore, kerpt 0 ρq is locally finitely generated. By surjectivity of the map (10.6), Γpker ds 0 q X Γpker dt 0 q ": F 1 is also locally finitely generated on V 1 , in particular F 1 is a singular foliation on V 1 Ă B 1 (we may assume that B 1 " V 1 .). Thus, one can take a path holonomy bi-submersion pB 2 , s 1 , t 1 q over pB 1 , F 1 q. The proof continues the same as the previous step.

Let us make a step further for clarity. The composition ΓpE ´2q

3 3 3 3

d p2q / / impd p2q q " ker ρ / / / / F 1
together with E ´2 is an anchored bundle over F 1 . Let φ " t 0 ˝t1 . Just like in the first step, define the surjective C 8 pB 2 q-linear map, Γpφ ˚E´2 q ÝÑ t 1 Γpker ds 1 q Ă XpB 2 q (10.7)

φ ˚e Þ ÝÑ Ð Ý e .
By Lemma 10.1.17 (2), the map 10.7 restricts (upon taking B 2 smaller) to a surjective map ker ˆΓpφ ˚E´2 q φ ˚dp2q ÝÑ Γpφ ˚E´1 q ˙ÝÑ ker `dt 1 | Γpker ds 1 q ˘" Γpker ds 1 q X Γpker dt 1 q Ă XpB 2 q (10.8)

2 We shall understand that the degree of elements of φ ˚E´i´j is ´j.

By exactness in degree ´2, the C 8 pM q-module ker ˆΓpE ´2q d p2q

ÝÑ ΓpE ´1q ˙" d p3q pΓpE ´3qq is (locally) finitely generated, hence F 2 :" Γpker ds 1 q X Γpker dt 1 q is a singular foliation on B 2 . Thus, one can take a path holonomy bi-submersion pB 3 , s 2 , t 2 q over pB 2 , F 2 q. By recursion on i ě 1, we use a path holonomy bi-submersion pB i`1 , s i , t i q over pB i , F i q and construct an anchor bundle over F i by the composition

Γpφ ˚E´i´1 q 2 2 2 2 φ ˚dpi`1q
/ / impφ ˚dpi`1q q " kerpφ ˚dpiq q / / / / F i with φ " t 0 ˝t1 ˝¨¨¨˝t i´1 : B i ÝÑ M and show as for i " 0, 1 that F i`1 :" Γpker ds i q X Γpker dt i q is a singular foliation on B i`1 . The proof follows.

p2q ñ p1q is proven by Lemma 10.1.8 and Remark 10.1.9 below.

Lemma 10.1.8. Let F be a singular foliation on M . Assume that there exists a bi-submersion tower

T B " pB i , t i , s i , F i q iě0 over F. Then, ¨¨¨/ / ker ds 2 dt 2 / / ker ds 1 dt 1 / / ker ds 0 dt 0 / / T M ¨¨¨/ / B 3 t 2 / / B 2 t 1 / / B 1 t 0 / / M.
(10.9) is a complex of vector bundles, which is exact on the sections level 3 if T B is an exact bi-submersion tower, i.e. if F i " Γpker ds i´1 q X Γpker dt i´1 q for all i ě 1.

Proof. For any element b P B i`1 and any vector v P ker ds i Ă T b B i`1 one has dt i pvq P T t i pbq F i , (since Γpker ds i q Ă t ´1 i pF i q). ùñ dt i pvq P pker ds i´1 X ker dt i´1 q | t i pbq by Definition 10.1.3.

ùñ dt i pvq P ker ds i´1 and dt i´1 ˝dt i pvq " 0, for all i ě 1.

This shows the sequence (10.9) is a well-defined complex of vector bundles.

Let us prove that it is exact when F i " Γpker ds i´1 qXΓpker dt i´1 q for all i ě 1. Let ξ P Γ pker ds i´1 q be a t i´1 -projectable vector field that projects to zero, i.e. dt i´1 pξq " 0. This implies that ξ P Γpker ds i´1 q X Γpker dt i´1 q " F i . Since t i is a submersion, there exists a t i -projectable vector field ζ P t ´1 i pF i q that satisfies dt i pζq " ξ. The vector field ζ can be written as ζ " ζ 1 `ζ2 with ζ 1 P Γ pker dt i q and ζ 2 P Γ pker ds i q, because t ´1 i pF i q " Γpker ds i q`Γpker dt i q. One has, dt i pζ 2 q " ξ. A similar argument shows that the map, Γpker ds 0 q dt 0 ÝÑ t 0 F, is surjective. This proves exactness in all degree.

Remark 10.1.9. One of the consequence of Lemma 10.1.8 is that:

3 Let us explain the notion of exactness at the level of sections when the base manifolds are not the same: what we mean is that for all n ě 0, Γpker dtnq X Γpker dsnq " ptn`1q˚pΓpker dsn`1qq.

Equivalently, it means that the pull-back of the vector bundles in (10.13) to any one of the manifold Bm with m ě n is exact at the level of sections, i.e Γpt n`1,m ker dsn`1q dt n`1 / / Γpt n,m ker dsnq dtn / / Γpt n´1,m ker dsn´1 q (10.10) is a short exact sequence of C 8 pBmq-modules, with tn,m " tn ˝¨¨¨˝tm for all m ě n.

1. If there exists a sequence of maps

M / / ε 0 / / B 1 / / ε 1 / / B 2 / / ε 2 / / ¨¨¨(10.11)
where for all i ě 0, ε i is a section for both s i and t i then the pull-back of (10.9) on M through the sections pε i q iě0 i.e. ¨¨¨d t 3 / / ε 2,0 ker ds 2 dt 2 / / ε 1,0 ker ds 1 dt 1 / / ε 0 ker ds 0 dt 0 / / T M (10.12) is a complex of vector bundles, with the convention ε n,0 " ε n ˝¨¨¨˝ε 0 . If T B is an exact bi-submersion tower then, (10.12) is a geometric resolution of F.

2. In case that T B is an exact path holonomy bi-submersion tower, such a sequence (1) always exists, since the bi-submersions pB i`1 , s i , t i q are as in Example 10.1.2. For such bi-sumersions, the zero section x Þ Ñ px, 0q is a section for both s i and t i .

Corollary

dt 1 / / T B n`1 ˆT M ker ds 0 pr 1 / / T B n`1 v v B n`1 6 (10.13)
is a geometric resolution of the pull-back foliation t ´1 0,n pFq Ă XpB n`1 q, where pr 1 is the projection on T B n`1 and for i ě 1, t i,j is the composition t i ˝¨¨¨˝t j : B j`1 Ñ B i .

Proof. By Lemma 10.1.8, the complex in Equation (10.13) is exact. By construction, the projection of the fiber product T B n`1 ˆT M ker ds 0 to T B n`1 induces the singular foliation t ´1 0,n pFq.

Lift of a symmetry to the bi-submersion tower

Let us investigate what an action ϱ : g Ñ XpM q of a Lie algebra g on pM, Fq would induce on a bi-submersion tower T B over F.

We start with some vocabulary and preliminary results.

Definition 10.1.11. Let pB, s, tq be a bi-submersion of a singular foliation F on a manifold M . We call lift of a vector field X P XpM q to the bi-submersion pB, s, tq a vector field r X P XpBq which is both s-projectable on X and t-projectable on X.

The coming proposition means that the notion of lift to a bi-submersion only makes sense for symmetries of the singular foliation.

Proposition 10.1.12. If a vector field on M admits a lift to a bi-submersion pB, s, tq, then it is a symmetry of F. Proposition 10.1.16. Let pB, s, tq be a bi-submersion over a singular foliation F on a manifold M . Let pA, ρq be an anchored bundle over F, i.e., A ÝÑ M is a vector bundle and ρ : A ÝÑ T M is a vector bundle morphism such that ρpΓpAqq " F. There exists two maps

ΓpAq ÝÑ XpBq a Þ ÝÑ Ð Ý a a Þ ÝÑ Ý Ñ a (10.14)
fulfilling the following conditions:

1. the vector field Ý Ñ a P XpBq (resp. Ð Ý a P XpBq) is t-related (resp. s-related) with ρpaq P F, 2. the vector field Ý Ñ a (resp. Ð Ý a ) is tangent to the fibers of s (resp. t),

3.

Ý Ñ f a " t ˚pf q Ý Ñ a and Ð Ý f a " s ˚pf q Ð Ý a for all a P ΓpAq, f P C 8 pM q.

The vector fields Ð Ý a (resp. Ý Ñ a ) for a P ΓpAq are called left-invariant (resp. right-invariant) vector fields of pB, s, tq.

Proof. By Proposition 10.1.15, given a section a P ΓpAq the vector field ρpaq P F admits a lift Ą ρpaq P XpBq on pB, s, tq of the form Ą ρpaq :" ρpaq t s `ρpaq s t with ρpaq t s P Γpkerpdsqq and ρpaq s t P Γpkerpdtqq. Also, dtpρpaq t s q " ρpaq and dspρpaq s t q " ρpaq. Let b P B and U b an open neighborhood of b. Let pa 1 , . . . , a r q be a local trivialization of A on the open subset U " spU b q Ă M . We define a map R U on local generators by

R U : Γ U b pt ˚Aq ÝÑ Γ U b pkerpdsqq (10.15) t ˚ai Þ ÝÑ ρpa i q t s
and extend by C 8 pU b q-linearity. These maps can be glued using partitions of unity. More precisely, let pχ λ q λPΛ be a partition of unity subordinate to an open cover pU λ q λPΛ by open sets that trivialize the vector bundle A. We define a map R on Γpt ˚Aq as The following lemma is important in the proof of Theorem 10.1.6. Lemma 10.1.17. Let pB, s, tq be any bi-submersion over a singular foliation F on a manifold M , and pA, ρq an anchored bundle over F. 1. There exists vector bundle morphisms R : t ˚A ÝÑ ker ds and L : s ˚A ÝÑ ker dt inducing (10.14).

ÿ λPΛ χ λ R U λ .
2. Let x P M . If pB, s, tq is a path holonomy bi-submersion over F near px, 0q then, every b P B such that tpbq " x admits a neighborhood V such that every t-projectable vector field of Γ V pker dsq is of the form Rpξq for some ξ P Γ V pt ˚Aq.

Remark 10.1.18. In item 1 of Lemma 10.1.17, by " inducing (10.14)"we mean that for every a P ΓpAq, Ý Ñ a :" Rps ˚aq and Ð Ý a :" Lpt ˚aq.

Proof. Let pa 1 , . . . , a r q be a local trivialization of A on the open subset U " tpU b q Ă M . One has by definition of right-invariant vector fields of pB, s, tq that dtp Ý Ñ a i q " t ˚ρpa i q for i " 1, . . . , rkpAq. The vector fields ρpa i q are generators of F on U. We necessarily have n ď rkpAq. Since the ρpa i qpxq 1 s are generators of F x , without loss of generality we can assume that ρpa 1 qpxq, . . . , ρpa n qpxq is a basis of F x . Since rkpker dsq " n, p Ý Ñ a i pbqq i"1,...,n form a basis of ker ds| x . Therefore, the Ý Ñ a i 's are independent at every point of some neighborhood V Ă U b of b i.e., they form a local trivialization of the vector bundle ker ds ÝÑ B. As a result, vector fields of Γ V pker dsq are of the form ř n i f i Ý Ñ a i with f i P C 8 pV q for i " 1, . . . , n. This ends the proof.

We can now state one of the important results of this section. It uses several concepts introduced in [AS09], which are recalled in the proof.

ř d i"1 y i X i . Since dφ Y
1 pXq " pφ Y 1 q ˚pX q P X `F, there exists Z y P F depending in smoothly on y such that dt 0 pX, 0q " X `Zy . Take r Z y P t ´1 0 pFq such that dt 0 p r Z y q " Z y . One has, dt 0 ´pX, 0q ´r Z y ¯" X " ds 0 pX, 0q.

We can write r Z y " r Z 1 y `r Z 2 y , with r Z 1 y P Γpker ds 0 q, r Z 2 y P Γpker dt 0 q. By construction, r X :" pX, ´r Z 1 y q is a lift of X to the bi-submersion pW, s 0 , t 0 q. This proves item 1.

If X B P XpBq and X B 1 P XpB 1 q are two lifts of the symmetry X on the path holonomy bisubmersions pB, s, tq and pB 1 , s 1 , t 1 q respectively, then pX B , X B 1 q is a lift of X on the composition bi-submersion B ˝B1 . This proves item 2, since the path holonomy atlas is made of fibered products and inverse of holonomy path holonomy bi-submersions [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF]. Also X a is a symmetry of pB, t, sq.

Item 2 in Proposition 2.10 of [START_REF] Androulidakis | The holonomy groupoid of a singular foliation[END_REF] states that if the identity of M is carried by pB, s, tq at some point v P B, then there exists an open neighborhood V Ă B of v that satisfies s | V " s 0 ˝g and t | V " t 0 ˝g, for some submersion g : V ÝÑ W, for W of the form as in item 1. Thus, for all X P spFq there exists a vector field r X P XpV q fulfilling ds | V p r Xq " dt | V p r Xq " X. This proves item 3.

Definition 10.1.21. A symmetry of the tower of bi-submersion T B " pB i`1 , s i , t i , F i q iě0 is a family X " pX i q iě0 , with the i-th component X i in spF i q, such that ds i´1 pX i q " dt i´1 pX i q " X i´1 for all i ě 1. We denote by spT B q the Lie algebra of symmetries of T B .

The next theorem gives a class of bi-submersion tower to which any symmetry of the base singular foliation F lifts.

Theorem 10.1.22. Let F be a foliation. Let T B be a path holonomy bi-submersion tower (or an exact path holonomy atlas bi-submersion tower). A vector field X P XpM q is a symmetry of F, i.e. rX, Fs Ă F, if and only if it is the component on M of a symmetry of T B .

Proof. It is a direct consequence of Proposition 10.1.12 and of item 1. resp. item 2. in Proposition 10.1.19. It is due to the fact that the tower T B is generated by path holonomy bi-submersions, and then we can lift symmetries at every stage of the tower T B .

Remark 10.1.23. Let pX i q iě0 be a lift of X 0 :" X P spFq. For i ě 1, ∇ i X :" ad X i preserves Γpker ds i´1 q, since X i is s i´1 -projectable. Altogether, they define a chain map p∇ i X q iě0 at the section level of the complex (10.9), on projectable vector fields in (10.10), since for every i ě 0 and any t i -projectable vector field ξ P ker ds i , dt i prX i`1 , ξsq " rdt i pX i`1 q, dt i pξqs " rX i , dt i pξqs, that is dt i ˝∇i`1 X " ∇ i X ˝dt i .

Remark 10.1.24. In [START_REF] Garmendia | Quotients of singular foliations and Lie 2-group actions[END_REF], under some assumptions, it is shown that if a Lie group G acts on a foliated manifold pM, Fq, then it acts on its holonomy groupoid. It is likely that this result follows from Theorem 10.1.22, this will be addressed in another study.

Lifts of actions of a Lie algebra on a bi-submersion tower

We end the section with the following constructions and some natural questions.

Let T B " pB i`1 , s i , t i , F i q iě0 be an exact path holonomy bi-submersion tower over a singular foliation pM, Fq.

By Theorem 10.1.22, any vector field X P spFq lifts to a symmetry pX i q iě0 of T B . Once a lift is chosen, we can define a linear map, X P spFq Þ Ñ pX i q iě1 P spT B q.

Let ϱ : g Ñ XpM q be a strict symmetry action of a Lie algebra g on pM, Fq. For x P g, there exists pϱpxq i q iě0 , with ϱpxq i P spF i q Ă XpB i q a symmetry of T B such that ϱpxq 0 " ϱpxq P spFq, by Theorem 10.1.22. Consider the composition, x P g Þ ÝÑ ϱpxq P spFq Þ ÝÑ pϱpxq i q iě0 P spT B q Þ Ñ ϱpxq 1 P XpB 1 q.

(10.17)

Lemma 10.1.25. For all x, y P g, rϱpxq, ϱpyqs 1 ´rϱpxq 1 , ϱpyq 1 s " dt 1 pC 1 px, yqq with C 1 px, yq P Γpker ds 1 Ñ B 2 q a t 1 -projectable vector field, for some bilinear map

C 1 : ^2 g ÝÑ Γpker ds 1 Ñ B 2 q.
Proof. This follows from Lemma 10.1.8, because rϱpxq, ϱpyqs 1 ´rϱpxq 1 , ϱpyq 1 s P Γpker ds 0 q X Γpker dt 0 q.

Theorem 10.1.26. The map C 1 : ^2 g ÝÑ Γpker ds 1 Ñ B 2 q of Lemma 10.1.25 satisfies for all

x, y, z P g, C 1 prx, ys g , zq `∇2 ϱpxq pC 1 py, zqq`ö px, y, zq " dt 2 pC 2 px, y, zqq (10.18)

for some tri-linear map C 2 : ^3 g ÝÑ Γpker ds 2 Ñ B 3 q. Here, ∇ 2 is as in Remark 10.1.23.

Proof. For x, y, z P g, dt 1 pC 1 prx, ys g , zqq `ö px, y, zq " rϱprx, ys g q, ϱpzqs 1 ´rϱprx, ys g q 1 , ϱpzq 1 s`ö px, y, zq " ( ( ( ( ( ( ( ( ( ( rrϱpxq, ϱpyqs, ϱpzqs 1 ´rrϱpxq, ϱpyqs 1 , ϱpzq 1 s`ö px, y, zq " ´(( ( ( ( ( ( ( ( ( ( rrϱpxq 1 , ϱpyq 1 s, ϱpzq 1 s `rdt 1 pC 1 px, yqq, ϱpzq 1 s`ö px, y, zq " dt 1 prC 1 px, yq, ϱpzq 2 sq`ö px, y, zq.

We have used Jacobi identity and dt 1 pϱpzq 2 q " ϱpzq 1 . This implies that dt 1 `C1 prx, ys g , zq ´rC 1 px, yq, ϱpzq 2 s`ö px, y, zq ˘" 0. (10.19) Again Lemma 10.1.8 implies the result.

Here is a natural question:

Question: Can we continue the construction in Theorem 10.1.26 to a Lie 8-morphism?

There is another natural type of questions:

Question: Can a strict symmetry action ϱ : g Ñ XpM q of a Lie algebra g on a singular foliation pM, Fq lift to a given geometric resolution pE ' , d, ρq of F?

Discussion

• Can we answer this question with what we have?

• We know by Theorem 8.3.1 that ϱ lifts on any universal Lie 8-algebroid pE, Qq of F to a Lie 8-morphism Φ : g ÝÑ X ' pEq. If we can choose at least the polynomial-degree zero of the Taylor coefficient Φ 1 : ^2 g Ñ X ´1pEq to be zero, then the answer is yes.

• If yes, can we assume that the previous action to preserve the dg-almost Lie algebroid? Again, if yes, then the polynomial-degree `1 can be chosen to be zero.

It seems that being able to suppress the higher Taylor coefficients has a strong geometric meaning.

We intend to study that problem in a subsequent paper.

Question: Can a bi-submersion tower be equipped with a (local) Lie 8-groupoid structure?

Appendices A.2 The tensor algebra of a linear space

Let V be an O-module. For k P N 0 , the k-th tensor power T k O V over O of V (elements of polynomialdegree k) is the tensor product of V with itself k times, namely

T k O V :" V b O ¨¨¨b O V loooooooomoooooooon k times
we also adopt the convention T 0 O V » O, and V » T1 O V. This leads to consider the O-module constructed as the direct sum of the tensor powers, i.e.

T ' O V :" 8 à k"1 T k O V " O ' V ' pV b O Vq ' pV b O V b O Vq ' ¨¨P
T k O V ˆT ℓ O V ÝÑ T k O V b O T ℓ O V » T k`ℓ O V,
that is extended by bilinearity to all T ' O V. We denote this product by b. The unit element is 1 P O » T 0 O V, in particular, we have 1 b v " 1 ¨v " v for every v P V.

A.2.1 T ' O V as a co-algebra

In this section, we consider an O-module V which can be possibly graded. However, the construction is independent whether the module is graded or not.

The notion of co-algebra structure is important in the context of the thesis, since it allows dealing with infinite dimension objects. It appears all along the thesis. For this concept, see Definition 1.2.1. A natural way to construct a co-product (see e.g [START_REF] Vallette | Algebraic Operads[END_REF][START_REF] Kassel | Quantum Groups[END_REF]) structure

1 on T ' O V ∆ : T ' O V ÝÑ T ' O V â T ' O V
is to define it on elements v P V » T 1 O V of polynomial-degree 1, and on the unit element 1 P O » T 0 O V and extend it to a (degree 0) O-algebra morphism to the whole

T ' O V, namely for v 1 b ¨¨¨b v k P T k O V,
∆pv 1 b ¨¨¨b v k q " ∆pv 1 q ¨¨¨∆pv k q.

The one which is defined by

v Þ Ñ v b 1 `1 b v 1 Þ Ñ 1 b 1
endows T ' O V with a co-associative (co-commutative) co-algebra structure, that is, it satisfies the axioms of Definition 1.2.1. Indeed, the maps ∆ b id, id b ∆ :

T ' O V b T ' O V Ñ T ' O V b T ' O V b T ' O V
are algebra morphisms, so are p∆ b idq ˝∆ and pid b ∆q ˝∆, therefore it suffices to check that they coincide on V. Let us check that. For v P V, We have used Formula (A.2) and 1 b v " 1 ¨v " v, each time b is the tensor symbol in T ' O V. If V is not graded, i.e., concentrated in degree zero, there is no sign p´1q |v||w| . More generally, for every v 1 , . . . , v n P V, Definition B.1.1. A complex of O-modules pV ' , dq is a graded module V " À iPZ V i together with a squared to zero O-linear map d : V Ñ V of degree `1 called the differential map. In other words, it is a sequence

¨¨¨ÝÑV i´1 d ÝÑ V i d ÝÑ V i`1 ÝÑ ¨¨¨(B.2)
O-linear maps such that d 2 " d ˝d " 0. 1. For every i P Z, elements of V i are called cochains of degree i.

2. We say that (B.2) is bounded below/above if V i " 0 for i ď n{i ě n, for some n P Z.

3. A subcomplex of a complex pV, dq a collection of O-modules pV 1 i Ď V i q iPZ such that dpV 1 i q Ă V i`1 for each i P Z. In particular pV 1 , d 1 " d| V 1 q is a complex of O-modules.

In particular, it induces a complex `V{V 1 , d ˘called the quotient complex where for i P Z, pV{V 1 q i :" V i {V 1 i and d :

V i {V 1 i ÝÑ V i`1 {V 1 i`1
is determined uniquely by the universal property of the quotient.

Remark B.1.2. Let pV, dq be a complex of O-modules. Denote by d i : V i Ñ V i`1 the restriction of d to V i . Then d 2 " 0 means that d i ˝di´1 for each i P Z. In particular, Imd i´1 Ď ker d i for every i P Z. This leads us to the next definition. 2. a i-coboundary of pV, dq is an element of ImpV i´1 d ÝÑ V i q;

3. the i-th cohomology group of pV, dq is the quotient H i pVq :"

kerpV i d ÝÑ V i`1 q ImpV i´1 d ÝÑ V i q .
The complex pV, dq is exact at i if H i pVq " t0u. It is said to be exact or acyclic if it is exact at every degree i P Z.

Definition B.1.4. Let pV, d V q and pW, d W q be complexes of O-modules.

1. A chain map or complex of O-modules morphism between the complexes pV, dq and pW, dq is a O-linear map L : V Ñ W of degree 0, which commutes with the differentials, that is a collection of O-linear map L ' : V ' ÝÑ W ' , such that the following diagram commutes

¨¨¨/ / V i L i d V / / V i`1 L i`1 / / ¨¨/ / W i d W / / W i`1 / / ¨¨¨( B.3)
i.e. d W ˝Li " L i`1 ˝dV for every i P Z.

2. A homotopy between two chain maps K ' , L ' : V ' ÝÑ W ' is the datum th i : V i ÝÑ W i´1 u iě1 of O-linear maps, that satisfies for each i P Z, K i ´Li " d W ˝hi `hi´1 ˝dV . These maps are displayed in the following diagram as

¨¨¨/ / V i´1 K i´1 ´Li´1 d V / / V i h i | | K i ´Li d V / / V i`1 K i`1 ´Li`1 h i`1 | | / / ¨¨/ / W i´1 d W / / W i d W / / W i`1 / / ¨¨¨( B.4) (a)
When there is a homotopy between two chain maps, L ' , K ' : V ' ÝÑ W ' , we often write L " K. One can check that " is indeed an equivalence relation.

(b) Two complexes of O-modules pV, d V q and pW, d W q are said to be homotopy equivalent, if there exist chain maps L ' : V ' ÝÑ W ' and K ' : W ' ÝÑ V ' such that L ˝K " id W' and K ˝L " id V' . Likewise, one can check that homotopy equivalence between complexes of O-modules is an equivalence relation.

Remark B.1.5. Note that in particular, if L ' : V ' ÝÑ W ' is a chain map that is an O-linear isomorphism, then its inverse is also a chain map. Thus, they define a homotopy equivalence between pV, d V q and pW, d W q.

Remark B.1.6. Let L ' : V ' ÝÑ W ' be a chain map. For every i P Z, we have

L ˆkerpV i d V ÝÑ V i`1 q ˙Ď kerpW i d W
ÝÑ W i`1 q, and L ˆImpV i´1

d V ÝÑ V i q ˙Ď ImpW i´1 d W ÝÑ W i q.
L induces naturally a well-defined O-linear map HpLq : HpVq Ñ HpWq, rvs Þ Ñ rLpvqs: for every v P kerpV i d V ÝÑ V i`1 q and v 0 P V i´1

HpLqprv `dV pv 0 qsq " rLpv `dV pv 0 qqs " rLpvq `dW ˝Lpv 0 qqs " rLpvqs " HpLqprvsq.

Notice that 1. homotopic chain maps induce the same map on cohomology groups.

2. if the complexes of O-modules pV, d V q and pW, d W q are homotopy equivalent through L, then

HpLq : HpVq Ñ HpWq is an isomorphism.

Proposition B.1.7. If pV 1 , d 1 q is an acyclic subcomplex of a complex pV, dq, then

H ' pV{V 1 q » H ' pVq.
Proof. The projection p ' : V ÝÑ V ' {V 1 ' is a chain map from pV, dq and pV{V 1 , dq. We claim p ' induces an isomorphism on the cohomology groups. To show injectivity of Hppq : H ' pVq ÝÑ H ' pV{V 1 q, let e P ker d such that ppeq " e P Im d, i.e. there is u P V |e|´1 such that e " dpuq. It follows that e ´du P V 1 |e| . This implies, d 1 pe ´duq " dpe ´duq " 0. (B.5) By Exactness of pV 1 , d 1 q, ùñ e ´du " d 1 pvq, (for some v P V 1 |e|´1 Ă V |e|´1 ) ùñ e " dpu `vq P Im d. This proves injectivity. Surjectivity goes as follows: let e P V i such that dpeq " 0. dpeq " 0 ùñ dpeq P V 1 i`1

We have, d 1 pdpeqq " d ˝dpeq " 0. By exactness of pV 1 , d 1 q, we can write dpeq " d 1 pvq for some v P V 1 i . This implies that, e ´v " u P ker d. Therefore, Hppqprusq " rppuqs " rppeq ´ppvqqs " rppeqs " res.

This completes the proof.

The Chevalley-Eilenberg complex

The following example of complex is important. We have used it several times in this thesis. Especially in Chapter 9 to define obstruction classes. Let us recall the definition. We refer the reader e.g. to [START_REF] Wagemann | Introduction to Lie algebra cohomology with a view towards BRST cohomology[END_REF] for more details.

Let pg, r¨, ¨sg q be a Lie algebra and V a K-vector space. Definition B.1.8. A representation or action of g on V is a Lie algebra morphism ν : pg, r¨, ¨sg q ÝÑ pEndpV q, r¨, ¨sq (B.6)

where pEndpV q, r¨, ¨sq denotes the vector space EndpV q of endomorphisms of V together with the Lie bracket r¨, ¨s which is the commutator: rα, βs " α ˝β ´β ˝α, @ α, β P EndpV q. In this case, V is then called a g-module (w.r.t to ν). In the literature, the action ν is often denoted by ¨.

Equation (B.6) means that for all x, y P g, νprx, ys g q " rνpxq, νpyqs " νpxq ˝νpyq ´νpyq ˝νpxq.

Example B.1.9. Here are two important examples of g-modules that we often use.

• The adjoint action: g acts on itself by the Lie bracket, i.e. νpxqpyq :" rx, ys g for all x, y P g. Indeed, ν : g ÝÑ Endpgq, x Þ ÝÑ rx, ¨sg ": ad x is a Lie algebra morphism: ad rx,ysg " rrx, ys g , ¨sg " ´rry, ¨sg , xs g ´rr¨, xs g , ys g , pby identity of Jacobiq " rx, ry, ¨sg s g ´ry, rx, ¨sg s g " ad x ˝ad y ´ad y ˝ad x " rad x , ad y s.

• The trivial action: i.e. K is a g-module through the action νpxqpλq :" 0 for all x P g and all λ P K.

'

Now let us recall the definition of the Chevalley-Eilenberg complex.

Definition B.1.10. Let ν : g ÝÑ EndpV q be an action of g on V . The Chevalley-Eilenberg complex of g valued in V is the complex

¨¨¨ÝÑHom K p^i ´1g, V q d CE ÝÑ Hom K p^ig, V q d CE
ÝÑ Hom K p^i `1g, V qÝÑ ¨¨¨(B.7)

whose i-th cochains space is defined to be Hom K p^ig, V q, the vector space of i-linear skew-symmetric linear maps from g ˆ¨¨¨ˆg looooomooooon i-times to V , under the convention Hom K p^0g, V q » V . The differential map is defined for µ P Hom K p^ig, V q by ´dCE µ ¯px 1 , . . . , x i`1 q " i`1 ÿ k"1 p´1q k´1 νpx k qpµpx 1 , . . . , p x k , . . . , x i`1 qq `ÿ 1ďkălďi`1 p´1q k`l µprx k , x l s g , x 1 , . . . , p x kl , . . . , x i`1 q where p x k means x k is missing in the k-th place also p x kl means that x k , x l are missing in the k-th and l-th place respectively. Remark B.1.11.

1. It follows from the identity of Jacobi that d CE ˝dCE " 0.

2. For all x, y, z P g we have,

• d CE pxq " νpxq.

• For µ P Hom K pg, V q, `dCE µ ˘px, yq " νpxqpµpyqq ´νpyqpµpxqq ´µprx, ys g q.

• For η P Hom K p^2g, V q, ´dCE η ¯px, y, zq " νpxqpηpy, zqq ´νpyqpηpx, zqq `νpzqpηpx, yqq ´ηprx, ys g , yq `ηprx, zs g , yq ´ηpry, zs g , xq.

Remark B.1.12. When g is of finite dimension, the Chevalley-Eilenberg complex (B.7) is canonically isomorphic to the complex p^'g ˚b V, dq and for ξ " ξ 1 ^¨¨¨^ξ k P ^kg where ξ 1 , . . . , ξ n is a basis of g. In the formula (B.8), d g is the Chevalley-Eileberg differential of g w.r.t the trivial action on K.

B.1.1 Operations on complexes

We have used and adapted the following lemma many times in this thesis, e.g. Section 4.3.1.

Lemma B.1.13. Let pV ' , d V q and pW ' , d W q be complexes of O-modules. Then, the pair pHom ' O pV, Wq , Bq is a complex of O-modules, where the differential map is given by BpF q :" d W ˝F ´p´1q |F | F ˝dV , (B.9) for all homogeneous element F P Hom O pV, Wq. In particular, the chain maps between pV, d V q and pW, d W q are the 0-cocyles of pHom ' O pV, Wq , Bq. 3. when pV ' , d V q " pW ' , d W q, we have `Hom ' O pV, Vq, B " rd V , ¨s˘.

Direct sum and tensor product of complexes

Let pV ' , d V q and pW ' , d W q be complexes of O-modules. Then, the tensor product for all homogeneous elements v, w P V b O W, is complex of O-modules. The operator B is indeed of degree `1. It is easily checked that B 2 " 0.

Bi-complex

Definition B.1.15. A bi-complex or double complex is a collection of O-modules V " pV i,j q i,jPZ together with two families of O-linear maps 1. d h i,j : V i,,j Ñ V i`1,j , such that d h i,j ˝dh i´1,j " 0, for i, j P Z called horizontal differential map 2. d v i,j : V i,j Ñ V i,j`1 , such that d v i,j ˝dv i,j´1 " 0, for all i, j P Z called vertical differential map that obey for all i, j P Z the identity

d h i,j`1 ˝dv i,j " d v i,j ˝dh i`1,j .
In this thesis we only consider first quadrant bi-complexes that is, V i,j " 0 for all i P Z ď0 and j P N 0 . These can be represented as the commutative diagram . . . . . . . . .

Ò Ò Ò ¨¨¨Ñ V i,j d h Ñ V i,j d h Ñ V i,j Ñ 0 d v Ò d v Ò d v Ò ¨¨¨Ñ V i,j d h Ñ V i,j d h Ñ V i,j Ñ 0 d v Ò d v Ò d v Ò ¨¨¨Ñ V i,j d h Ñ V i,j d h Ñ V i,j Ñ 0 Ò Ò Ò 0 0 0
"-2 column" "-1 column" "last column" (B.10)

One associate to the bi-complex (B.10) the so-called total complex which is defined by the antidiagonals of (B.10), namely ´Tr :" À i`j"r V i,j ¯r with total differential D : T r Ñ T r`1 defined by Dpτ ij q :" d h pτ ij q ´p´1q r d v pτ ij q, for τ ij P V i,j .

Indeed, D 2 " pd h ´p´1q r`1 d v q ˝pd h ´p´1q r d v q " pd h q 2 lo omo on "0

´ p´1q r d h ˝dv ´(( ( ( ( ( ( ( p´1q r`1 d v ˝dh ´pd v q 2 lo omo on "0 " 0.

Proposition B.1.16 (Acyclic Assembly Lemma [START_REF] Charles | An introduction to homological algebra / Charles A. Weibel[END_REF]). Let pV i,j q i,jPZ be a first quadrant bi-complex like in the notation above such that the rows are exact, then the total complex pT ' , Dq is exact.

Mapping cone. Let pV, d V q and pW, d W q be two complexes and L : V ' Ñ W ' a chain map. The mapping cone is the complex pC, Bq whose degree i is given by V i´1 ' W i and whose differential is defined as ) By construction, we have kerpd n q " Impd n`1 q. Therefore, we have built an exact sequence up to length n `1. This completes the proof.

B " ˜´d V 0 ´L d W ¸ 0 / / ker
One shall notice that the process (B.14) may be continued forever without reaching a free kernel.

Here is an important operation on complexes. The following Proposition states the localization in the sense of item 2 of Section 3.2.1 preserves exactness. More precisely, Proposition B.2.6 ([Sta22], Section 10.9 or [And]). It is well-known that a submodule of a finitely generated module is not finitely generated in general. The following assertion assures that for module over Noetherian rings (see Proposition 1.4 of [START_REF] Eisenbud | Commutative algebra. With a view toward algebraic geometry[END_REF], Page 28). of a finitely generated R-module A is said to be minimal if the differential map satisfies dpV ´iq Ď mV ´i`1 for all i ě 2.

Proposition B.2.14. [START_REF] Eisenbud | The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra[END_REF] Every finitely generated R-module A admits a minimal free resolution.

Proof. The proof goes just like in Proposition B.2.5, one just need to take minimal set of generators in the construction. By doing so, it suffices to check that dpV ´2q Ă V ´1: Let r " dimpA{mAq. By Nakayama Lemma, we can take V ´1 " R r . One has a short exact sequence, 0 ÝÑ dpV ´2q " ker π ãÝÑ V ´1 ÝÑ A ÝÑ 0.

It induces an exact sequence dpV ´2q{m dpV ´2q ãÝÑ V ´1{mV ´1 ÝÑ A{mA ÝÑ 0, by tensorizing with K " R{m. Since by construction V ´1{mV ´1 » A{mA » K r , the image of the map dpV ´2q{m dpV ´2q ãÝÑ V ´1{mV ´1 is zero, i.e., dpV ´2q Ď mV ´1. Likewise, by Noetheriality of R, ker π Ă V ´1 is finitely generated. One can repeat the procedure by starting with a minimal set of generators of ker π and so on. The proof continues by recursion.

Remark B.2.15. Minimal resolutions and the Tor complex: given a minimal projective resolution as in (B.15), its quotient by the maximal ideal m corresponds to the complex

¨¨¨/ / V ´3 b O K dbid / / V ´2 b O K dbid / / V ´1 b O K πbid / / A b O K / / 0
whose cohomology compute Tor ' pA, Kq. By minimality of pV ' , d, πq one has d b id " 0, therefore Tor ´ipA, Kq » V ´i b O K for every i ě 2. In particular, the ranks of the V ´i's for i ě 2 are independent of the choices made in the construction of the minimal projective resolution.

Koszul complex

Assume that V is a free O-module of finite rank n, which is concentrated in degree ´1. Here, we denote by Ź ' V the graded symmetric algebra of V. Let F : V Ñ O be a O-linear map. Given a free basis e 1 , . . . , e n of V, F is completely determined by n-uplet pf 1 , . . . , f n q Ă O.

Definition B.2.16. [START_REF] Eisenbud | Commutative algebra. With a view toward algebraic geometry[END_REF][START_REF] Hideyuki | Commutative ring theory[END_REF] The Koszul complex associated to pf 1 , . . . , f n q is the complex which form a (chain) complex, i.e. d piq ˝dpi`1q " 0 and ρ ˝dp2q " 0.

0 ÝÑ n ľ V d ÝÑ n´1 ľ V d ÝÑ ¨¨¨ÝÑ 2 ľ V d ÝÑ V F ÝÑ O. (B.
Remark B.3.2. Two main cohomology groups can be associated to a complex of vector bundles over F:

1. Cohomology at the level of sections. The complex of vector bundles (B.18) induces a complex of sheaves of modules over functions. More explicitly, for every open subset U Ă M , there is a complex:

¨¨¨ÝÑΓ U pE ´i´1 q d pi`1q

On existence of geometric resolutions

Geometric resolutions of a singular foliation F as defined in Remark B.3.2(a) are not guaranteed in all contexts. However, there always exists a projective resolution of F as a O-module (see B.2). But these resolutions do not induce always a geometric resolution of F, since the projective modules of a projective resolution may not correspond to vector bundles because they may not be locally finitely generated. When the latter condition is satisfied, the Serre-Swan theorem [START_REF] Swan | Vector Bundles and Projective modules[END_REF][START_REF] Archana | Note on the Serre-Swan theorem[END_REF] states that there is a one-to-one correspondence between locally finitely generated projective modules and sections of vector bundles. Under the assumptions of the latter theorem, geometric resolutions of singular foliations are exactly projective resolutions at the sections level in the category of chain complexes by O-modules, since sections of vector bundles over M are projective O-modules.

The following proposition summarizes some contexts where geometric resolutions exist, see [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] for their proofs. The same holds for a real analytic of holomorphic singular foliation, but only in a neighborhood of a point.

A locally real analytic singular foliation on a manifold of dimension d admits a geometric resolution of length ď d `1 on any relatively compact open subset of M .

Here we have some examples of geometric resolutions of singular foliations.

Example B.3.5. Let F 0 " tX P XpV q | Xp0q " 0u be the singular foliation made of all vector fields vanishing at the origin of a vector space V (e.g. think of C N or R N ). The contraction by the Euler

vector field Ý Ñ E " N ÿ i"1
x i B Bx i gives rise to a complex of trivial vector bundles

¨¨¨ÝÑ ^3 T ˚V ι Ý Ñ E ÝÑ ^2T ˚V ι Ý Ñ E ÝÑ T ˚V ι Ý Ñ E ÝÑ C ˆV ": C, (B.19)
whose complex on the sections level is pΩ ' pV q, ιÝ Ñ E q. Here px 1 , . . . , x N q are the canonical coordinates on V . The latter is the Kozul complex associated to the coordinate functions x 1 , . . . , x N of V . Since the x i 's form a regular sequence, it is well known that pΩ ' pV q, ιÝ Ñ E q is exact.

The following complex of vector bundles over

V ¨¨¨ÝÑ ^3 T ˚V b T V ι Ý Ñ E bid ÝÑ ^2T ˚V b T V ι Ý Ñ E bid ÝÑ T ˚V b T V ι Ý Ñ E bid ÝÑ C b T V. (B.20)
is a geometric resolution of F 0 since ´Ω' pV q b XpV q, ιÝ Ñ E b id ¯is also exact (here Ω i pV q :" Γp^iT ˚V q stands for the sheaf of i-forms on V ).

More generally, the construction we have made in (B.20) is still possible by contracting with any vector field X " N ÿ i"1 X i B Bx i P XpV q. The latter yields a complex of vector bundles that covers the singular foliation F X generated by the X i B Bx j 's. For instance, if X is a polynomial vector field and pX 1 , . . . , X N q form a regular sequence, we get a geometric resolution of F X .

Example B.3.6. Let F 2 " I 2 0 XpK 2 q Ă F 0 be the sub-singular singular foliation made of vector fields vanishing at order 2 at the origin of K 2 , where I 2 0 Ă OpK 2 q is the ideal generated by the monomials x 2 , xy, y 2 . Note that the ideal I 2 0 admits a free resolution of the form 0 ÝÑ OpK 2 q ' OpK 2 q δ 1 ÝÑ OpK 2 q ' OpK 2 q ' OpK 2 q δ 0 ÝÑ I 2 0 ÝÑ 0, (B.21

)
where for all f, g, h P OpK 2 q, δ 0 pf, g, hq " x 2 f `xyg `y2 h and δ 1 pf, gq " pxf, xf ´yg, xgq.

The free resolution (B.21) has to take the form 0 ÝÑ ΓpI ´2q δ 1 ÝÑ ΓpI ´1q δ 0 ÝÑ I 2 0 ÝÑ 0, for sum trivial vector bundles I ´1, I ´2 on K 2 . Thus, the following complex

0 ÝÑ I ´2 b T K 2 δ 1 bid ÝÑ I ´1 b T K 2 δ 0 bid ÝÑ I 2 0 b T K 2 ÝÑ 0 (B.22)
is a geometric resolution of F 2 . Note that I ´1 can be identified with the tivial vector bundle S 2 ppK 2 q ˚q.

More generally, let F k be the singular foliation made of vector fields vanishing at order k at the origin of a vector space V of dimension N over R or C. The Hilbert's syzygy theorem assures the existence of a free resolution of length N `1 of the ideal I k 0 made of functions on V vanishing to order k at the origin. This resolution is of the form

¨¨¨ÝÑ ΓpI ´2q δ 1 ÝÑ ΓpI ´1q δ 0 ÝÑ I 2 0 ÝÑ 0,
for some family of trivial vector bundles pI ´iq iě1 over V . We obtain a geometric resolution of F k of the form

¨¨¨ÝÑ ΓpI ´2 b T V q δ 1 bid ÝÑ ΓpI ´1 b T V q δ 0 bid ÝÑ I 2 0 b XpV q " F k .

1.

  Nous montrons qu'il existe une équivalence de catégories entre les algèbres de Lie-Rinehart sur une algèbre commutative O et classes d'équivalence d'homotopie d'algébroïdes de Lie infinie acycliques graduées négativement. Par conséquent, ce résultat donne un sens à l'algébroïde de Lie infinie universel d'un feuilletage singulier, sans hypothèse supplémentaire, et pour les algébroïdes de Lie singulières d'Androulidakis-Zambon. Ceci étend à un cadre purement algébrique la construction de la Q-variété universelle d'un feuilletage singulier localement réel analytique de [LLS20, Lav17]. Aussi, à tout idéal I de O préservé par l'application d'ancre d'une algèbre de Lie-Rinehart A, on associe une classe d'équivalence d'homotopie d'algébroïdes de Lie infinie négativement graduée sur des complexes calculant Tor O pA, O{Iq. Plusieurs exemples explicites sont donnés.

Definition 1.2. 1 .

 1 A coassociative graded cocommutative co-algebra structure on an graded O-moduleC " ' iPZ C i is • a linear map of degree zero called (graded) coproduct ∆ : C ÝÑ C b C with ∆pC k q Ă à i`j"k C i b C jis such that ∆pC k q has non-zero 1 intersection with only finitely many spacesC i b C j • p∆ b idq ˝∆ " pid b ∆q ˝∆ : C Ñ C b C b C, called graded coassociativity.• τ ˝∆ " ∆, called graded cocommutativity, where τ : C b C Ñ C b C is defined by τ pa b bq " p´1q |a||b| b b a.

  ϵpσqx σp1q d ¨¨¨d x σpiq b x σpi`1q d ¨¨¨d x σpnq (1.1) for every x 1 , . . . , x n P E. Example 1.2.5. For small n. The formula of Equation (1.1) on homogeneous elements x, y, z P E means that • ∆pxq " 0 • ∆px d yq " x b y `p´1q |x||y| y b x • ∆px d y d zq " x d y b z `p´1q |x||z| x d z b y `p´1q |x|p|z|`|y|q y d z b x x b y d z `p´1q |x||y| y b x d z `p´1q |z|p|x|`|y|q z b x d y

  Lie algebroids over O Definition 2.1.3. [Lav17, LLS20] A (negatively graded) almost differential graded Lie algebroid

  Both sides are obviously equal. Likewise, a straightforward computation shows that ∆ ˝Bpv b αdtq " pB b id `id b Bq ˝∆pv b αdtq. Hence ∆ ˝B " pB b id `id b Bq ˝∆ on S ' K pEq b K Ω ' pra, bsq.

  Example 3.2.3. Sections of a Lie algebroid that have values in the kernel of the anchor map form a Lie-Rinehart algebra Kerpρ A q for which the anchor map is zero. The next two examples are part of our the main source of motivation to study Lie-Rinehart algebras. The second is more general than the first one. Example 3.2.4. Singular foliations. [AS09, AZ14, Cer79, Deb01, LLS20, LGLR22] A singular foliation on a smooth, real analytic, or complex manifold M or Zariski open subset U Ď C d is a subsheaf F Ď XpM q that fulfills the following conditions • Stability under Lie bracket: rF, Fs Ď F.

  1. the objects are Lie 8-algebroids over O, 2. arrows are homotopy equivalence classes of morphisms of Lie 8-algebroids over O. and by Lie-Rhart-alg/O the category of Lie-Rinehart algebra over O. We want to study the functor,

  Corollary 4.2.12. Let A be a Lie-Rinehart algebra over O. Let I Ă O be a Lie-Rinehart ideal. Then any two Lie 8-algebroids of I are homotopy equivalent, and there is a distinguished class of homotopy equivalences between them. Taking under account Remark 4.2.11, here is an alternative manner to restate this corollary. Corollary 4.2.13. Let A be a Lie-Rinehart algebra over O. Let I Ă O be a Lie-Rinehart ideal. Then the complex computing Tor ' O pA, O{Iq comes equipped with a natural Lie 8-algebroid structure over O{I, and any two such structures are homotopy equivalent in a unique up to homotopy manner.

•

  from the category Lie-8-alg-oids/O, • to the category Lie-Rhart-alg/O Theorem 4.2.1 gives a right inverse of this functor. In particular, this functor becomes an equivalence of categories when restricted to homotopy equivalence classes of acyclic Lie 8-algebroids over O, i.e: Corollary 4.2.14. There is an equivalence of categories between: (i) Lie-Rinehart algebras over O, (ii) acyclic Lie 8-algebroids over O.

pkq'

  pEq instead of Page pkq ' pE, Eq. The following diagram recapitulates the whole picture of Page pkq ' pE 1 , Eq:

.

  module for all pj, mq P Z ´ˆN 0 , and pE, d, πq is a resolution, all the lines of the above bicomplex are exact. This proves the first item. The second and the third are obtained by diagram chasing (see Appendix B.2 for more details).

  is mapped by P to R ´i for all possible indices. In such case, it defines by restriction to Ä ' R an element ι RP in the graded O-module Page pkq j pRq :" ' iěn Hom O ´Äk`1 R | ´j´i , R ´i¯.

For

  the sake of clarity, let us denote by D E and D R the respective differentials of the bi-complexes Page pkq j pEq and Pagepkq j pRq and by D E h , D R h and D R v , D R v the horizontal differential resp. vertical differential, of their associated bi-complexes. Also, ι RP stands for the restriction of P P Page

  pkq

1.

  For every k ě 0, a D E -cocycle P P Page pkq j pEq which preserves R is the image through D E of some element Q P Page pkq j´1 pEq which preserves R if and only if its restriction ι RP P Page pkq j pRq is a D R -coboundary. 2. In particular, if the restriction of d E and π E to R makes it a resolution of π E pR ´1q Ă V, then any D E -cocycle P P Page pkq j pEq which preserves R is the image through D E of some element Q P Page pkq j´1 pEq which preserves R. Proof. Let us decompose the element P P Page pkq j pEq as P " ř iě1 P i with, for all i ě 1, P i in Hom O ´Äk`1 E | ´j´i , E ´i¯. Assume P P Page pkq j pEq is a D E -cocycle which preserves R. Let us prove one direction of item 1. If P is the image through D E of some element Q P Page pkq j´1 pEq which preserves R, then D R pι RQq " ι RD E pQq " ι RP , with ι RQ P Page pkq j´1 pRq. Thus, the restriction ι RP P Page pkq j pRq of P is a D R -coboundary. Conversely, let us assume that ι RP P Page pkq j pRq is a D R -coboundary, i.e. ι RP " D R Q R for some Q R P Page pkq j´1 pRq. Take Q P Page pkq j´1

  2. a degree `1 graded symmetric operation l2 on E as follows: all i P I k , j P I l with k ě 2 or l ě 2.

  4.4. EXAMPLES OF UNIVERSAL LIE 8-ALGEBROIDS OF LIE-RINEHART ALGEBRAS 71 Proposition 4.4.4. Let W be an affine variety with functions O W . For every point a P W and any Lie-Rinehart A over O W , the germ at a of the universal Lie 8-algebroid of A is the universal Lie 8-algebroid of the germ of A at a. Therefore, the germ at a of a Lie-Rinehart algebra or a Lie 8-algebroid is simply its localization w.r.t the complement of m a .

Proposition 4.4. 6 .

 6 If χ is not a zero-divisor in O, and pE, t¨¨¨u kě1 , ρ, πq is a universal Lie 8algebroid of A, then the Lie 8-structure described in the four items of 4.4.1 is the universal Lie 8-algebroid of χA.

  An affine variety W Ď C d is a the zero locus of an ideal I Ď O, i.e., W :" ZpIq :" ta " pa 1 , . . . , a d q P C d | f paq " 0, @f P Iu. It admits a topology, induced by the Zariski topology in C d . Remark 5.1.2. In Definition 5.1.1, we do not exclude irreducible varieties, e.g. W " tpx, yq P C 2 | xy " 0u is an affine variety. The following facts and remarks are important. For W Ď C d an affine variety in the notation of Definition 5.1.1, 1. we denote by I W the vanishing ideal of W , namely,

  g [LGPV13], Appendix B.2) any pointwise derivation δ a of O at a P C d has the form δ a "

  of the lecture on singular foliations, Poisson 2022 [LGLR22]. Example 5.1.21. Let W " tpx, y, zq P C 3 | xypx `yqpx `yzq " 0u Ă C 3 .

  of graded commutative algebras over K such that every m P M admits an open neighborhood U Ă M on which the sheaf structure takes the formEpUq " O U b K SpE ˚'q

  Lemma 7.1.5. Let pM, Fq be a singular foliation. Let pE, Qq be a universal Lie 8-algebroid of F and let pE, d, ρq : ¨¨¨d p4q all m P M , we have kerpρmq impd p2q m q

  The projection map π : Ă M Ă Gr ´ℓpE ´1q ÝÑ M, pV, mq ÝÑ m (7.10) fulfills the following properties 1. For each point m P M , the set

p0q 1 "

 1 Ψp0q . In polynomial-degree one, we have, dΘ p1q t dt " Qp0q ˝Hp1q t " rQ, H p1q s " Ψp1q ´Ξp1q 1 .

  all i, j P I ´1, (b) l2 ´ξi , e p´1q j ¯" χ ´ξi , e p´1q j

  roposition A.2.1. The O-module T ' O V comes equipped with a graded unital O-algebra structure, which is induced by the canonical map

  p∆ b idq ˝∆pvq " ∆ b idpv b 1 `1 b vq " ∆pvqb1 `∆p1qbv " pv b 1 `1 b vq b 1 `p1 b 1q b v " pv b 1q b 1 `p1 b vq b 1 `p1 b 1q b v " pid b ∆q ˝∆pvq, by associativity of b. By extending ∆ to an algebra morphism, one gets explicit expressions as follows. For v b w P T 2 O V, one has ∆pv b wq " ∆pvq∆pwq " pv b 1 `1 b vqpw b 1 `1 b wq " pv b wq b 1 `v b w `p´1q |v||w| w b v `1 b pv b wq.

∆pv 1 b ¨¨¨b v n q "

  v σp1q b ¨¨¨b v σpiq â v σpi`1q b ¨¨¨b v σpnq , (A.3)where σ P Spi, n ´iq is a pi, n ´iq-shuffle and ϵpσq is the Koszul sign associated to the n-uplet v 1 , . . . , v n . See Section 1.1 for more details.

Definition B.1. 3 .

 3 Let pV, dq be a complex of O-modules. For each i P Z, 1. a i-cocycle of pV, dq is an element of kerpV i d ÝÑ V i`1 q;

  b pξ i ¨vq ´dg pξq b v (B.8)

Proof.

  The O-linear map, B : Hom O pV, Wq Ñ Hom O pV, Wq, is clearly of degree `1, since the maps d V and d W are. Moreover, for every F P Hom O pV, WqB 2 pF q " Bpd W ˝F ´p´1q |F | F ˝dV q " d W ˝dW˝F loooooomoooooon ˝F ˝dV ´p´1q |F | ˜(( ( ( ( ( d W ˝F ˝dV ´p´1q |F |`1 F ˝dV ˝dV looooomooooon It's worth it to notice that 1. the cocycles F of pHom ' O pV, Wq, Bq are those that satisfy d V ˝F " p´1q |F | F ˝dW .

2

  . two chain maps F ' , G ' : V ' ÝÑ W ' are homotopic if and only if G ´F is 0-coboundary of pHom ' O pV, Wq , Bq, that is, there exists a O-linear map H P Hom ´1 O pV, Wq of degree ´1 such that F ´G " d W ˝H `H ˝dV .

  V ' b O W ' together with the grading pV b O Wq k " à redi`j"k V i b O W jfor k P Z, comes equipped with a differential map classically defined byB " d V b id `id b d W , namely, Bpv b wq " d V pvq b w `p´1q |v| v b d W pwq,

  Let A be an O-module and S Ă O a multiplicative subset. For any free resolution of A ¨¨¨d/ / V ´3 d / / V ´2 d / / V ´1 π / / A , the complex ¨¨¨S ´1d / / S ´1V ´3 S ´1d / / S ´1V ´2 S ´1d / / S ´1V ´1 S ´1π / / S ´1Ais a free resolution of S ´1A by S ´1O-modules.

Proposition B.2. 7 .

 7 If O is Noetherian as a ring, then all submodules of an O-module V are finitely generated if and only if V is finitely generated.The following theorem assures existence of free resolution of finite length in a special case. Theorem B.2.8 (Hilbert Syzygy Theorem).[START_REF] Peeva | of Algebra and Applications[END_REF][START_REF] Eisenbud | The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra[END_REF] Assume O " Crx 1 , . . . , x d s. Any finitely generated (graded) O-module A admits a finite graded free resolution by finitely generated O-modules¨¨¨d / / V ´3 d / / V ´2 d / / V ´1 π / / A of length N ď d `1.One can construct free resolutions of finitely generated modules over R like in the proof of Proposition B.2.5 by taking a minimal set of homogeneous generators at each step. This can be formalized as follows,

  Proposition B.3.4. 1. Any algebraic singular foliation 2 on K d admits geometric resolutions by trivial vector bundles and of length ď d `1.

  The map px, y, zq Þ Ñ Jacpx, y, zq is O-trilinear: indeed, it is clear if x, y, z P E ď´2 . By graded symmetry, we need to verify this point when the at least one of the variables is of degree´1. Let f P O,Case 1: x P E ´1 and y, z P E ď´2 Jacpx, y, f zq " ℓ 2 pℓ 2 px, yq, f zq `p´1q |z||y| ℓ 2 pℓ 2 px, f zq, yq `p´1q |z|`|y| ℓ 2 pℓ 2 py, f zq, xq " f ℓ 2 pℓ 2 px, yq, zq `p´1q |z||y| pℓ 2 pf ℓ 2 px, zq, yq `ℓ2 pρpxqrf sz, yqq `p´1q |z|`|y| f ℓ 2 pℓ 2 py, zq, xqp ´1q |z|`|y| p´1q |z|`|y|`1 ρpxqrf sℓ 2 py, zq

	Remark 2.1.5. " f Jacpx, y, zq `(( ( ( ( ( ( ( ( ( ( ( p´1q

Remark 2.1.4. For any almost graded Lie-algebroid pE ' , ℓ 1 , ℓ 2 , ρq define the Jacobiator Jacpx, y, zq " ℓ 2 pℓ 2 px, yq, zq `p´1q |y||z| ℓ 2 pℓ 2 px, zq, yq `p´1q |x|p|y|`|z|q ℓ 2 pℓ 2 py, zq, xq, x, y, z P E (2.2) For any triple of elements x, y, z P E ´1 of degree ´1. Equivalently: Jacpx, y, zq " ℓ 2 pℓ 2 px, yq, zq`ö px, y, zq, where ö px, y, zq means that we sum on the circular permutations of x, y, z with Koszul signs. It is graded symmetric of degree ´2. By axiom 3, the Jacobiator takes values in the kernel of the anchor map, since ρpJacpx, y, zqq " ρpℓ 2 pℓ 2 px, yq, zqq`ö px, y, zq " rρpℓ 2 px, yqq, ρpzqs`ö px, y, zq " rrρpxq, ρpyqs, ρpzqs`ö px, y, zq " 0, since r¨, ¨s satisfies Jacobi identity. It is easy to see that the bracket ℓ 2 goes to the quotient and endows, E ´1 ker ρ and E ´1 ℓ 1 pE ´2q with a Lie K-algebra bracket l2 . |z||y| ρpxqrf sℓ 2 pz, yq ´(( ( ( ( ( ( ( ρpxqrf sℓ 2 py, zq Case 2: x, y P E ´1 and z P E ď´2 Jacpx, y, f zq " ℓ 2 pℓ 2 px, yq, f zq `p´1q |z| ℓ 2 pℓ 2 px, f zq, yq `p´1q |z|`1 ℓ 2 pℓ 2 py, f zq, xq " f ℓ 2 pℓ 2 px, yq, zq `ρpℓ 2 px, yqqrf sz `p´1q |z| ℓ 2 pf ℓ 2 px, zq `ρpxqrf sz, yqqp ´1q |z|`1 ℓ 2 pf ℓ 2 py, zq `ρpyqrf sz, xq " f Jacpx, y, zq `ρpℓ 2 px, yqqrf sz `p´1q |z| pp´1q |z| ( ( ( ( ( ( ( ( ρpyqrf s ℓ 2 px, zq `(( ( ( ( ( ( ( ρpxqrf sℓ 2 pz, yq `p´1q |z| ρpyqrρpxqrf sszq `(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( p´1q |z|`1 pp´1q |z| ρpxqrf s ℓ 2 py, zqq `(( ( ( ( ( ( ( ρpyqrf sℓ 2 pz, xq `p´1q |z| ρpxqrρpyqrf sszq " f Jacpx, y, zq `(( ( ( ( ( ( ( ( ( ( ( ( pρpℓ 2 px, yqq ´rρpxq, ρpyqsqrf sz " f Jacpx, y, zq.

  When O " K we have only the axiom piq of 2.1.8, since the other axioms are trivial. Therefore, we recover the axioms that define Lie 8-algebras[START_REF] Stasheff | Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras[END_REF][START_REF] Lada | Introduction to SH Lie algebras for physicists[END_REF]. Geometrically, we are in the case where the projective modules pE ´iq iě1 of Definition 2.1.8 are finitely generated. Serre-Swan theorem[START_REF] Swan | Vector Bundles and Projective modules[END_REF] assures for every i ě 1, E ´i " ΓpE ´iq for some vector bundle E ´i Ñ M over a manifold M . Hence, Definition 2.1.8 is rewritten word by word as follows: A (finitely generated) negatively graded Lie 8-algebroid pE, pℓ k q kě1 , ρq over a manifold M is 1. a collection of vector bundles E " pE ´iq iě1 over M 2. together with a sheaf of Lie 8-algebra structures pℓ k q kě1 over the sheaf of sections of E 3. that comes with a vector bundle morphism ρ : E ´1 ÝÑ T M , called the anchor,

	Remark 2.1.13.
	`ρpℓ 2 px, yqqrf sz
	"0
	`ρpyqrρpxqrf ssz ´ρpxqrρpyqrf ssz `(( ( ( ( ( ( ( ρpyqrf sℓ 2 px, zq `(( ( ( ( ( ( ( ρpyqrf sℓ 2 pz, xq ´(( ( ( ( ( ( ( ρpxqrf sℓ 2 pz, yq
	´(( ( ( ( ( ( ( ρpxqrf sℓ 2 py, zq.

yq, zq `ρpℓ 2 px, yqqrf sz ´ℓ2 pf ℓ 2 px, zq, yq ´ℓ2 pρpxqrf sz, yq `ℓ2 pf ℓ 2 py, zq, xq `ℓ2 pρpyqrf sz, xq " f pℓ 1 pℓ 3 px, y, f zqq `ℓ2 pℓ 2 px, yq, zq ´ℓ2 pℓ 2 px, zq, yq `ℓ2 pℓ 2 py, zq, xqq loooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooon This implies, pρpℓ 2 px, yqqrf s ´rρpxq, ρpyqsrf sqz " 0. Therefore, when O has no zero divisors on E ´1, we obtain that ρpℓ 2 px, yqq " rρpxq, ρpyqs.

Definition 2.1.11. A Lie 8-algebroid is said to be acyclic if the complex (2.4) has no cohomology in degree ď ´2.

Example 2.1.12.

4. such that the k-ary-brackets are all O-multilinear except when k " 2 and at least one of the arguments is of degree ´1. The 2-ary bracket satisfies the Leibniz identity ℓ 2 px, f yq " ρpxqrf sy `f ℓ 2 px, yq, x P ΓpE ´1q, y P ΓpEq.

(2.5)

  Differential Graded Lie Algebras. We come back to Example 2.1.6. Any differential graded Lie algebra pg " ' PZ g i , r¨, ¨s , dq is Lie 8-algebroid with trivial anchor, where the unary bracket

Remark 2.2.5. Note that, e.g. if O " K, we recover the equivalence between Lie 8-algebras and co-differentials. Convention 2.2.6. From now, when relevant, we sometime denote an underlying structure of Lie 8-algebroid pE ' , ℓ ' , ρq on E by pE, Q E , ρq instead. Remark 2.2.7. Notice that Q E : S ' K pEq ÝÑ S ' K pEq does not induce a co-derivation on Ä ' E unless ρ " 0. Let us make the correspondence given by Proposition 2.2.4 explicit on the following examples. Example 2.2.8.

  P ra, bs, pJ s , H s q sPra,ts is a homotopy that joins Φ and J t .

										ρ 1 q are homotopy equivalent
	to the identity map of respective space.
	The following proposition justifies Definition 2.3.14.
	Proposition 2.3.19. Let Φ be a Lie 8-algebroid morphism from pE 1 , Q E 1 , ρ 1 q to pE, Q E , ρq. For all
	t P ra , bs, let H t : S n`1 pnq K pE 1 q Ñ E be a family O-multilinear piecewise rational maps indexed by
	n P N 0 . Then,
	1. There exists a unique piecewise rational continuous family of co-algebra morphisms J t such that
	(a) J a " Φ
	(b) pJ t , H t q is a solution of the differential equation (2.12), where H t is the J t -co-derivation
	whose n-th Taylor coefficient is H t pnq	for all n ě 0.
	2. Moreover, for all t Proof. Let us show item 1. We claim that equation (2.12) is a differential equation that can be solved
	recursively. In polynomial-degree zero, it reads,
						p0q t dJ dt	" Q p0q E	˝Hp0q t	`Hp0q t	˝Qp0q E 1	(2.15)
	and			J	p0q t	" Φ p0q `ż t a	´Qp0q E	˝Hp0q s	`Hp0q s	˝Qp0q E 1 ¯ds	(2.16)
	is defined for all t P ra, bs. Also, d dt J t pn`1q	: S n`2 pE 1 q Ñ E is an algebraic expression of Q p0q E , . . . , Q	pn`1q E	,
	Q p0q E 1 , . . . , Q	pn`1q E 1	J t , . . . , J p0q	pnq t , H t , . . . , H p0q	pn`1q t	. But J	pn`1q t

  Let us prove the last part of the statement. By assumption, there exists a ď b n ď b such that for all t P rb n , bs, we have gptq ě n, so that J

								dJ dt " pQ E ˝Ht `Ht ˝QE 1 q pnq depends pnq t
	only on H t for i " 0, . . . , n ´1, we have by assumption piq	pnq t dt " 0 for all t ě n. As a consequence, dJ
	J t pnq	is constant on rn, `8r. It follows from Proposition 2.3.19 that J 8 is a Lie 8-algebroid morphism
	since for every n P N 0 and t P rn, `8r
			pQ E ˝J8 ´J8 ˝QE 1 q pnq "	ÿ	pQ	piq E	˝Jpiq t	´Jpjq t	˝Qpjq E 1 q
					i`j"n
				" 0 (since J t is a Lie 8-algebroid morphism).
					pnq gptq " J 8 and g 1 ptqH pnq gptq " 0 on rb n , bs. The function J pnq t pnq
	(resp. H t ) being piecewise rational continuous (resp. piecewise rational) on rc, ns, the same holds pnq
	for J	pnq gptq (resp. g 1 ptqH	pnq gptq

  pn`1q ¯˝∆ 1 px 1 d¨¨¨dx k q. (2.19)This proves the first item. Since both Φ and Ξ are O-multilinear, pΞ ´Φq pn`1q is O-multilinear. This proves the second item. Since, Φ and Ξ are Lie 8-morphisms:

	pΞ ´Φq ˝QE 1 ´QE ˝pΞ ´Φq " 0.	(2.20)
	By looking at the component of polynomial-degree n `1, one obtains, pΞ ´Φq pn`1q ˝Qp0q E 1	´Qp0q E ˝pΞ	Φq
	pn`1q " 0. This proves the third item.

Lie-Rhart-alg/O Remark 3.1.2. A

  B Ď A is a said to be a Lie-Rinehart subalgebra of A if it carries a Lie-Rinehart algebra structure over O whose Lie bracket and anchor map are the restriction of the bracket r¨, ¨sA and the anchor ρ A respectively to B. Lie-Rinehart algebra is said to be a Lie algebroid if A is a projective O-module. See Example 3.2.2 for the relation with usual Lie algebroid as vector bundles. For every Lie 8-algebroid pE ' , ℓ ' , ρq over O, the quotient space E ´1 ℓ 1 pE ´2q comes equipped with a natural Lie-Rinehart algebra over O. The 2-ary bracket ℓ 2 goes to quotient to E ´1 ℓ 1 pE ´2q to define a Lie algebra since for all x P E ´2 and y P E ´1 we have ℓ 2 pℓ 1 pxq, yq " ℓ 1 pℓ 2 px, yqq.

	Remark 3.1.3.

Lie-Rinehart algebras over O form a category that we denote by

  then there exist an acyclic Lie 8-algebroid pE ' , ℓ ' , ρq that covers A through a hook π such that 1. E contains E 1 as a subcomplex, 2. the Lie 8-algebroid morphism from E 1 to E announced in Theorem 4.2.4 can be chosen to be the inclusion map E

1 ãÑ E (i.e. a Lie 8-morphism where the only non-vanishing Taylor coefficient is the inclusion E 1 ãÑ E). The following Corollary follows immediately from Proposition 4.2.6: Corollary 4.2.7. Let A be a Lie-Rinehart algebra over O and B be a Lie-Rinehart subalgebra of A. Any universal Lie 8-algebroid of B can be contained in a universal Lie 8-algebroid of A. Induced Lie 8-algebroids structures on Tor O pA, O{Iq. Let pA, r¨, ¨sA , ρ A q be a Lie-Rinehart algebra over O. Definition 4.2.8. We say that an ideal I Ă O is a Lie-Rinehart ideal of A if ρ A paqrIs Ă I for all a P A. Remark 4.2.9. For any Lie-Rinehart ideal IO of A, 1. we have rIA, As A Ă IA. Therefore, the quotient space A{IA comes equipped with a natural Lie-Rinehart algebra structure over O{I. 2. For pE ' , ℓ ' , ρq an acyclic Lie 8-algebroid that covers A, the quotient space E ' {I » O{I b O E ' comes equipped with an induced Lie 8-algebroid structure: the n-ary brackets for n ‰ 2 go to quotient by linearity, while for n " 2, the 2-ary bracket goes to the quotient in view of the relation ρ E pE ´1qrI s Ă I. Also, π goes to the quotient to a Lie-Rinehart algebra morphism E ´1{I Ñ A{IA. Definition 4.2.10. Let A be a Lie-Rinehart algebra over O. For every Lie-Rinehart ideal I Ă O, we call Lie 8-algebroid of I the quotient Lie 8-algebroid E ' {I, with pE ' , ℓ ' , ρq a universal Lie 8-algebroid that covers A. Remark 4.2.11. The complexes on which the Lie 8-algebroids of the ideal I are defined compute Tor O pA, O{Iq by construction (see B.2).

  Every free resolution pE, d, πq of a Lie-Rinehart algebra pA, r¨, ¨sA , ρ A q comes equipped with a binary bracket ℓ 2 that makes it an almost differential graded Lie algebroid of A.

	We start by proving this lemma.
	Lemma 4.3.10. Proof. For all k ě 1, let us denote by pe	p´kq i

modules equipped a graded almost differential graded Lie algebroid pE ' , ℓ 1 , ℓ 2 , ρq over O such that 1. ρ " ρ A ˝π : E ´1 ÝÑ DerpOq, 2. π is a morphism, i.e. for all x, y P E ´1 πpℓ 2 px, yqq " rπpxq, πpyqs A .

  RN is a degree `2 element in the total complex Page p1q pEq. By construction rd, l2 s RN has no component on the last column. Since πprd, l2 s RN | E ´1 q " 0 and also rd, rd, l2 s RN s RN | E ď´2 " 0, the O-bilinear operator rd, l2 s RN is D-closed in Page p1q pEq. By virtue of the first item of Proposition 4.3.3, the operator rd, l2 s RN is then a D-coboundary, so there exists τ 2 P ' jě2 Hom O ´Ä2 E | ´j´1 , E ´j ¯such as Dpτ 2 q " ´rd, l2 s RN . Upon replacing l2 by l2 `τ2

The map defined for all homogeneous x, y P E by rd, l2 s RN px, yq " d ˝l 2 px, yq `l 2 pdx, yq `p´1q |x| l2 px, dyq , is a graded symmetric degree `2 operation pE b Eq ' ÝÑ E '`2 , and rd, l2 s RN | E ´1 " 0. Let us check that it is O-bilinear, i.e. for all f P O, x, y P E: rd, l2 s RN px, f yq ´f rd, l2 s RN px, yq " 0. 1. if x P E ´1, this quantity is zero in view of rd, l2 s RN px, f yq " f rd, l2 spx, yq `dρpxqrf s y ´ρpxqrf s dy loooooooooooooomoooooooooooooon "0

2. if x P E ´2, one has rd, l2 s RN px, f yq ´f rd, l2 spx, yq " l2 pdx, f yq ´f l2 pdx, yq " ρpdxqpf q y " 0 since ρ ˝d " ρ A ˝π ˝d " 0, 3. if x P E ´i with i ě 3, it is obvious by O-linearity of l2 on the involved spaces.

As a consequence, rd, l2 s

  Let us check that it is D-closed: for this purpose we have to check that both conditions in Lemma 4.3.2 hold:

x, y, z P E by Jacpx, y, zq :" ℓ 2 pℓ 2 px, yq, zq `p´1q |y||z| ℓ 2 pℓ 2 px, zq, yq `p´1q |x||y|`|x||z| ℓ 2 pℓ 2 py, zq, xq

is O-linear in each variable, hence is a degree `2 element in À jě1 Hom O p Ä 3 E | ´j´2 , E ´j q Ă Page p2q pEq.

For degree reason, its component on the last column of diagram (6.16) is zero, i.e. it belongs to z Page p1q pEq.

  Proof (of Proposition 4.2.3). This is a consequence of Proposition 4.2.1 and the third item of the Proposition 4.3.3: If the component of Jac on the column ´1 is zero, we can choose ℓ 3 with no component on the last column and in column ´1 (see Proposition 4.3.3), i.e. the restriction of ℓ 3 to Ä 3 E ´1 is zero. Then ℓ 3 has no component on the last column, the column ´1 and the column ´2. so rℓ 2 , ℓ 3 s RN has no component in the last column, ´1 and ´2 columns as well. Hence, ℓ 4 can be chosen with no component on column ´1, ´2 and ´3 by the third item of Proposition 4.3.3. The proof continues by recursion.

	Proof of Proposition 4.2.3 and Proposition 4.2.6

seen as an element in Page pi`j´2q pEq by Remark 4.3.6, is a cocycle and for degree reason it has no element on the last column, and the columns ´1, . . . , 3 ´n in 6.16. The third item of Proposition 4.3.3 gives the existence of an pn `1q-ary bracket ℓ n`1 such as rd, ℓ n`1 s RN " ´ÿ i`j"n`2 iďj rℓ i , ℓ j s RN .

This completes the proof.

  yq ´ℓ2 pΦ p0q pxq, Φ p0q pyqq. This map is in fact O-bililinear, i.e. belongs to Hom O p Ä 2 E 1 , Eq, hence to Page p1q pE 1 , Eq, see Equation (6.16). Let us check that it is a D-cocycle:

	(4.16)

  are constant and equal to Ψ pnq for t ě n. By Lemma 2.3.22, these homotopies can be glued to a homotopy on r0, 1s. Explicitly, since t Þ Ñ t 1´t maps r0, 1r to r0, `8r and by Lemma 2.3.22, the pair ´J t

	pnq
	t
	1´t	, 1 p1´tq 2 H k, t 1´t	¯is a homotopy between Φ and Ψ. This proves the
	second item of the Theorem 4.2.4.

  be considered as a S ´1O-module homomorphism ´1E ; 4.4. EXAMPLES OF UNIVERSAL LIE 8-ALGEBROIDS OF LIE-RINEHART ALGEBRAS 70 2. The anchor map ρ 1 is defined by

	S ´1φ : S ´1E ÝÑ S ´1T with S ´1φ	´x s	¯:"	φpxq s	, x P E, pf, sq P O ˆS,

called the localization of φ. Given a Lie 8-algebroid structure pE ' , ℓ ' , ρq that covers A through π. The triplet pE 1 ' , ℓ 1 ' , ρ 1 q is a Lie 8-algebroid structure that covers S ´1A through the hook π 1 where 1. E 1 " S

  e. replace O Y by O Y {I X . This operation has already been defined in Section 4.2, and here is an immediate consequence of Corollary 4.2.13: Let I Ă O be a Lie-Rinehart ideal, i.e. an ideal such that ρ A pAqrIs Ă I. The quotient of a universal Lie 8-algebroid of A with respect to an ideal I is a Lie 8-algebroid that terminates in A{IA. It is universal if and only if pp E ´i IE ´i q iě1 , l1 , πq is exact, i.e. if Tor ' O pA, O{Iq " 0. Note that the anchor map ρ : E ´1 Ñ DerpOq goes to quotient to E ´1 IE ´1 Ñ DerpOq as an O-linear map, but needs the extra condition ρpE ´1qrI s Ă I to induce an O{I-linear map

	Proposition 4.4.2. Remark 4.4.3. E ´1 IE ´1 Ñ DerpO{Iq.

  Recall that for O a unital algebra with no zero divisor, derivations of O induce derivations of its field of fractions O. For every Lie-Rinehart algebra A over O whose anchor map takes values in derivations of O preserving Õ, then 1. any Lie 8-algebroid structure pE, pℓ k q kě1 , ρ, πq that terminates at A extends for all i " 0, . . . , n to a Lie 8-algebroid structure on Õ b O E, 2. and this extension Õ b O E is a Lie 8-algebroid that terminates at the Lie-Rinehart algebra Õ b O A.Proof. Since they are O-linear, the hook π, the anchor ρ, and the brackets ℓ k for k ‰ 2 are extended to Õ-linear maps. Since the image of ρ is the image of ρ A , it is made of derivations preserving O, which is easily seen to allow an extension of ℓ 2 to Õ b O E using the Leibniz identity.

Proposition 4.4.7. Let O be an unital algebra with no zero divisor, O its field of fractions, and Õ an algebra with O Ă Õ Ă O. Remark 4.4.8. Of course, the Lie 8-algebroid structure obtained on Õ b O E is not in general the universal Lie 8-algebroid of Õ b O A, because the complex p Õ b O E, ℓ 1 , πq may not be a resolution of Õ b O A (see Example 4.4.10).

Remark 4.4.9. Since any module over a field is projective, any Lie-Rinehart algebra over a field is a Lie algebroid. If we choose Õ " O therefore, the Lie-Rinehart algebra O b O A is a Lie algebroid, so is homotopy equivalent to any of its universal Lie 8-algebroid. Unless A is a Lie algebroid itself, the Lie 8-algebroid in Proposition 4.4.7 will not be homotopy equivalent to a Lie 8-algebroid whose underlying complex is of length one, and is therefore not a universal Lie 8-algebroid of O b O A.

  the vector fields ∆ ij , for i, j P t0, . . . , N u, w.r.t affine chart U N . The vector fields r ∆ ij , j ‰ N , belong to the O U N -module generated by the vector fields r jN ). Said differently, the vector fields r ∆ iN , i " 0, . . . N ´1 are generators of Ă F φ . Since they are independent, the singular foliation Ă F φ is a free O U N -module. A universal Lie 8-algebroid for it is therefore concentrated in degree ´1 and is given by the O U N -module generated by some set pe i , i " 0 . . . N ´1q, equipped with the Lie bracket:

				∆ iN ,
	(explicitly r ∆ ij " z 2 j	r ∆ iN	´z2
			re i , e j s :" 2z N	`z2 i e j	´z2 j e i ˘(4.23)

i r ∆

  obtained by 4.4. EXAMPLES OF UNIVERSAL LIE 8-ALGEBROIDS OF LIE-RINEHART ALGEBRAS 76 concatenation of n multi-indices I 1 , . . . , I n . For every i 1 P I 1 , . . . , i n P I n , ϵpi 1 , . . . , i n q is the signature of the permutation which brings i 1 , . . . , i n to the first n slots of I 1 ' ¨¨¨' I n . Last, for i s P I s , we define I is s :" I s zi s .

  |B I k |`1 `φi tB j , B I 2 . . . , B In u 1 n ´φj tB i , B I 2 . . . , B In u 1 PI 2 ,...,inPIn ϵpi 2 , . . . , i n qι φ pB i ^Bj qrφ i 2 ¨¨¨in s B I i 2 PI 2 ,...,inPIn ϵpi 2 , . . . , i n qρpB i ^Bj qrφ i 2 ¨¨¨in s B I i 2 2 '¨¨¨'I in n since ρ " ´ιφ . Hence, the quantities in lines (4.38) and (4.37) add up, when we re-write them in terms of the new brackets t¨¨¨u k , to yield precisely the higher Jacobi identity for this new bracket. It is then not difficult to see this is still the case if there is more than one bivector field, by using many times the same computations.We keep the convention and notations of the previous section. Let us consider the restriction i W F φ of the Lie-Rinehart algebra F φ to the zero-locus W of a Koszul polynomial φ. Since all vector fields in F φ are tangent to W , this restriction is now a Lie-Rinehart algebra over O W " O Oφ , see Section 3.2.1 (1). Let φ be a Koszul Polynomial. The restriction of the universal Lie 8-algebroid of Proposition 4.4.17 to the zero-locus W of φ is a universal Lie 8-algebroid of the Lie-Rinehart algebra i W F φ .

	4.4. EXAMPLES OF UNIVERSAL LIE 8-ALGEBROIDS OF LIE-RINEHART ALGEBRAS 77
	and
	␣	tB i ^Bj u 1 1 , B I 2 . . . , B In	( 1 n " p´1q	k"2 n ř n	"
			´p´1q	ř n k"2 |B I k |	ÿ i 2 2 '¨¨¨'I in n
			" p´1q	ř n k"2 |B I k |	ÿ
	i 2 Proposition 4.4.19.
						2 '¨¨¨'I in n

¨¨¨, B In u n´1 , B i ^Bj ( 2 ´ÿ i 2 PI 2 ,...,inPIn ϵpi 2 , . . . , i n qρpB i ^Bj qrφ i 2 ¨¨¨in s B I i 2 4.4.7 Restriction to φ " 0 of vector fields annihilating φ

  Local coordinates at a P C d generate the maximal ideal m a of O a . Indeed, let t 1 , . . . , t d P O a be local coordinates at a. By applying Nakayama Lemma (B.2.11) to R " O a Ě m a and V " m a : the basis t 1 , . . . , t d P m a {m 2 a lifts to a (minimal) generating set t 1 , . . . , t d for m a .

. . , t r P O W,a are called local coordinates of W at a, if they vanish at a (i.e. t i P m W,a for i " 1, . . . , r), and if the classes of t 1 , . . . , t r P m W,a {m 2 W,a form a basis. Example 5.1.13. If a " pa 1 , . . . , a d q P C d , then x 1 ´a1 , . . . , x d ´ad are local coordinates of C d at a. Remark 5.1.14. Notice that:

  Theorem 4.2.4 implies the existence of a unique up to homotopy Lie 8-algebroid morphism Ψ : U O pI W XpC d qq ÝÑ U O pX W pC d qq. In view of Proposition 4.2.6, the morphism Ψ may be represented by the inclusion map for well-chosen representation of U O pI W XpC d qq and U O pX W pC d qq. 2. The anchor map of U O pX W pC d qq is tangent to W , hence the restriction i W U O pX W pC d qq to W of U O pX W pC d qq exists, and is a Lie 8-algebroid over XpW q.

	In general, it does not need to
	be a universal one, but Theorem 4.2.4 implies the existence of a unique up to homotopy Lie
	8-algebroid morphism:
	Φ

O W is a Noetherian ring, XpW q admits a free resolution by finitely generated O W -modules (by Proposition B.2.7).

In view of Theorem 4.2.1, these three resolutions do admit Lie 8-algebroid structures over their respective algebras, and those are universal Lie 8-algebroids. We denote by U O W pXpW qq, U O pX W pC d qq and U O pI W XpC d qq the universal Lie 8-algebroids associated to the three Lie-Rinehart algebras above.

Remark 5.1.23. There exist natural Lie 8-algebroid morphisms between these structures:

1. Since I W XpC d q Ă X W pC d q,

  `1 ě 3. Let O H " ´2hpxqy be the coordinate ring of H. Let XpHq " DerpO H q be the Lie-Rinehart algebra of derivations of O H , i.e. of vector fields on H. As an O H -module, XpHq is the submodule algebroid is given by the Lie-Rinehart algebra E ´1 " O H X Ă O H In the singular case, i.e., gcdphpxq, h 1 pxqq " dpxq ‰ 1, the O H -module XpHq is not free and has two generators,

	Crx,ys
	xy 2 " f B Bx `g B By | f, g P O H , yg ´h1 pxqf " 0	*	Ă O H	B Bx	' O H	B By	.
					Bx	`h1 pxq B By . A universal
	Lie 8-B Bx ' O H By . X " y B B Bx `h1 pxq B B By and Y " 2 hpxq dpxq Bx

The curve H is non-singular if and only if gcdphpxq, h 1 pxqq = 1. In that case, the Lie algebra of vector fields XpHq is a free O H -module of rank 1 generated by the vector field X " y B `y h 1 pxq dpxq B By with a relation yX " dpxqY (see

  Let U O W pXpW qq " pE, ℓ ' , ρq be a universal Lie algebroid of W . Let us choose a P W . As stated in Section 4.4.1, the Lie 8-algebroid structure of W restricts at a (i.e. goes to the quotient with respect to the maximal ideal m a ) if and only if ρrm a s Ď m a , (i.e. m a is a Lie-Rinehart ideal).

Lie 8-algebra of an affine variety at a point: Define the O W -submodule Sker a pρq :" te P E ´1 | ρpeqrOs Ď m a u Ď E ´1.

  . for every k ě 3, the k-ary brackets ℓ k : ΓpS k K pEqq ÝÑ ΓpEq and the polynomial-degree k´1 component Q pk´1q : ΓpE ˚q Ñ ΓpS k K pE ˚qq of Q are dualto each other, i.e. A Q pk´1q rαs, e 1 d ¨¨¨d e k E " xα, ℓ k pe 1 , . . . , e k qy , for α P ΓpE ˚q and e 1 , . . . , e k P ΓpEq. From now on, unless otherwise mentioned, we shall simply say "Lie 8-algebroids over M " for "finitely generated 8-algebroids over M ". Also, Lie 8-algebroids pE, pℓ k q kě1 , ρq over M shall be denoted as pE, Qq. Whether we see Lie 8-algebroids as N Q-manifolds or as co-differentials, these two approches have in common to give the same notion of Lie 8-morphism of 8-algebroids. This can be seen directly by writing the conditions (2.3.1) and (6.8) in terms of the k-ary brackets.

	Convention 6.1.17. Remark 6.1.18.
	A	E
	Q p0q rαs, e	" xα, ℓ 1 peqy ,
	3. for all homogeneous elements e 1 , e 2 P ΓpEq and α P ΓpE

2 , 1. for all f P O, e P ΓpE ´1q xQ p1q rf s, ey " ρpeqrf s, 2. for all α P ΓpE ˚q and e P ΓpEq: ˚q A Q p1q rαs, e 1 d e 2

E

" ρpe 1 qrxα, e 2 ys ´ρpe 2 qrxα, e 1 ys ´xα, ℓ 2 pe 1 , e 2 qy, with the understanding that the anchor ρ vanishes on E ´i when i ě 2.

2 Our sign's convention are those of

[START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF]

.

4

  (resp. vertical differential B v ) are given by left composition (resp. right composition with Q p0q ), namely, for all Ψ pkq P Hom ' O ´ΓpE ˚q, Γ ´Sk`1

					K pE	˚q¯¯o ne has,
			$
	B h ´Ψpkq ¯:"	& %	Ψ pkq ˝Qp0q Ψ pkq ˝ρ˚w hen Ψ pkq restricts to ΓpE ˚1q, when Ψ pkq restricts to ΓpE ˚iq, with i ‰ 1,	(6.14)
	and
		$
	B v ´Ψpkq ¯:"	& % ρ ˚˝Ψ pkq Q p0q ˝Ψpkq	when Ψ pkq is of degree i and restricts to ΓpE ˚i´1 q, otherwise.	(6.15)
	With the understanding that Γ `S0 K pE	˚q˘»

' O ´ΓpE ˚q, Γ ´Sk`1 K pE ˚q¯¯w here the horizontal differential B h

  ´1 » slp2, Rq ˆR2 is a trivial vector bundle of rank 3, and E ´2 a trivial vector bundle of rank 1. Here, the bracket between two constant sections of E ´1 is defined as being their bracket in slp2, Rq and all other brackets between generators is trivial. Also, Q takes the form ´2, and pe ˚, f ˚, h ˚, µ ˚q is the dual basis of ppe, f, h, µqq.

						By	,	f " y	B Bx	,	h " x	B Bx	´y B By	,
	where e, f, h are the standard basis elements of slp2, Rq. The singular foliation F admits a universal
	Lie 8-algebroid pE, Qq built on a geometric resolution ([LLS20], Example 3.31) of length 2
								0ÝÑE ´2 d ÝÑ E ´1 ρ ÝÑ T M,
	where E Q " xe ˚B By	`yf	˚B Bx	`h˚p x	B Bx	´y B By	q`e ˚df	˚B Bh ˚´2e ˚dh ˚B Be	˚`2f ˚dh	˚B Bf	˚`px	˚q B 2 f ˚`y 2 e ˚´xyh Bµ
	In this case, one has
									XpR 2 q F	» R 2 ' R
	is generated by the classes of
								B	B Bx	,	B By	, x	B Bx	`y B By	F	.

ẘhere µ is the generator of E

  Fpmq is a Lie ideal, For A a Lie-Rinehart algebra over O, the same construction can be done for any maximal ideal m. Let A| m :" ta P A | ρpaqrOs Ď mu. quotient A|m mA is a Lie algebra over O{m. For O " C 8 pM q and m " I m for a point m P M , the Hompm{m 2 , O{mq is the image of A Ñ Hompm{m 2 , O{mq, a Þ Ñ ρpaq. Proposition 7.0.4 is however subtle to extend to this setting. For a finitely generated Lie-Rinehart A over C 8 pM q, there exists a Lie algebroid pA, ρ A , r¨, ¨sA q such that A » ΓpAq for some vector bundle A Ñ M if and only if A ImA , as a vector space, has constant rank at all points. In that caseApmq ImA is kerpρ A | m q for all m P M . In particular, it is the case in a neighborhood of every point where the dimension of A ImA is minimal.

	The following sequence
	0	/ / A| Im ImA	/ / A ImA	/ / / / T m ρpAq	/ / 0	(7.1)
	is exact, where T m ρpAq Ă Proposition 7.0.5.
	Definition 7.0.2. the quotient space g m " algebra of F at m.	Fpmq I m F	is a Lie algebra, which is called the isotropy Lie

Remark 7.0.3. The isotropy Lie algebras detect singular and regular points of pM, Fq. It measures how far F is of being regular in neighborhood of a point. This can be stated as follows (see Lemma 1.1 of

[START_REF] Androulidakis | Smoothness of holonomy covers for singular foliations and essential isotropy[END_REF]

). Proposition 7.0.4. For any point m P L of a leaf L Ă M , g m " t0u if and only if, L is a regular leaf. Also, the g m 's are all isomorphic as Lie algebras while m P L.

Proof. This is a consequence of Nakayama Lemma B.2.11. The germ O m of functions on a neighborhood of U Ď M of m is a local ring

[START_REF] Requejo | Localization in the rings of continuous functions[END_REF]

with maximal ideal still denoted by I m . The tensor product

O m b O A is a module over O m . Let r " dimp A

ImA q. By Nakayama Lemma B.2.11 any basis pe 1 , . . . , e r q of A ImA lifts to minimal generators on

  Lie 8-morphisms are homotopy equivalent over the identity of M , 3. any two such lifts of any two equivalent weak symmetry actions of g on F are homotopy equivalent. Again, Lie 8-morphisms in item 1 of Theorem 8.3.1 are g-actions on pE, Qq in[START_REF] Rajan | L 8 -algebra actions[END_REF]. The item 1 in Theorem 8.3.1 means that there exists a linear map Φ 0 : g ÝÑ X 0 pEq such that Φ 0 pxqrf s " ϱpxqrf s, and rQ, Φ 0 pxqs " 0, @x P g, f P O. (8.13)This morphism is not a graded Lie algebra morphism, but there exist a linear map Φ 1 : ^2g ÝÑ X ´1pEq such that for all x, y, z P g, Φ 0 prx, ys g q ´rΦ 0 pxq, Φ 0 pyqs " rQ, Φ 1 px, yqs. prx, ys g , zq ´rΦ 0 pxq, Φ 1 py, zqs`ö px, y, zq " rQ, Φ 2 px, y, zqs (8.15)

	Remark 8.3.2. Remark 8.3.3. (8.14)
	Also,
	Φ 1

  pE 1 qr1s, r¨, ¨s , ad Q 1 ˘.Proof. The composition of Φ with the horizontal map in the diagram (8.16) is a lift of the action ϱ. It is necessarily homotopy equivalent to Ψ by item p2q in Theorem 8.3.1.

	(8.16)

  Let us show Item 1. Note that Lemma 8.3.7 gives the existence of a linear map Φ 0 : g ÝÑ X 0 pEq such that, rQ, Φ 0 pxqs " 0 for all x P g. For x, y P g, consider Λpx, yq " Φ 0 prx, ys g q ´rΦ 0 pxq, Φ 0 pyqs.

	taken in polynomial-degree n `1 yields,
		0 " `Q ˝p Φ ˝Qg ´Q ˝Φq	˘pn`1q	" rQ, p Φ ˝Qg ´Q ˝Φq pn`1q s,
	since Q	p0q g " 0 and `Φ ˝Qg ´Q ˝Φ ˘piq	" 0 for i P t0, . . . , nu. It is clear that for all n ě 0 the map
	`Φ ˝Qg ´Q ˝Φ ˘pn`1q : S n`2 K g ÝÑ X ´npEqr1s take value in vertical vector fields on E. By virtue of
	Lemma 6.2.11 there exists a map ζ : S n`2 K g ÝÑ X ´n´1 pEqr1s such that
		rQ, Φpn`1q `ζs " Φpnq ˝Qp1q g ´Q p1q ˝Φ pnq .	(8.18)
	By redefining the polynomial-degree n`1 of Φ as Φpn`1q :" Φpn`1q `ζ. One obtains a Lie 8-morphism
	up to polynomial-degree n `1. The proof continues by recursion.
	Proof of Theorem 8.3.1. (8.19)
				The identity,
		Q ˝`Φ ˝Qg ´Q ˝Φ ˘``Φ	˝Qg ´Q ˝Φ ˘˝Q g " 0

  yq pαq :" xα, ηpx, yqy for all α P ΓpE ˚q, and extended it by derivation on the whole space. For every f P O, `Λpx, yq ´rQ, ι ηpx,yq s ˘pf q " pϱprx, ys g q ´rϱpxq, ϱpyqs ´ρpηpx, yqq rf s (by definition of Φ 0 )It is clear that Λpx, yq `rQ, ι ηpx,yq s is a ad Q -cocycle. Also, `Λpx, yq `rQ, ι ηpx,yq s ˘p´1q : for every α P ΓpE ˚q, rQ, ι ηpx,yq s p´1q pαq " rQ p0q , ι ηpx,yq spαq "

	" 0	(by definition of η)
		( ( ( ( ( ( ( ( ( Q p0q rxα, ηpx, yqys `(( ( ( ( ( ( ( ( xQ p0q rαs, ηpx, yqy " 0,

  where pΞ t q tPr0,1s is as in Definition 8.2.6, and for t P r0, 1s, H t is the unique Ξ t -co-derivation where the only non-zero polynomial-degree is H p0q " H 0 . Equation (8.25) gives a homotopy between Φ and Ξ 1 . When we consider the polynomial-degree zero component in Equation (8.25), one obtains

		p0q dΞ t dt	" Qp0q ˝Hp0q t	`Hp0q t	˝Qp0q g
			" rQ, H 0 s
			" Ψ 0 ´Φ0 " Ψp0q ´Φ p0q .
	Therefore, Ξ	p0q t " Φp0q `tp Ψp0q ´Φ p0q q, and Φ " Ξ 1 with Ψp0q " Ξ p0q 1 . Using Lemma 8.3.9, the image of
	any element through the map Ψp1q ´Ξp1q 1 : S 2 K g ÝÑ X ´1pEqr1s is a ad Q -coboundary. Thus, Ψp1q ´Ξp1q 1
	can be written as
		Ψp1q ´Ξp1q 1 " rQ, H p1q s, with H p1q : S 2 K g ÝÑ X ´2pEqr1s.	(8.26)
	Let us go one step further by considering the differential equation on r0, 1s given by
		$
		& % Ξ 0 dΘt dt	" Q ˝Ht `Ht ˝Qg " Ξ1	(8.27)
					(8.24)
					0 pEqr1s
	Consider the following differential equation
		$
		& % Ξ 0 " dΞt dt " Q ˝Ht `Ht ˝Qg , Φ	t P r0, 1s	(8.25)

  with ρ 1 pe, uq " p Ð Ý u , ϱpuq `ρpeqq, is a geometric resolution of pXpGqˆϱ Fq. Here, p : GˆM ÝÑ M is the projection on M .

		ϱpuqq u P g		(8.35)
		p0, Xq	X P F
	is a singular foliation that we denote by pXpGq ˆϱ Fq.
	2. If pE, d, ϱq is a geometric resolution of pM, Fq, then
	¨¨¨ÝÑ p ˚E´3	d ÝÑ p ˚E´2	d ÝÑ g ' p ˚E´1	ρ 1 ÝÑ T pG ˆM q	(8.36)

  38)where ΓpE ´1q " K ´i, comes equipped with a Lie 8-algebroid structure 1. whose unary bracket is d and whose anchor map ρ 1 , sends an element x ' e P g ' E ´1 to ϱpxq `ρpeq P ϱpgq `T F,2. the binary bracket satisfiesℓ 2 pΓpE ´1q, ΓpE ´1qq Ă ΓpE ´1q and ℓ 2 pΓpE ´1q, Γpgqq Ă ΓpE ´1q,3. the g-component of the binary bracket on constant sections of g ˆM is the Lie bracket of g. When we have ϱpgq X T m F " t0u for all m in M , Equation (8.38) is a free resolution of the singular foliation C 8 pM qϱpgq `F and we can apply directly the Theorem 4.2.1. Otherwise, we need to show there is no obstruction in degree ´1 while doing the construction of the brackets if the result still needs to hold.

	Remark 8.4.5.

  Thus, rd, l2 s RN |R ´2ˆR ´1 P dR ´2, because ppE ´iq iě1 , d, ρq is a geometric resolution. Therefore, rd, l2 s RN is a degree `2 element in the total complex z Page The O-bilinear operator rd, l2 s RN is D-closed in z Page

p1q pRq. p1q pRq, since rd, rd, l2 s RN s RN | R ď´2 " 0. So there exists τ 2 P ' jě2 Hom O

  Thus rrQ, Φ 1 px, yqs , ι e s p´1q " ι ∇ rx,ysg peq ´ι∇x˝∇ypeq `ι∇y˝∇xpeq .for some linear map γpx, yq : ΓpE ´'q ÝÑ ΓpE |e|´1 q depending linearly on x, y. Therefore, rrQ, Φ 1 px, yqs , ι e s p´1q " px, yq p0q , rQ, ι e s p´1q ı " ι ℓ 2 pηpx,yq,eq `"Q p0q , ι γpx,yqpeq ı´"Φ 1 px, yq p0q , ι ℓ 1 peq ı " ι ℓ 2 pηpx,yq,eq `ιℓ 1 pγpx,yqpeqq ´ιγpx,yqpℓ 1 peqq By equating the latter with (9.3) we can recapitulate as follows: ΓpE ´2q ùñ xα, ηprx, ys g , zqy ´´Φ 0 pxq p0q rxα, ηpy, zqys ´xΦ 0 pxq p0q pαq, ηpy, zqy ¯`ö" xQ p0q rαs, ζpx, y, zqy, In particular, if m P M is such that ℓ 1 | m " 0 and ℓ 2 | m " 0, then the map x Þ Ñ ∇ x defines an action on the isotropy Lie algebra g m of F, since ∇ x preserves the kernel of ρ. If in addition ηpx, yq P ker ρ m , then η| m : ^2 g ÝÑ g m is a cocycle of Chevalley-Eilenberg.

							ff	ff p´1q
	""	Q, ι ηpx,yq	‰	, ι e	‰ p´1q `«« Q,	ÿ	Φ 1 px, yq piq	, ι e
						iě0
					«		ffff p´1q		ff p´1q
	" ι ℓ 2 pηpx,yq,eq `«Q, Remark 9.1.2. Notice that by using the duality which is given in Proposition 8.4.1, the linear map ÿ Φ 1 px, yq piq , ι e ´«ÿ Φ 1 px, yq piq , rQ, ι e s
					iě0			iě0
					«		ff p´1q	fi
	" ι ℓ 2 pηpx,yq,eq `» -Q p0q , fl ´"Φ 1 ùñ ÿ iě0 Φ 1 px, yq piq , ι e	(9.3)
	On the other hand, Φ 1 px, yq admits a polynomial decomposition
	Φ 1 px, yq p´1q `ÿ iě0 xα, ηprx, ys g , zqy ´`( ( ( ( ( ( ( ( ( ϱpxqrxα, ηpy, zqys ´(( ( ( ( ( ( ( ( Φ 1 px, yq piq " ι ηpx,yq ϱpxqrxα, ηpy, zqys `xα, ∇ x ηpy, zqy ˘`ö px, y, zq " `ÿ iě0 Φ 1 px, yq piq xα, ℓ 1 pζpx, y, zqqy.
	and that We have used Equations (8.13) and (8.9) in the last line. Hence,
	«
	ÿ

iě0

Φ 1 px, yq piq , ι e ff p´1q " " Φ 1 px, yq p0q , ι e ı " ι γpx,yqpeq ,

Conclusion:

In general, the map g ÝÑ DerpEq, x Þ Ñ ∇ x is not a Lie algebra morphism even when the action ϱ is strict. In fact, there exists a bilinear map γ : ^2 g ÝÑ EndpEqr1s of degree 0 that satisfies ∇ rx,ysg ´r∇ x , ∇ y s " γpx, yq ˝ℓ1 ´ℓ1 ˝γpx, yq `ℓ2 pηpx, yq, ¨q, here ℓ 2 is the corresponding 2-ary bracket of pE, Qq, and η : ^2 g ÝÑ ΓpE ´1q is such that ϱprx, ys g q ´rϱpxq, ϱpyqs " ρpηpx, yqq.

4. Also, Equation (8.15) taken in polynomial-degree ´1 implies, that, for all α P ΓpE ˚1q Φ 1 prx, ys g , zq p´1q ´rΦ 0 pxq p0q , Φ 1 py, zq p´1q s`ö px, y, zq " rQ p0q , Φ 2 px, y, zq p´1q s.

ùñ ι ηprx,ysg,zq ´rΦ 0 pxq p0q , ι ηpy,zq s`ö px, y, zq " rQ p0q , ι ζpx,y,zq s, with ζ : ^3 g Ñ xα, ηprx, ys g , zq ´∇x ηpy, zqy`ö px, y, zq " xα, ℓ 1 pζpx, y, zqqy.

Since α is arbitrary, one obtains ∇ x ηpy, zq ´ηprx, ys g , zq`ö px, y, zq " ℓ 1 pζpx, y, zqq.

(9.4)

  Now for a P Γpaq we define Ý Ñ a :" Rps ˚aq. The map Ð Ý ' is defined similarly. Item 1, 2 and 3 hold by

	construction.
	Assume that pA, ρq is equipped with an almost Lie algebroid bracket r¨, ¨sA . For all a, b P ΓpAq
	one has
	ds	´Ð ÝÝÝ Ý ra, bs A ´rÐ Ý a ,	Ð Ý b s ¯" ρpra, bs A q ´rρpaq, ρpbqs
			" 0,

because Ð Ý a is s-projectable to ρpaq and ρ is a morphism of brackets. Since left-invariant vector fields are tangent to the fibers of t, one has dt ´Ð ÝÝÝ Ý ra, bs A ´rÐ Ý a , Ð Ý b s ¯" 0. The proof is similar for Ý ÝÝÝ Ñ ra, bs A ´rÝ Ñ a , Ý Ñ b s. This ends the proof.

  Item 1 is an immediate consequence of Proposition 10.1.16 and is in fact used in its proof. Let us prove item 2. By assumption, B is a neighborhood of p0, xq in M ˆRn with n " rk x pFq " dimpF

x :" F{I x Fq near x P M (see Example 10.1.2). Let b P B and U b an open neighborhood of b.

  d 1 ´1 π / / / / A .Once again, ker d 1 does not need to be a free O-module, therefore we can continue the procedure. Inductively, assume that we have constructed a O-linear map, d n : V ´n´1 Ñ V ´n, for n ě 2. Thus, there exists a free O-module V ´n´2 together with a surjective O-linear map π n`1 :V ´n´2 Ñ V ´n´1 such that V ´n´2 { ker π n`1 » ker d n .Joining this to the previous sequence, one get / / V ´n´1 dn / / / / V ´n / / ¨¨¨/ / V ´2 d 1 / / V ´1 π / / / / A (B.14

				/ / V ´2	π 1 / / / / ker π	; ; ; ; / / V
					d 1
	0	/ / ker d n`1	/ / V ´n´2	π n`1 / / / / ker d n	9 9 9 9
				d n`1

  It is easily checked that this gives a well-defined complex. An ordered sequence of elements f 1 , . . . , f r P O is called a regular sequence on a module V if the following conditions are satisfied 1. f i is not a zero divisor on the quotient V{xf 1 , . . . , f i´1 yV, 2. xf 1 , . . . , f r yV ‰ V. If pf 1 , . . . , f k q is regular sequence on O, then the Koszul complex (B.16) has no cohomology in degree less equal to ´1.Proof. Theorem 16.5. of[START_REF] Hideyuki | Commutative ring theory[END_REF]. For O " Krx 1 , . . . , x d s be the polynomial ring in d indeterminates, the Koszul complex which is associated to px 1 , . . . , x d q induces a free resolution of K » O{xx 1 , . . . , x d y. Let F be a singular foliation on M . A complex of vector bundles over F consists of a triple pE ' , d ' , ρq, where 1. E ' " pE ´iq iě1 is a family of vector bundles over M , indexed by negative integers.2. d pi`1q P HompE ´i´1 , E ´iq is a vector bundle morphism over the identity of M called the differential map 3. ρ : E ´1 ÝÑ T M is a vector bundle morphism over the identity of M called the anchor map with ρpΓpE ´1qq " F.

	Definition B.2.17. Proposition B.2.18. Example B.2.19. such that
	¨¨¨/ / E ´i´1 d pi`1q / / E ´i d piq / / E i´1	/ /	d p2q / / E ´1	ρ / / T M	(B.18)
		M	M	M		M	M
						16)
	whose differential d :	Ź ' V

d

ÝÑ

Ź '´1 V is the unique derivation of Ź ' V such that, d| V " F. In other words,

dpe i 1 ^¨¨¨^e i k q " k ÿ j"1 p´1q j´1 F pe i j qe i 1 ^¨¨¨ê i j ¨¨¨^e i k .

(B.17)

B.3 Geometric resolutions of a singular foliation

Let us start with this definition.

Definition B.3.1.

i.e. for every f P O the linear map E´1 f / / E´1 given by multiplication by f , is injective.

It should be understood that e.g, pQE b id `id b d dR qpαptqv b 1q " pQE pvq b α `v b d dR αq ptq.

Φ ˝QE 1 ´QE ˝Φ being a Φ-co-derivation, its component of polynomial-degree i is zero for 0 ď i ď n if only if its i-th Taylor coefficient is zero for 0 ď i ď n.

It should be understood that ι f pOq :" 0.

A vector field P P XpEq is said to be vertical if it is linear with respect to functions on M , in other words if P rf s " 0 for all f P O.

In particular, when the

2-ary bracket ℓ2 is zero at m, on elements of degree ´1 we have, Zpgmq " gm.

In real analytic case, the module Γpker ds1q X Γpker dt1q is locally finitely generated because of the noetherianity of the ring of germs of real analytic functions[START_REF] Frisch | Points de platitude d'un morphisme d'espaces analytiques complexes[END_REF][START_REF] Siu | Noetherianness of rings of holomorphic functions on stein compact subsets[END_REF].

here b is an "outer" tensor product, it should not be confused with the internal tensor product in T ' O V that denotes also its graded algebra structure.

the Krull dimension of R is by definition the supremum of lengths of all chains of prime ideals in R [Hid89], p. 30

A singular foliation which is generated by polynomial vector fields on K d .

ACKNOWLEDGEMENTS

I wish to express my gratitude to God and to several people for their help on the accomplishment of this thesis.

2. When generators of degree ´1 are involved, the higher Jacobi identities are obtained by doing the same procedure as in the proof of Proposition 4.4.18, that is, we first consider the higher Jacobi identities for the Poisson 8-structure (4.40), and we put aside the terms where t¨u 1 is applied to these degree ´1 generators. We then check that the latter terms are exactly the terms coming from an anchor map when the 2-ary bracket is applied to generators of degree ´1 and the pn ´1q-ary brackets of the remaining generators.

Vector fields vanishing on a complete intersection

Proposition 4.4.23. Let W Ă C n be an affine variety defined by a regular sequence φ 1 , . . . , φ k P O.

Then, the Lie 8-algebroid described in Proposition 4.4.22 is the universal Lie 8-algebroid of the singular foliation of vector fields vanishing along W .

Proof. In the notation of the proof of Proposition 4.4.22, K ' equipped with the derivation B "

Bµ k is a free O-resolution of the ideal I W of functions vanishing along W , since φ 1 , . . . , φ k is a regular sequence. Since XpC d q is a flat O-module, the sequence

(4.42) is a free O-resolution of the singular foliation I W XpC d q. The Lie 8-algebroid structure of Proposition 4.4.22 is therefore universal.

Example 4.4.24. As a special case of the Proposition 4.4.23, let us consider a complete intersection defined by one function, i.e. an affine variety W whose ideal xφy is generated by a regular polynomial φ P CrX 1 , . . . , X d s. One has a free resolution of the space of vector fields vanishing on W given as follows:

where µ is a degree ´1 variable, so that µ 2 " 0. The universal Lie 8-algebroid structure over that resolution is given on the set of generators by :

and t¨¨¨u k :" 0 for every k ě 3. It is a Lie algebroid structure. Notice that this construction could be also be recovered using Section 4.4.2.

Conclusion:

This chapter described the 1-1 correspondence "Lie-Rinehart algebras ÐÑ Lie 8-algebroids on acyclic complexes". It extends greatly [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] for singular foliations. The functor "ÐÝ" consists in the 1-truncation of the Lie 8-agebroid structure. The converse functor consists in taking any free resolution, and constructing the brackets by recursion.

We prove that it is unique by proving it is universal. Notice that we need the "complicated" notion of homotopy given in Definition 2.3.14.

Last, some examples of [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF] are conceptually understood, and new examples are given. Some algebraic constructions (blow-up, localization, germs, quotient) are also given.

Obstruction classes to the existence of a Lie algebroid with a surjective morphism onto the Lie-Rinehart algebra are also described.

We never assume finite rank here!

Part II

Geometric Applications

Lemma 5.1.17. The image of the map

3. If W is a complete intersection (i.e. I W " pφ 1 , . . . , φ r q and dim W " d ´r), then T a X W pC d q " T a W for all a P W reg .

4. More generally, for an arbitrary affine variety W , T a X W pC d q " T a W for all a P W reg .

Proof. Item 1. is given by the construction in item 2. of Remark 5.1.16. Let us show item 2: let pv 1 , . . . , v d q P C d . For a R W , there exists φ P I W such that φpaq ‰ 0. The vector field

Now we prove item 3. Let pv 1 , . . . , v d q P T a W " kerpJpaqq, with J :" ´Bφ i Bx j ¯i,j . By assumption, we have rkpJpaqq " r. Thus, J admits pr, rq-minor µ such that µpaq ‰ 0. We can assume that µ is the determinant of the first r-columns of J. Consider the vector fields

, for j P tr `1, . . . , du, understood as the cofactor expansion along the first row. Since for each i P t1, . . . , ru, H j rφ i s has two repetitive lines, therefore it vanishes. Therefore H j 's are tangent to W . We claim that the following vector fields does the job, namely

(5.4) Indeed, if we denote by µ 1 , . . . , µ r the minors associated to the partials B Bx 1 , . . . , B Bxr , respectively, then for every j the decomposition of H j reads

Hence,

¸

We also need to define the following Definition 6.2.5. A vector field X P XpM q is said to be an infinitesimal symmetry of F, if rX, Fs Ă F. The Lie algebra of infinitesimal symmetries of F is denoted by, spFq Example 6.2.6. Here are some examples.

1. Vertical vector fields on a graded manifold are longitudinal for any singular foliation.

2. For any Q-manifold pE, Qq over a manifold M . The homological vector field Q P XpEq is a longitudinal vector field for F :" ρpΓpE ´1qq: in local coordinates pU, x 1 , . . . , x n q on M and a local trivialization ξ 1 , ξ 2 , . . . of graded sections in ΓpE ˚q. The vector fields Q are of the form:

Bx j P FpUq for every k. One has for all f P C 8 pUq, Qpf q "

3. For pE, Qq a Q-manifold and F :" ρpΓpE ´1qq its basic singular foliation. For any extension of a symmetry X P spFq of F to a degree zero vector field p X P XpEq, the degree `1 vector field rQ, p Xs is longitudinal for F.

Let us show this last point using local coordinates px 1 , . . . , x n q on M and a local trivialization ξ 1 , ξ 2 , . . . of graded sections in ΓpE ˚q. The vector fields Q and p X take the form:

. By using Equation (6.20) we note that all the terms of rQ, p Xs are vertical except maybe for the ones where the vector field X appears. For k ě 1, the vector field

Bξ j , Xs is vertical; and for every fix k, one has

Thus,

CHAPTER 9

On weak and strict symmetries: an obstruction theory

In this chapter, we apply the main theorems of 8.2 of Chapter 8 to define a class obstructing the existence of strict symmetry action equivalent to a given weak symmetry action.

Introduction

Recall that Theorem 8.3.1 assures that any weak symmetry action ϱ : g Ñ XpM q of a Lie algebra g on a singular foliation F admits a lift to a Lie 8-morphism Φ : pg, r¨, ¨sg q ù pX ' pEqr1s, r¨, ¨s , ad Q q.

(9.1)

To understand the compatibility conditions between the low terms of Φ see Remark 8.3.3. By playing with the definition of Φ we make the following observation.

Remark 9.1.1. What does Φ induces on the linear part of pE, Qq? We have seen in Lemma 8.2.5 and Proposition 8.2.10 that the 0-th Taylor coefficient of the lift Φ induces a linear map x P g Þ ÝÑ p∇

x : E ´i ÝÑ E ´iq for every i ě 1 that satisfies

1. Also, for every x P g and e P ΓpE ´1q, ρp∇ x peqq " rϱpxq, ρpeqs.

Definition 9.2.10. A point p P W is said to be strongly singular if for all f P I W , d p f " 0 or equivalently if for all f P I W and X P XpC d q, one has Xrf sppq P I p .

Example 9.2.11. Any singular point of a hypersurface W defined by a polynomial φ P Crx 1 , . . . , x d s is strongly singular.

The lemma below is immediate.

Lemma 9.2.12. In a strongly singular point, the isotropy Lie algebra of the singular foliation F "

Corollary 9.2.13. Let ϱ : g ÝÑ XpW q be a Lie algebra morphism.

1. Any extension r ϱ as in Example 8.1.11 is a weak symmetry action for the singular foliation F " I W XpC d q.

2. For any strongly singular point p in W if the class clpηq does not vanish the strict action ϱ : g ÝÑ DerpO W q can not be extended to the ambient space.

Let us give an example of a Lie algebra action on an affine variety that do not extend to the ambient space.

We hope to construct an example as follows Example 9.2.14. Let W Ă C d be an affine variety generated by a regular homogeneous polynomial φ P O " Crx 1 , . . . , x d s of degree ě 2. Assume there exists two vector fields X, Y P XpC d q that satisfy Xrφs " f φ, Y rφs " gφ, with f, g P I 0 , and such that rX, Y s " φZ, for some Z P XpC d q. Consider the action of the trivial g " R 2 on W that sends its canonical basis to X, and Y respectively. It is a weak symmetry action on the singular foliation F φ :" xφyXpC d q and induces a Lie algebra map, g ÝÑ XpW q.

Notice that the universal Lie algebroid of F φ is a Lie algebroid (see Example of [START_REF] Laurent | Lie-Rinehart algebras » acyclic Lie 8algebroids[END_REF]) because,

Here µ is a degree ´1 variable, so that µ 2 " 0. The universal algebroid structure over that resolution is given on the set of generators by:

and ℓ k :" 0 for every k ě 3. Write Z "

where e 1 , e 2 is the canonical basis of R 2 . This Lie 8-algebroid structure satisfies all the assuptions of Proposition 9.2.6. Assume that the vector field Z does not vanish at zero. Since the action is trivial at zero and η | 0 ‰ 0, therefore its class is non-zero. By consequence, such action cannot be extended to ambient space.

Let us make it explicit Example 9.2.15. Let W Ă C 2 be the affine variety generated by the polynomial φ " F G with F, G P Crx, ys ": O. We consider the vector fields U " F X G , V " GX F P XpC 2 q, where X F and X G are Hamiltonian vector fields w.r.t the Poisson structure tx, yu :" 1. Note that U, V are tangent to W , i.e. U rφs, V rφs P xφy. It is easily checked that rU, V s " φX tF,Gu .

The action of the trivial Lie algebra g " R 2 on W that sends its canonical basis pe 1 , e 2 q to U , and

V respectively, is a weak symmetry action on the singular foliation F φ :" xφyXpC 2 q, and induces a Lie algebra map, ϱ : g ÝÑ XpW q.

(9.10)

Here µ is a degree ´1 variable, so that µ 2 " 0. The universal algebroid structure over that resolution is given on the set of generators by: for some linear map β : g ÝÑ g p0,0q . Therefore, Equation (9.14) is impossible by Equation (9.13) and since η | p0,0q ‰ 0. In orther words, its class clpηq does not vanish. By Corollary 9.2.13 (2), the action ϱ given in Equation (9.10) cannot be extended to ambient space.

Conclusion:

We applied the existence of a Lie 8-morphism to the question of lifting to M an g-action on M {F, i.e. of making strict a weak g-action.

We apply this question to several geometric issues, e.g. neighbourhood of leaves, Lie algebra actions on an affine variety.

CHAPTER 10

Bi-submersion towers

Symmetries of bi-submersions

In this chapter, we introduce the notion "bi-submersion towers". The work contained in this chapter is entirely original, except for the notion below that arose in a discussion between C. Laurent-Gengoux, L. Ryvkin, and I, and will be the object of a separate study. • B is a manifold,

• s, t : B Ñ M are submersions, respectively called source and target, such that the pull-back singular foliations s ´1F and t ´1F are both equal to the space of vector fields of the form ξ `ζ with ξ P Γpkerpdsqq and ζ P Γpkerpdtqq. Namely, s ´1F " t ´1F " Γpkerpdsqq `Γpkerpdtqq.

(10.1)

In that case, we also say that pB, s, tq is a bi-submersion over pM, Fq.

Example 10.1.2. Let F be a singular foliation over a manifold M . For x P M and X 1 , . . . , X n P F inducing generators for F x :" F{I x F. We know from [AS09] that there is an open neighborhood W of px, 0q P M ˆRn such that pW, t, sq is a bi-submersion over F, where spx, yq " x and tpx, yq " exp x ˜n ÿ

i"1

where for X P XpM q, φ X 1 denotes the time-1 flow of X. Such bi-submersions are called path holonomy bi-submersions [START_REF] Androulidakis | Smoothness of holonomy covers for singular foliations and essential isotropy[END_REF].

Proof. Let r X P XpBq be a lift of X P XpM q. Since r X is s-projectable, r r X, Γpker dsqs Ă Γpker dsq. Since r X is t-projectable, r r X, Γpker dtqs Ă Γpker dtq. Hence: r r X, s ´1pFqs " r r X, Γpker dsq `Γpker dtqs " r r X, Γpkerpdsqs `r r X, Γpker dtqs Ă Γpker dsq `Γpker dtq " s ´1pFq.

In words, r X is a symmetry of s ´1F. Since r X projects through s to X, X is a symmetry of F.

We investigate the existence of lifts of symmetries of F to bi-submersions over F.

Remark 10.1.13. For a given X P spFq, 1. the lift r X to a given bi-submersion is not unique, even when it exists. However, two different lifts of a X P spFq to a bi-submersion pB, s, tq differ by an element of the intersection Γpkerpdsqq X Γpkerpdtqq.

r

X is a symmetry of Γpkerpdsqq X Γpkerpdtqq: r r X, Γpkerpdsqq X Γpkerpdtqqs Ă Γpkerpdsqq X Γpkerpdtqq, since r X is s-projectable and t-projectable.

As the following example shows, the lift of a symmetry to a bi-submersion may not exist.

Example 10.1.14. Consider the trivial foliation F :" t0u on M . For any diffeomorphism ϕ : M ÝÑ M , pM, id, ϕq is a bi-submersion over F. Every vector field X P XpM q is a symmetry of F. If it exists, its lift has to be given by, r X " X since the source map is the identity. But r X " X is t-projectable if and only if X is ϕ-invariant. A non ϕ-invariant vector field X therefore admits no lift to pM, id, ϕq. However, internal symmetries, i.e. elements in F admit lifts to any bi-submersion.

Proposition 10.1.15. Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M . Every internal symmetry, i.e. every vector field in F, admits a lift to pB, s, tq.

Proof. Let X P F. Since s : B ÝÑ M is a submersion, there exists X s P XpBq s-projectable on X. Since t is a submersion, there exists X t P XpBq t-projectable on X. By construction X s P s ´1pFq and X t P t ´1pFq. Using the property (10.1) of the bi-submersion pB, s, tq, the vector fields X s and X t decompose as

By construction, X s t is s-projectable to X and t-projectable to 0 while X t s is s-projectable to 0 and t-projectable to X. It follows that, r X :" X t s `Xs t , is a lift of X to the bi-submersion pB, s, tq.

Let us make the Proposition 10.1.15 more precise. The proof we give for Theorem 10.1.6 uses the notion of left-invariant, resp. right-invariant, vector fields on a bi-submersion over a singular foliation. We define the latter in the next proposition. It uses the notion of anchored bundle over a singular foliation and almost Lie algebroid, see [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF][START_REF] Laurent-Gengoux | Geometry of singular foliations: a draft of an introduction[END_REF] for more details.

APPENDIX A

Tensor algebra

We have used almost everywhere in the thesis, modules or vector spaces that arise as the quotient of the tensor algebra. It is worth it to dedicate a section to recall the construction and some basic facts on tensor algebras. In this chapter we assume that the reader is familiar with the notion of O-modules, graded O-modules, and vector spaces.

It is known that many algebras such as the exterior algebra, symmetric algebra, Clifford algebras [START_REF] Todorov | Clifford algebras and spinors[END_REF], universal enveloping algebras [Bek] and many other algebras are the quotient of tensor algebras. These make the tensor algebra a fundamental and a very useful notion. In this thesis we have dealt a lot with O-multilinear maps, it simply means that these maps are O-linear w.r.t each argument while we fix the other arguments. The tensor algebra is used to characterize multilinear relations between algebraic objects related to modules or vector spaces. There are several ways to construct the tensor algebra, we refer the reader e.g. to chapter 16 of [START_REF] Lang | Algebra. Graduate Texts in Mathematics[END_REF] or [And,[START_REF] Keith | Tensor products[END_REF] for more details on the matter.

When O " K the reader may replace "O-module" by "K-vector space" and the construction of the tensor algebra that we give below works the same.

A.1 Tensor product

2. Furthermore, if there is another O-module Q and a O-bilinear map, p 1 : V ˆW Ñ Q with the property (A.1), then there exists a unique isomorphism i :

Proof. We first show "uniqueness" when such O-module exits, then show existence. Uniqueness: Suppose that there exist two pairs pVbW, p : VˆW Ñ Vb O Wq and pV bW, p : VˆW Ñ V bO Wq satisfying the property A.1.1. By using two times A.1.1, we deduce the existence of a unique O-linear map i : V b O W Ñ V bW, and another one j : V bO W Ñ VbW such that i ˝p " p and j ˝p " p. This implies that j ˝i ˝p " p and i ˝j ˝p " p. By unicity of the factorization in A.1.1, we must have i ˝j " id and j ˝i " id. This proves item 2.

Existence: Let us consider the free O-module P generated by elements of the Cartesian product V ˆW, i.e. elements of P are formal finite sums of elements of the form f pv, wq, with f P O and v P V, w P W. Next, we divide by the submodule R Ă P of P generated by elements of the following types:

pv 1 `v2 , wq ´pv 1 , wq ´pv 2 , wq, pv, w 1 `w2 q ´pv, w 1 q ´pv, w 2 q, pf v, wq ´f pv, wq, and pv, f wq ´f pv, wq which are the relations that elements of the tensor product must satisfy with v, v 1 , v 2 P V and w, w 1 , w 2 P W and f P O. In those notations, the tensor product of V and W over O is defined as the quotient

For v P V, w P W, we denote the class of pv, wq P V ˆW by v b w P V b O W. This quotient comes with the natural map, p :

For any O-linear map B : V ˆW Ñ Z, the O-linear map q : P Ñ Z which is given on the basis of P by px, yq Þ Ñ qppx, yqq :" Bpx, yq clearly goes to quotient to define a O-linear map s q : V b O W Ñ Z, since e.g.

q ppv 1 `v2 , wq ´pv 1 , wq ´pv 2 , wqq " qppv 1 `v2 , wqq ´qppv 1 , wqq ´qppv 2 , wqq " Bpv 1 `v2 , wq ´Bpv 1 , wq ´Bpv 2 , wq " 0, by bilinearity of B also, qppf v, wq ´f pv, wqq " qppf v, wqq ´f qppv, wqq, by O-lineraity of q " Bpf v, wq ´f Bpv, wq " 0, by O-bilineraity of B.

We did everything to get, s q ˝p " B. Therefore, we can take γ " s q to satisfy (A.1).

Proposition A.1.2. Let V, W, Z be O-modules. We have the following isomorphisms

Proof. For item 1, the map pv, wq

For item 2, the map pf, vq P O ˆV Ñ f v also induces an isomorphism whose inverse is

A similar map gives the second clause.

For item 3, the isomorphism is, obviously, pv

This is trivially extended to more O-modules.

Remark A.1.3. A similar construction as in Theorem A.1.1 can be done for a finite family of Omodules V 1 , . . . , V r , as in the case of bilinear maps. Then V 1 b O ¨¨¨b O V r is a universal object that factorizes r-multilinear maps, defined on V 1 ˆ¨¨¨ˆV r . We naturally have,

So, in this thesis, we make no difference in how we denote the elements pv

for homogeneous elements c 1 , c 2 P C and c 1 1 , c 1 2 P C 1 . Moreover, if C and C 1 are unitary with units 1 C and 1 C 1 , respectively, then the algebra C b O C is also unitary with unit 1 C b 1 C 1 . Here, to avoid explosion of notations, we denote the product of two elements a, b by ab. Also, we will not make notational distinctions between the unit elements.

Homological algebra

The goal of this chapter is to introduce some important results on homological algebra, which are used in this thesis. Although, these are classical notions in commutative algebra, I think it is important to make a brush-up on them for the readability of the thesis. Most of the notions of this chapter can be found in [Cha14, Eis95, Hid89, Mic07], I also have learned a lot from the Lecture notes [START_REF] Lassueur | Cohomology of Groups[END_REF].

B.1 Complexes of modules

We recall that a module over O is like a vector space in the sense that all the axioms still hold, except that the underlying field is replaced by O. In this section, when O " K, the reader may replace "module" by "vector space".

For us, a Z-graded module over O is a module V endowed with a direct sum decomposition V " ' iPZ V i of O-modules. We simply say "O-modules" for graded modules which are concentrated in degree zero. For every i P Z, elements of V i are said to be of degree i. Let V and W be graded

such that the V ´i's are free/projective modules.

Remark B.2.4. The complex (B.12) may be of infinite length. When the length is finite, we say that we have a finite free/projective resolution. In that case, the sequence

is exact in every degree. In particular, the sequence 0 Ñ

Proposition B.2.5. Every O-module A admits a free/projective resolution.

Proof. Firstly, notice that every module is isomorphic to a quotient of a free module: to see this, choose a set of generators tv But ker π does not need to be a free O-module, but there is a free O-module V ´2 together with a surjective O-linear map V ´2 π 1 / / / / ker π such that V ´2{ ker π 1 » ker π. This is added to the previous sequence as follows

Tor complex

Let pV ' , d V , πq be a projective resolution of an O-module A. For any O-module B, we consider the complex

We define for i ě 0 Tor ´ipA, Bq :" H ´ipV ' b O Bq.

Here are some properties of Tor [START_REF] Lassueur | Cohomology of Groups[END_REF].

Proposition B.2.9. The functor Tor satisfies

2. If A is projective, then Tor ´ipA, Bq " 0 for every i ě 1.

3. If B is flat, then Tor ´ipA, Bq " 0 for every i ě 1.

4. For all i, Tor ´ipA, Bq " Tor ´ipB, Aq.

5. The construction of TorpA, Bq is independent of the choice of the resolution, i.e. any other projective resolution of A or B yields the same Tor groups.

Minimal resolutions

Detailed definitions on local rings can be found in [START_REF] Masayoshi | Interscience tracts in pure and applied mathematics[END_REF].

Definition B.2.10. 1. A ring R is said to be a local ring if it has a "unique maximal ideal", i.e., a proper ideal m Ă R such that m contains every other ideal of R.

2. A local ring R with maximal ideal m is called regular if m can be generated by n elements, where n " dim R is the krull dimension 1 of R.

Assume now that R is a local ring with maximal ideal m, and let K :" R{m. Geometrically, local rings correspond to germs of functions on a manifolds or affine variety at a point.

We assume that pR, mq is a local Noetherian commutative ring. Here is an important lemma that uses definition of local rings. Lemma B.2.11 (Nakayama). Let V be a finitely generated R-module such that r " dimpV{mVq ă 8.

Then, any basis of the vector space V{mV lifts to a (minimal) generating set for V as a R-module. In particular, V can be generated by r elements.

Proof. [START_REF] Eisenbud | The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra[END_REF] Lemma 1.4, p. 6. Remark B.2.12. By Nakayama Lemma, a local ring R is regular if the dimension of the R{m-vector space m{m 2 is dim R.

Since Impd pi`1q q Ď ker d piq for every i P N, we are allowed to define the quotient spaces, They are modules over functions on U. For each i ě 1, H ´ipE ' , Uq is called the i-th cohomology of pE ' , d ' , ρq at the level of sections on U.

(a) We way then pE ' , d ' , ρq is a geometric resolution of F if for every open set U Ă M and i ě 1, if it induces an exact complex on the level of sections, that is

for every open subset U 1 Ă U. We can look at the quotient vector spaces: Here we call H ´ipE ' , mq the i-th cohomology of pE ' , d ' , ρq at the point m.

It is important to notice the following: even if pE ' , d ' , ρq is a geometric resolution of F, there are no reasons for pE ' , d ' , ρq to be exact at m. For example, if the resolution is minimal at some point m P M , then H ´ipE ' , mq » E ´i|m for each i ě 2 and H ´1pE ' , mq » kerpρ |m q.

Remark B.3.3. Most of the definitions and operations on chain complexes of modules are adapted in a obvious manner to complexes of vector bundles over M , both on the level of the sections or at a point. See also [START_REF] Laurent-Gengoux | The universal Lie 8algebroid of a singular foliation[END_REF][START_REF] Lavau | Lie 8-algebroids and singular foliations[END_REF] for more.