Anne Sénéchal

Françoise Crestey

DO254 User group, an industry initiative

Keywords: certification, safety, DO-254/ED-80, Hardware Design and Verification

Facing the increasing use of complex electronic hardware for most of the safety critical aircraft functions, a joint RTCA/EUROCAE committee elaborated and released, in 2000, the first standard applicable to hardware development : DO-254/ED-80 « Design assurance guidance for airborne electronic hardware » standard. This document, providing guidance on design assurance activities, became the standard to design hardware components like FPGAs, ASICs and other PLD components, mandatory for certification of civil aircraft programs.

In 2004, several European avionic companies decided to collaborate on application of DO-254/ED-80, and created DO-254/ED-80 User Group. Objectives shared by members were to clarify DO-254/ED-80 recommendations, define "industry" practices compliant with certification objectives, share good or bad experiences and practices, and also propose industrial feedback to certification authorities.

This paper presents the DO254 User Group. Two members companies Barco and Rockwell Collins are sharing their experiences with DO254 and the User group, and develop their interest through such an initiative.

Introduction

The use of increasingly complex electronic hardware for most of the safety critical aircraft functions generates new safety and certification challenges. These challenges arise from a concern that aircraft functions may be more and more vulnerable to the adverse effects of hardware design errors which become difficult to manage due to the increasing complexity of the hardware. To counteract this perceived increased risk it has become necessary to ensure that the potential for hardware design errors is addressed in a more consistent and verifiable manner during both the design and certification processes. In April 2000, a joint RTCA/EUROCAE committee released the DO-254/ED-80 "Design assurance guidance for airborne electronic hardware" document, in which guidance resides on conducting design assurance activities along hardware development process. Today, the DO-254/ED-80 compliance is mandatory for aircraft programs, DO-254/ED-80 application being required by Airworthiness Authorities (for example by FAA AC 20-152) and customer directives. This document is now the standard for the design of hardware components like FPGAs, ASICs and other PLD components. European aircraft manufacturers have been applying DO-254/ED-80 during their development activities on most of the recent programs. As mentioned before, the standard, released in 2000, was the first standard applicable to hardware.

DO-254 and User Group origins

Although giving recommendations for hardware design process, the DO-254 standard created quite a number of clarification needs, which were emphasized by the fact that misinterpretations could finally result in safety issues, and in unacceptable extra costs.

Founding of DO254 User Group

In 2004 a number of companies decided to collaborate and join efforts to climb the learning curve. The initiative started with 13 companies around a common objective "Have a clear and common understanding of the DO-254/ED-80 objectives and processes".

For avionic industrials it is indeed essential to be capable of converting standards into industrial processes compliant for certification activities. Actually the DO254 User Group created the opportunity to exchange good and "less good" experiences, challenge own practices and techniques, while respecting each other's technology discretion. After few meetings to "get in touch", the companies involved quickly understood the benefits of such initiative. Sharing is consequently giving an opportunity to learn from others' experience, but also procuring a chance to provide feedback.

From time to time, members have shared feedback from audits with customers and Airworthiness Authorities, thus having access to the kind of request or concerns that they would have to be prepared for.

Barco sharing experience and interest for DO254 User Group

Barco joined DO254 User Group since its creation in 2004 and is as such a founding member.

At start, for Barco, Avionics Display specialist, the main objective when new standard got released was to fully understand the new requirements, avoiding misinterpretation of terminology in order to obtain full compliance of its development process to the new standard. It is indeed essential on aircraft programs as also in other domains like automotive, to be first time right when designing the hardware as well as on certification aspects. Sharing feedbacks and clarification from authorities between User group members helped and accelerated each other's in getting knowledge and awareness.

The challenging part when having new standards for Barco as for other companies is to translate standards into concrete steps in the design and verification flow. Starting from plans elaboration, requirement capture & validation up to the details of formulating dedicated questions in checklists, all of these activities need to address the core essence or spirit of DO-254 standard.

For Barco, it was a crucial target at the time of User group's creation as its first DO-254 project was a level A Primary Flight Display for Honeywell. Flying today on Pilatus Aircraft PC12, this primary Flight Display has been successfully developed and certified with highest Design Assurance Level (DAL A) 1 .

Understanding and anticipating the evolution of certification requests are key assets for Barco to prepare its future, upgrade its hardware development process, train its personnel.

Over the past few years, Barco has collected a successful DO-254 experience on various avionics programs like for instance on its multi-purpose display development in DO-254 DAL B, selected by Thales for ATR 600, Sikorsky S76D, Lockheed C130 and Dassault ATL2. Moreover with its Primary Flight Displays DAL A on Falcon900 and some coming platforms ongoing in certification process, Barco has 1 Note: Highest Design Assurance Level DAL A is requested to equipments or subsystems whose failure or anomalous behavior would result in a catastrophic failure condition for the aircraft. There are five system development assurance levels, A through E, corresponding to the five classes of failure conditions: catastrophic, hazardous/severe-major, major, minor and no effect. consolidated its hardware "DO-254 compliant" process.

DO254 User group is also offering the possibility for experienced industrial companies to provide some feedbacks on DO-254 implementation, which (when converging) can result in proposal of new practices.

Distinction between Simple and Complex electronic

Hardware is an example of topic on which User Group members expressed feedbacks, challenged their opinions and finally proposed some methodology by defining specific criteria for Simple/Complex electronic Hardware classification, used in the certification process.

Establishing a process and having it performed efficiently needs efforts for development teams, which have to adapt and constantly align their processes with technology evolution and also certification concerns.

DO-254 objectives and what is beyond the DO-254 document itself (Certification Review Items, Issue Papers, CAST guidance...) can represent important challenges for equipment suppliers and their subcontractors.

Exchanging feedback between industrial companies on audits with customers and Airworthiness Authorities eases independent Avionic display provider Barco, to speed up its process evolution towards the requests of aircraft manufacturers and certification authorities.

In addition to the civil aviation requests development process compliant to DO-254 has become attractive for military programs for safety aspects as well as for reliability objective. Barco is currently providing DO-254 compliant equipments on various military programs, such as the multifunctional display "Touch-screen Unit" developed for Agusta Westland selected by UK Royal Navy for its helicopter Merlin fleet..

3-DO254 User Group Evolution

User Group organization DO254 User Group is currently composed of 35 members from aerospace companies, certification experts, components manufacturers. An exhaustive list of participants can be found on DO254 User Group Website (www.do254.com).

User Group meetings take place about four times a year, and are hosted by one company member. It is the occasion to share the progress per topic in plenary sessions. Sub-groups have been constituted to particularly address specific hardware topics, being currently:

• Verification coverage in FPGA/ASICs development -What are the current verification practices and various combinations to fulfil 100% of functional coverage?

• IP/SOC -How to include them in a DO-254 project, what is needed from IP provider?

• Simple/Complex devices -What are the criteria for classifying a component as simple? What is the "minimum" verification flow associated for simple devices?

• COTS -How to get design assurance on these components?

• Reliability -How to know if components are produced in a reliable way? What are the inevitable key features to survey? The hardware design process, which was already in place, has been evolving from that time, to reach DO-254 objectives, especially in term of process formalization and documentation production. For these equipments, demonstration for certification used product service experience and TSO authorization, and moreover, DO-254 process was applied on FPGA modification. Certification approvals were obtained for them in 2006-2007, from DGAC for Eurocopter Dauphin, and from ENAC for Agusta A109.

While not currently a requirement for all TSO equipments, since the 2005 time frame, Rockwell Collins has been proactively anticipating the introduction of DO-254 requirement in development of TSO equipments by including DO-254 compliance artifacts with the TSO applications.

Today, various hardware developments (around display, radar, navigation, communication systems) are in progress at RCF and/or RCI premises, applying DO-254 guidance. These projects concern DAL A, B, C equipments, with use of COTSs, and development or reuse of ASICs and HPLDs.

Over the past ten years, these various projects involving hardware development have contributed to give to Rockwell Collins a solid experience in term of DO-254 hardware development.

One big interest for a company like Rockwell Collins to participate to DO-254 user group is obviously the possibility to share experience with other avionic suppliers. It allows highlighting good practices (in term of both efficiency and compliance with DO-254), and defining some generic acceptable means to apply DO-254 on hardware development.

Discussing with hardware components/tools vendors is also profitable. On our side, we have the opportunity to present them our needs with regard to technical performance and application of DO-254. This can be then collated with their own requirements and constraints.

Linked to our own experiences (use of new technology, application of new regulation …), we can propose new hardware design topics. Working together with sharing of experiences and opinions on the introduction of these new topics in our hardware design can help us defining appropriate practices. This can result in production of technical/process notes which will be then introduced into methodologies put in place by companies.

Finally, establishment of a liaison between DO-254 users Group and Airworthiness Authorities is very important. Giving them our problematic and our proposals in term of hardware development, and collecting their feedback with regard to the certification requirements and their potential evolution, will allow industries to move forward in an acceptable and efficient way in their conduction of hardware development processes compliant with DO-254.

New phase & activities for DO254 User Group

The DO254 User Group, having acquired a certain maturity in hardware certification process, is now entering a new phase. From fruitful discussions and work collaboration, industrial companies are elaborating some proposals on various technical subjects.

These proposals can be either oriented towards industry or more intended to certification organizations (CAST, certification authorities…) depending on the subject they address.

One pending proposal relates to the IP/SoC paper, intended for IP/SoC providers. Target of the paper is to ease their use in DO-254 projects by promoting somehow IP/SoC design practices. This paper will help to perform a gap analysis, promote design and verification activities and encourage the production of some necessary output data. Indeed, to use IPs in a DO-254 context, IP providers could contribute by giving access to IP-related data and/or by preparing DO254-oriented packages.

Thus the IP/SoC paper will also help IP providers to better understand what their avionic customer's challenges are. Some other subjects are on the roadmap such as The prepared proposals can be "challenged" when relevant by US industry, giving a wider impact and collection of feedbacks through European and US DO254 User Groups.

The work performed on HDL design rules, collecting practices and formulating a proposal of design rules set illustrates the cross-collaboration between the US and European User Groups. This package has already been implemented by some tools, what helps to perform effective verification of these rules. These coding guidelines have also been recently submitted to the CAST.

4-DO254 Standard in few words

The main concern which is at the origin of certification principle is Safety. The question is: how to ensure that avionics systems have been safely designed and manufactured? I.e. how to avoid design and manufacturing errors that could lead to occurrence of hazardous events? To cover problematic of design errors, different standards were jointly issued by US RTCA and European EUROCAE committees: ARP4754 at system level, DO-178B at software level and DO-254 at hardware level. These guidance documents propose structured development processes allowing meeting the safety requirements emitted by Airworthiness Authorities.

The DO-254 document describes the hardware design life cycle processes, detailing objectives and activities associated to each phase of the life cycle (see Figure below).

One could say that among DO-254 processes, requirement is the kernel. Then each process is defined to "serve requirement".

Announcing upfront how product will be developed

Starting with planning process, DO-254 requests to define the methodology and the means to produce hardware items, which ones will have to satisfy System and Certification requirements. All development plans are the foundations of development and certification, and need to be released at start of product development cycle.

Defining "WHAT" has to be developed

Once having collected the system requirements, how will these requirements be flowed down to hardware level? The work consists in identifying and documenting the hardware requirements in accordance with the system requirements. The resulting hardware specification contains requirements inferred from system requirements, and derived requirements which are resulting from design decisions.

Requirements need to be carefully reviewed and validated to guarantee from start that they are adequate with the system safety and well responding to the target application of the product. This validation allows identifying errors or omissions early in the development cycle, and thus reducing exposure to subsequent redesign or inadequate system performance.

During requirements validation, traceability is established between hardware requirements and system requirements, providing trace of how system requirements are covered at hardware level.

Describing Product architecture at Concept level

It is essential prior to rush upon HDL coding or board design that HW architecture is defined early in development phase. This concept phase obviously focuses on defining an architecture that allows to fulfil simultaneously all hardware requirements, while identifying reliability, maintenance and test features. A way to get this confidence is to review HDL design with regard to hardware requirements, and to establish the traceability between these two steps.

Unused functions are also to be considered according to DO-254. It is mandatory to identify the potential effects on safety of the unused functions implemented in the product. The next step called "implementation process" builds the physical hardware item according to its associated design data.

At the end of the hardware life cycle, the production transition phase establishes the baseline that includes all design and manufacturing data needed to support the consistent replication of the hardware item, in line with the key attributes of the unit on which the certification is based. DO-254 also identifies several supporting processes, which are of a matter of importance to establish design assurance of the final hardware product.

Does the designed hardware meet its requirements?

The supporting process to answer this question is obviously called the verification process, whose objective is getting assurance that each level of the hardware design life cycle meets its specified requirements. It provides confirmation that the intended functions have been correctly implemented.

Verification is one of the key areas of concern within any project. As design grows, becoming more and more complex, verification is responsible for an increasing proportion of the design cycle. Moreover an ineffective verification methodology can result in months or even years of debugging work in laboratory. Thus, establishing a verification methodology that catches more bugs, earlier in the design process, and reduces debugging time in the lab, should be the goal for any project.

Today, elsewhere in industry, electronics companies are adopting a whole breed of verification technologies to ensure a high quality of results as well as more productive and efficient verification process.

Avionic domain with DO-254 adds a strong but powerful request: the independence of verification. Despite sometimes the effort it demands, getting "new eyes" in a review, using an independent verification or analysis tool in a flow, can procure a considerable added value. For DAL A&B Hardware development, independence is mandatory in verification activities.

To get insurance that performed verification correctly and completely covers the hardware requirements, reviews should be organized, and traceability should be established between hardware requirements and the verification procedures and results.

Are the development tools bug-free? Are the tools telling the truth?

Tool assessment is another challenging area of DO-254 that occurs alongside each step in the development methodology. Any time a life cycle task is automated, reduced or eliminated, the tool that takes over this work must go through a process called "tool assessment and qualification". The purpose of this assessment is to ensure that the tools used to design, generate or verify the hardware don't introduce any error in the hardware or don't corrupt verification results.

What is released? Which release is applicable? What are the changes since previous release?

Configuration management process provides a technical and an administrative control of the configuration. It allows controlling the changes of the equipment/component and associated relevant documentation, and gives assurance that physical archiving, recovery, and control are maintained for documentation. Especially, version management is used for hardware design files. Typically, tools used for configuration management run the entire project, including the hardware, software, and parts systems.

Who guarantee that the development has been done according to the plans?

Avionic company who pretends to certify products shall perform process assurance activities, which consists on verifying that each process has been performed according to the plan. Organization of audits, identification of non-compliances and/or inadequacies and follow up of actions contribute to this process.

Get the Hardware certified!

Certification liaison consists in establishing communication and understanding between the company (air framer or equipment supplier) and the certification authority. This includes approval by certification authority of certification data (like hardware certification plan).

The certification is the legal recognition by the certification authority that a product, service, organization or person complies with the requirements. For avionic certification, to reach this recognition, airworthiness authorities conduct audits at various steps of the development process. In this frame, specific audits are conducted on hardware processes to assess their compliance with DO-254 and associated regulations.

For a company manufacturing hardware equipments and components, compliance with DO-254 standard implies finding acceptable ways of meeting the objectives of each phase of the DO-254 life cycle. Indeed, the challenge for companies is to manage maintaining profitability, while setting up and applying a methodology compliant with DO-254 recommendations. One interest of DO254 User Group is precisely to help companies defining appropriate methodologies.

Conclusion

This DO254 User Group is a great success, and has been recognized internationally as a model to use when new standards are released. There is no fee to join, and each member contributes and benefits from the quality of the exchanges among a large number of stakeholders.

Work performed by the DO-254/ED-80 Users Group over the years has provided real interest for industrial companies (avionics suppliers, components manufacturers …).

Based on the success of the DO254 User Group, similar initiatives for the DO178C and ARP4754A are currently being prepared2 as these updates will be released soon.

  Designing and implementing Hardware in order to fulfil requirements One can say that detailed design and Implementation processes are very close to most of HDL design house standard hardware development flows. Indeed writing HDL code, synthesis and place & route steps are the heart of HDL design activity. Nevertheless, what guarantees that requirements have been completely and correctly implemented? Are the constraints files completely in line with requirements? Do they contain all conditions?

  Technical capabilities for the topics and contexts discussed In addition the company is supposed to allocate time for 3-4 meetings per year, in Europe, and to actively participate to subgroup discussions, through meetings and phone calls. The company member is also requested to write or review the different papers, presentations… for the User Group.In 2009, Rockwell Collins joined the DO254 User Group and is sharing its interest and motivation to contribute to this organization.In mid-2005, the FAA recognized the use of DO-254 via AC 20-152. Since this time, DO-254 findings of compliance have been generated for approximately 40 different equipment types including Air Data Computers, Displays, Radios and Navigation equipments. Among these, we can mention different versions of display for Eurocopter and Agusta Westland helicopters (equipments DAL A and C). These products, initially designed in years 1996-1997, were later modified under DO-254 application.

	The Group is exclusively open to industrial companies (equipment suppliers or aircraft manufacturers). Other contributors like tool vendors, IP providers… can be concerned but usually it is more effective to invite them for specific topics, after preliminary discussion with them about their objectives. The Group has reached a high level of maturity in the discussions. Newcomers are welcome when they can contribute and present: • Experience on several DO-254 DAL A projects • Experience of certification audits, and of several certification contexts like EASA/FAA, • Rockwell Collins sharing experience and interest for DO254 User Group One of the first experiences of Rockwell Collins in term of DO254 hardware design is the development of first DAL A Ethernet Switch unit (AFDX) for Airbus A380. This development has been submitted to several audits conducted on hardware by European authorities JAA/CEAT. Afterwards, final approval was given at end of 2003, with a good appraisal from Airbus and CEAT.

• Alternative methods -What are other current methods used for development and verification in a certification context?

• HDL design rules -Collecting careful practices to achieve safe design and avoid synthesis errors

Besides plenary meetings, each sub-group leader organizes necessary conference calls to work on specific guidance / lessons learnt papers. These working sessions, with additional e-mail exchanges among members, allow elaborating these papers, which are then submitted to the whole User Group during plenary meeting.

For more information, refer to Certification Together website( www.certification-together.com )

Acknowledgement

The authors acknowledge the contribution of Lionel Burgaud, chairman of DO254 User Group, who provided some input data for this paper.


Friedrich Wagemann
email: wagemann@math.univ-nantes.fr

Friedrich Wagemann Spectral

SPECTRAL SEQUENCES FOR COMMUTATIVE LIE ALGEBRAS

Keywords: Mathematics Subject Classification. Primary 17A32; Secondary 17B56 Leibniz cohomology, Chevalley-Eilenberg cohomology, spectral sequence, commutative Lie algebra, commutative cohomology

whether they are published or not. The documents may come

Introduction

The classification of finite-dimensional simple Lie algebras over a field F of characteristic 2 is still widely open. It is known from the classification of Lie algebras over a field of characteristic zero that cohomological methods may be powerful allies in the classification quest. In [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF], there has been recently introduced a new cohomology theory for Lie algebras in characteristic 2, called commutative cohomology [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF]. It is more generally defined on so-called commutative Lie algebras in characteristic 2, i.e. vector spaces g over F with a bracket which satisfies the usual Jacobi identity and on top of that [x, y] = [y, x] for all x, y ∈ g. The corresponding cohomology theory is then based on the symmetric tensor powers of g with the usual Chevalley-Eilenberg differential.

It has been illustrated in [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF] that commutative cocycles and commutative cohomology arise in the classification of finite-dimensional simple Lie algebras in characteristic 2, see [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF] and references therein. The authors of [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF] listed a whole catalogue of questions concerning commutative cohomology, and we will answer some of them in the present paper. We hope that the computational methods of the present paper will serve in future computations of commutative cohomology.

We focus in the present paper on two cohomological tools for the computation of commutative cohomology. In Sections 1-3, we develop a Hochschild-Serre-type spectral sequence for commutative cohomology. The construction is very close to the original construction by Hochschild and Serre [START_REF] Hochschild | Cohomology of Lie algebras[END_REF]. In Section 4, we perform some cohomology computations with the help of the spectral sequence, notably for the two dimensional commutative Lie algebra N generated by e and f with the only relation [f, f ] = e and for the two dimensional Lie algebra a generated by e and h with the relations [h, e] = [e, h] = e. On the other hand, we obtain a cohomology vanishing theorem for any commutative Lie algebra g with a 1-dimensional ideal with values in the 1-dimensional non-trivial g-module F 1 , see Theorem 4.1.

The second part of the paper in Sections 5-6 concerns three comparison spectral sequences which mediate between Chevalley-Eilenberg-and Leibniz cohomology, between Chevalley-Eilenberg-and commutative cohomology, and between commutative-and Leibniz cohomology, respectively. In order to illustrate the use of these comparison spectral sequences, we show for example in Theorem 6.4 that for any Lie algebra g with a 1-dimensional ideal with values in the 1-dimensional non-trivial g-module F 1 , the vanishing of commutative cohomology implies the vanishing of Leibniz-and Chevalley-Eilenberg cohomology as well.

Note that there are more spectral sequences which should transpose easily to commutative Lie algebras, as for example the Feigin-Fuchs spectral sequence for N-graded Lie algebras.

Commutative Lie algebras and their cohomology

Let F be a field of characteristic 2. Definition 1. A commutative Lie algebra over F is an F -vector space g with a bilinear bracket [, ] : g × g → g such that for all x, y, z ∈ g

[x, y] = [y, x], and [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.
Remark 1. In particular, a commutative Lie algebra is a left and right Leibniz algebra, i.e. a symmetric Leibniz algebra.

As usual, an F-vector space M together with an F-bilinear

map g × M → M , denoted (x, m) → x • m, is called a g-module in case for all x, y ∈ g and all m ∈ M , we have [x, y] • m = x • (y • m) + y • (x • m).
Following [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF], we next define a cochain complex for commutative Lie algebras which will have its applications in the study and classification of Lie algebras in characteristic 2.

Definition 2. Let g be a commutative Lie algebra and M be a g-module. We set

CS n (g, M ) := Hom F (S n g, M ),
where S n g denotes the usual symmetric algebra on the vector space g. The graded vector space CS • (g, M ) becomes a cochain complex with the usual Chevalley-Eilenberg differential

d n f (x 1 , . . . , x n+1 ) := n+1 i=1 x i • f (x 1 , . . . , xi , . . . , x n+1 ) + 1≤i<j≤n+1 f ([x i , x j ], x 1 , . . . , xi , . . . , xj , . . . , x n+1 ) (1.1)
for any f ∈ CS n (g, M ) and all elements x 1 , . . . , x n+1 ∈ g. The corresponding cohomology is called commutative cohomology of commutative Lie algebras and denoted by HS • (g, M ).

Remark 2.

Observe that this cohomology is different from the usual Lie algebra cohomology in characteristic 2, where one takes cochain spaces C q (g, M ) which vanish in case q exceeds the dimension of g.

We next show that we have the usual Cartan relation for the symmetric cohomology of commutative Lie algebras. Proposition 1.1. Let g be a commutative Lie algebra over F. Then we have for all x ∈ g

L x = d • i x + i x • d,
where L x is the Lie derivative, i.e. for all f ∈ CS n (g, M ) and all x 1 , . . . , x n ∈ g, we have

(L x f )(x 1 , . . . , x n ) := x • f (x 1 , . . . , x n ) + n i=1 f (x 1 , . . . , [x, x i ], . . . , x n ),
and i x is the usual insertion operator, i.e. for all f ∈ CS n (g, M ) and all x 1 , . . . , x n-1 ∈ g, we have

(i x f )(x 1 , . . . , x n-1 ) := f (x, x 1 , . . . , x n-1 ).
Proof. Let f ∈ CS n (g, M ) and x, x 1 , . . . , x n ∈ g. When computing

(i x (df ))(x 1 , . . . , x n ) = df (x, x 1 , . . . , x n ),
there are the terms involving x in the bracket terms or in the action term of the coboundary operator, and there are the terms where x is simply inserted as the first argument of f . These last terms constitute exactly d(i x f )(x 1 , . . . , x n ), because in d(i x f ), the element x is never involved in the action or in the bracket. The remaining terms constitute (L x f )(x 1 , . . . , x n ).

The spectral sequence associated to a subalgebra

In this section, we construct a Hochschild-Serre-type spectral sequence, closely inspired by [START_REF] Hochschild | Cohomology of Lie algebras[END_REF], which is designed to compute the commutative cohomology of a commutative Lie algebra g which admits a subalgebra h.

The filtration leading to the spectral sequence is defined as follows.

F p CS n (g, M ) := {c ∈ CS n (g, M ) | c(x 1 , . . . , x n ) = 0 if n-p+1 elements x i belong to h}.

Lemma 2.1. The filtration is compatible with the Chevalley-Eilenberg coboundary operator (1.1), i.e.

dF p CS n (g, M ) ⊂ F p CS n+1 (g, M ).
Proof. Consider Equation (1.1) with n + 1 -p + 1 = n -p + 2 elements in h. We have to show that all terms in this equation are zero in case the cochain f vanishes on n -p + 1 elements of h. The action terms vanish when n -p + 2 elements are in h, because there always remain at least n -p + 1 elements in h as arguments of f . The situation is similar for the bracket terms, because even if both x i and x j are in h, the bracket will also be in h as h is a subalgebra, thus also in this case the corresponding term vanishes.

The next step in the construction of the spectral sequence is the computation of the term E 0 which is by definition the associated graded space to the filtration, i.e. E p,q 0 := F p CS p+q (g, M ) / F p+1 CS p+q (g, M ).

Lemma 2.2. The vector space E p,q 0 is isomorphic to E p,q 0 ∼ = Hom(S q h, Hom(S p (g/h), M )).

Proof. The isomorphism is induced by the restriction map r : CS p+q (g, M ) → Hom(S q h, Hom(S p (g/h), M ))

which restricts the first q arguments of the cochain f to elements of h. Clearly, the kernel of r| F p CS p+q (g,M) is F p+1 CS p+q (g, M ). It remains to show that r| F p CS p+q (g,M) is surjective. This follows as in the proof of Theorem 1 (p.593) of [START_REF] Hochschild | Cohomology of Lie algebras[END_REF]: Denote by π : g → g/h the natural quotient map and by p : g → h a linear projection onto the subspace h ֒→ g. For a given f ∈ Hom(S q h, Hom(S p (g/h), M )), we define a preimage f ∈ F p CS p+q (g, M ) of f by

f (x 1 , . . . , x n ) := σ∈Sh(p,q)
f (p(x σ(1) ), . . . , p(x σ(q) ))(π(x σ(q+1) ), . . . , π(x σ(p+q) )),

where the sum extends over all (p, q)-shuffles. Observe that only one term of the sum contributes to r(f ), because, as the shuffles keep the order in the first q arguments, only one term has all its elements in h and for the other terms, there is π applied to at least one element of h which gives zero. Thus r(f ) = f as had to be shown. Now we identify the differential d 1 which is by definition induced from the Chevalley-Eilenberg coboundary operator (1.1) on E 0 . Observe for this that the quotient g/h (which is in general not a Lie algebra !) becomes naturally an h-module using the adjoint action, because h is a subalgebra.

Lemma 2.3. The differential d 0 on E 0 identifies to the Chevalley-Eilenberg coboundary operator on CS • (h, Hom(S p (g/h), M )), the commutative cohomology of the commutative Lie algebra h with values in the h-module Hom(S p (g/h), M ).

Proof. The differential

d 0 : F p CS p+q (g, M ) / F p+1 CS p+q (g, M ) → F p CS p+q+1 (g, M ) / F p+1 CS p+q+1 (g, M )
is by definition induced by the Chevalley-Eilenberg coboundary operator (1.1). For the identification of d 0 , we consider again Equation (1.1), but now with p + q + 1 -(p + 1) + 1 = q + 1 elements in h, because we want to see which terms lie in F p+1 CS p+q+1 (g, M ), and with a cochain f which vanishes on p + q -p + 1 = q + 1 elements of h.

The action terms do not vanish exactly if x i is extracted from the first q elements (under the identification of E 0 with Hom(S q h, Hom(S p (g/h), M ))), otherwise q + 1 elements in h stay as arguments of f and the resulting term is zero. Similarly for the bracket terms, they do not vanish exactly in case x i and x j are both in h, because the number of elements in h is then reduced by one. This shows the claim.

The spectral sequence associated to an ideal

One can go further in the identification of the terms in the spectral sequence in case h is not only a subalgebra, but an ideal in g. Suppose now we have a short exact sequence of commutative Lie algebras

0 → h → g → q → 0.
We have seen in the previous section that

E p,q 1 ∼ = Hom(S p (g/h), HS q (h, M )) ∼ = Hom(S p q, HS q (h, M )),
where we have exchanged arguments in order to express the cohomology with respect to h. Lemma 3.1. The differential d 1 on E 1 identifies with the Chevalley-Eilenberg differential of q with values in HS q (h, M ).

Proof. Note that the action of q on HS • (h, M ) is well-defined, because the Lie algebra h acts trivially on its cohomology by the Cartan relation, see Proposition 1.1.

It is clear (noting also that h acts trivially on q) that the differential d 1 is given by the remaining terms of the Chevalley-Eilenberg coboundary operator (1.1), thus the claim.

Corollary 3.2. The E 2 -term of the spectral sequence associated to an ideal reads E p,q 2 = HS p (q, HS q (h, M )).

Applications of the spectral sequence

Example D in [START_REF] Feldvoss | On Leibniz cohomology[END_REF] is the 2-dimensional commutative Lie algebra N := Fe ⊕ Ff generated by e and f with the relations [f, f ] = e. The subalgebra h = e F is an ideal. Example E in [START_REF] Feldvoss | On Leibniz cohomology[END_REF] is the 2-dimensional Lie algebra a := Fh ⊕ Fe with the bracket determined by [h, e] = -[e, h] = e which becomes commutative over F of characteristic 2. The subalgebra h = e F is an ideal in a. Observe that in these examples, the bracket on the subalgebra is zero, thus the coboundary operator on the complex CS • (h, F) for cohomology with trivial coefficients is zero. Thus for these examples and trivial coefficients, it will be not simply an easy argument of cohomology vanishing which computes the Hochschild-Serre spectral sequence, because HS q (h, F) = S q (h, F) is non-zero for all q ≥ 0. Note however that all the spaces S q (h, F) are 1-dimensional as h is 1-dimensional. We then have HS p (N/h, HS q (h, F)) = HS p (N/h, S q (h, F)) and the coboundary operator computing the latter cohomology is still zero, because N/h = f F is a trivial quotient Lie algebra and the action is also zero. Thus E p,q 2 = S p (N/h, F) ⊗ S q (h, F) is 1-dimensional in this case, and the computation of the spectral sequence has to continue with the computation of

d p,q 2 : S p (N/h, F) ⊗ S q (h, F) → S p+2 (N/h, F) ⊗ S q-1 (h, F), which is induced by the bracket [f, f ] = e.
For the latter coboundary operator, it matters only whether there is an even or an odd number of bracket terms. But for n = p + q, the coboundary operator has n(n+1) 2 bracket terms, which gives zero bracket terms for n = 0, one bracket term for n = 1, three bracket terms for n = 2, six terms for n = 4 and so on. It follows that d p,q 2 = 0 if and only if

(p+q)(p+q+1) 2
is even, and d p,q 2 is injective otherwise. But (p+q)(p+q+1) 2 is even if and only if 4 | (p + q)(p + q + 1). As only one of the numbers p + q or p + q + 1 can be even, there are thus two cases: Either 4 | (p + q) or 4 | (p + q + 1). In the first case, 2 | (p+q)(p+q-1)

2

, thus furthermore d p-2,q+1 2 = 0. In the second case d p-2,q+1 2 is injective. This permits to compute

E p,q 3 =    S p (N/h, F) ⊗ S q (h, F) if 4 | (p + q) {0} if 4 | (p + q + 1) {0} else .
Observe that in the case 4 | (p + q + 1), it is the quotient space

(S p (N/h, F) ⊗ S q (h, F)) / (S p-2 (N/h, F) ⊗ S q+1 (h, F))
which is zero. The spectral sequences has vanishing higher differentials d r for r ≥ 3 and the E 3 -term thus computes the commutative cohomology of N with trivial coefficients. We obtain

HS n (N, F)) = F n+1 if 4 | n {0} if not .
For the Lie algebra a and trivial coefficients, the situation is similar: HS q (h, F) = S q (h, F) is non-zero for all q ≥ 0. But here it is already the coboundary operator computing the cohomology HS p (a/h, S q (h, F)) which involves the original bracket [h, e] = -[e, h] = e in its action terms. As all these action terms are non-zero, it again depends only on the parity of the number of terms whether the coboundary operator is zero or injective. We obtain as before

E p,q 2 =    S p (a/h, F) ⊗ S q (h, F)) if 4 | (p + q) {0} if 4 | (p + q + 1) {0} else .
Again, all higher differentials d r for r ≥ 2 are zero and the E 2 -term computes the commutative cohomology of a with trivial values. We obtain again

HS n (a, F)) = F n+1 if 4 | n {0} if not .
In order to have an easier example, let us consider the following. Let h = F be the 1-dimensional abelian Lie algebra generated by x ∈ h. Let us take as a module the 1-dimensional h-module F λ generated by v ∈ F λ such that the action is given by x • v = λv for a fixed λ ∈ F. Then all symmetric or Leibniz cohomology of h with values in (the symmetric h-bimodule) F 1 is zero. Indeed, the complexes are 1-dimensional in every degree with a coboundary operator having zero bracket part, but the action part is the identity in even degree and zero in odd degree. This gives zero cohomology.

Therefore, we obtain the following theorem.

Theorem 4.1. Let g be commutative Lie algebra which has a 1-dimensional ideal h. Suppose that h acts non-trivially on the module F 1 . Then

HS • (g, F 1 ) = 0.
Proof. By the discussion before the statement of the theorem, we have HS • (h, F 1 ) = 0. Inserting this in the formula of Corollary 3.2 shows the theorem.

Using again the formula of Corollary 3.2, one can formulate conditions on a commutative Lie algebra g with an ideal h such that the 1-dimensional quotient g/h acts non-trivially on HS q (h, M ) ∼ = F 1 . In this case, we have also cohomology vanishing.

Comparison to Lie-and Leibniz cohomology

5.1. Leibniz algebras and bimodules. As already observed, a commutative Lie algebra is in particular a left-and right Leibniz algebra.

A left Leibniz algebra is an algebra L such that every left bracket operator L x : L → L, y → [x, y] is a derivation. This is equivalent to the identity

(5.1) [x, [y, z]] = [[x, y], z] + [y, [x, z]]
for all x, y, z ∈ L. There is a similar definition of a right Leibniz algebra. Leibniz algebras have been studied by Loday and Pirashvili, see [START_REF] Loday | Une version non commutative des algèbres de Lie: les algèbres de Leibniz[END_REF], [START_REF] Loday | Universal enveloping algebras of Leibniz algebras and (co)homology[END_REF] and [START_REF] Pirashvili | On Leibniz homology[END_REF]. Every left Leibniz algebra has an important ideal, its Leibniz kernel, that measures how much the Leibniz algebra deviates from being a Lie algebra. Namely, let L be a left Leibniz algebra over F. Then

Leib(L) := [x, x] | x ∈ L F
is called the Leibniz kernel of L. The Leibniz kernel Leib(L) is an abelian ideal of L. By definition of the Leibniz kernel, L Lie := L/Leib(L) is a Lie algebra which we call the canonical Lie algebra associated to L. In fact, the Leibniz kernel is the smallest ideal such that the corresponding factor algebra is a Lie algebra. As alternativity (i.e. [x, x] = 0 for all x ∈ g) and anticommutativity ([x, y] = -[y, x] for all x, y ∈ g) are not equivalent in characteristic two, we see that commutative Lie algebras are a class lying properly in between Leibniz algebras and Lie algebras:

Leib ⊃ CommLie ⊃ Lie.
Observe that in this notation, commutative algebras are not a special case of Lie algebras ! Next, we will briefly discuss bimodules of left Leibniz algebras. Let L be a left Leibniz algebra over a field F. An L-bimodule is an F-vector space M with Fbilinear left and right

L-operations L × M → M , (x, m) → x • m and M × L → M , (m, x) → m • x such that (LLM) [x, y] • m = x • (y • m) + y • (x • m), (LML) x • (m • y) = (x • m) • y + m • [x, y], (MLL) m • [x, y] = (m • x) • y + x • (m • y).
are satisfied for every m ∈ M and all x, y ∈ L. Observe that all this makes sense over the field F of characteristic 2 and gives a notion of bimodules for commutative Lie algebras which goes beyond the usual notion of modules. 5.2. The three relative complexes. In [START_REF] Pirashvili | On Leibniz homology[END_REF] (see also Section 2 of [START_REF] Feldvoss | On Leibniz cohomology[END_REF]), Pirashvili constructs a comparison spectral sequence between Chevalley-Eilenberg cohomology and Leibniz cohomology of a Lie algebra. Here, we can use the same construction to have three comparison spectral sequences: There is a comparison spectral sequence between Chevalley-Eilenberg cohomology and commutative cohomology for a Lie algebra (passage from Λ • to S • ), then a comparison spectral sequence from commutative cohomology to Leibniz cohomology (passage from S • to ⊗ • ) and also the usual Pirashvili comparison spectral sequence from Chevalley-Eilenberg cohomology to Leibniz cohomology (passage from Λ • to ⊗ • ). Observe that all cohomologies involving the Chevalley-Eilenberg complex make only sense for Lie algebras, while the comparison spectral sequence from S • to ⊗ • makes sense for the broader class of commutative Lie algebras.

Let us first of all describe the different complexes which lead to these comparison spectral sequences. In characteristic 2, there are two ways of expressing the idea of alternating cochains which are not equivalent (corresponding to alternativity and antisymmetry as above mentioned). The first one is:

(5.2) ∀x ∈ g : f (. . . , x, . . . , x, . . .) = 0.

This condition is equivalent to saying that f (x 1 , . . . , x n ) vanishes in case the inserted elements x 1 , . . . , x n of g are linearly dependent and describes cochains as maps on the usual exterior algebra Λ n g. These cochains have the property that if g is finitedimensional of dimension n and f has n + 1 arguments, then f is zero. We will denote the quotient space of the vector space ⊗ n g with respect to the subspace

I n := x 1 ⊗ . . . ⊗ x ⊗ . . . ⊗ x ⊗ . . . ⊗ x n-2 | x, x 1 , . . . , x n-2 ∈ g F
by Λ n g := ⊗ n g/I n and the corresponding Chevalley-Eilenberg cochain spaces by C n (g, M ) := Hom F (Λ n g, M ).

On the other hand, the condition

(5.3) ∀x 1 , x 2 ∈ g : f (. . . , x 1 , x 2 , . . .) = f (. . . , x 2 , x 1 , . . .)
describes cochains on the usual symmetric algebra and such a cochain does not have the vanishing property (5.2) in general. Observe that condition (5.2) implies condition (5.3) by replacing x = x 1 + x 2 . Here the n-th graded component S n g of the symmetric algebra S • g is defined as the quotient of ⊗ n g by the subspace

J n := x 1 ⊗. . .⊗x⊗y⊗. . .⊗x n-2 +x 1 ⊗. . .⊗y⊗x⊗. . .⊗x n-2 | x, y, x 1 , . . . , x n-2 ∈ g F
(and not as the invariants with respect to the symmetric group). We had already introduced CS n (g, M ) = Hom F (S n g, M ). The fact that condition (5.2) implies condition (5.3) can also be understood as the inclusion J n ⊂ I n . We thus have three inclusions of subcomplexes (in case g is a Lie algebra and the bimodule M is symmetric)

i 1 : C • (g, M ) ֒→ CL • (g, M ), i 2 : C • (g, M ) ֒→ CS • (g, M )
because condition (5.2) implies condition (5.3), and also

i 3 : CS • (g, M ) ֒→ CL • (g, M )
which works for general commutative Lie algebras g.

All three give rise to cokernel complexes andC • rel,S (g, M ) := Coker(i 3 )[-2] respectively (up to a degree shift), and we have the corresponding long exact sequences (like in Proposition 2.2 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF]) induced by the short exact sequences of complexes. In each case, the relative cohomology (i.e. the cohomology of the quotient complex C • rel,Λ (g, M ), C • rel,S (g, M ) or C • rel,Λ,S (g, M )) measures the discrepancy between H • (g, M ) and HL • (g, M ), resp. HS • (g, M ) and HL • (g, M ), resp. H • (g, M ) and HS • (g, M ) in the sense that if all relative cohomology vanishes, then the two cohomologies coincide.

C • rel,Λ (g, M ) := Coker(i 1 )[-2], C • rel,Λ,S (g, M ) := Coker(i 2 )[-2],

5.3.

Comparison Lie-to Leibniz cohomology. In this subsection, g is a Lie algebra (i.e. [x, x] = 0 for all x ∈ g). In order to obtain the comparison spectral sequences, we now introduce filtrations in these three complexes C • rel,Λ (g, M ), C • rel,S (g, M ) and C • rel,Λ,S (g, M ) according to the condition of being alternating or symmetric (in the first p arguments).

Observe that due to the degree shift, a representative of a class in C n rel,Λ (g, M ), C n rel,S (g, M ) or C n rel,Λ,S (g, M ) has n + 2 arguments. On the quotient complex C • rel,Λ (g, M ), there is the following filtration

F p C n rel,Λ (g, M ) = {[c] ∈ C n rel,Λ (g, M ) | c(x 1 , . . . , x n+2 ) = 0 if ∃ j ≤ p+1 : x j-1 = x j } .
Note that the condition is independent of the representative c the class [c]. This defines a finite decreasing filtration

F 0 C n rel,Λ (g, M ) = C n rel,Λ (g, M ) ⊃ F 1 C n rel,Λ (g, M ) ⊃ • • • ⊃ F n+1 C n rel,Λ (g, M ) = {0}
, whose associated spectral sequence converges thus in the strong (i.e., finite) sense to H n rel,Λ (g, M ). Like in Section 2 of [START_REF] Feldvoss | On Leibniz cohomology[END_REF] (due to Pirashvili [START_REF] Pirashvili | On Leibniz homology[END_REF] !), we have a product map

m 1 : Λ n g ⊗ g → Λ n+1 g, x 1 ∧ . . . ∧ x n ⊗ x → x 1 ∧ . . . ∧ x n ∧ x.
The map m 1 induces a monomorphism

m • 1 : C • (g, F) ֒→ C • (g, g * ),
where C

• (g, F) is the truncated cochain complex C 0 (g, F) := 0 and C n (g, F) := C n (g, F) for every integer n 0 .

Accordingly, we have a cochain complex

CR • Λ (g) := Coker(m • 1 )[-1], the cokernel of m •
1 (up to a degree shift). We then have a long exact sequence in cohomology from the short exact sequence of complexes like in Proposition 2.1 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF].

For this filtration/spectral sequence associated to C • rel,Λ (g, M ), the arguments of Section 2 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF] go through word by word in order to show the following theorem (this is Theorem A of Pirashvili [START_REF] Pirashvili | On Leibniz homology[END_REF]): Theorem 5.1. Let g be a Lie algebra, and let M be a left g-module considered as a symmetric Leibniz g-bimodule M s . Then there is a spectral sequence converging to H • rel,Λ (g, M ) with second term E p,q 2 = HR p Λ (g) ⊗ HL q (g, M s ) . 5.4. Comparison Lie-to commutative cohomology. In this subsection, g is still a Lie algebra (i.e. [x, x] = 0 for all x ∈ g). The next step is to alter the arguments in Section 2 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF] to apply to the filtration/spectral sequence associated to C • rel,Λ,S (g, M ). We introduce a filtration as follows. In addition to

I n = x 1 ⊗ . . . ⊗ x i ⊗ . . . ⊗ x j ⊗ . . . ⊗ x n | x i = x j ∀x 1 , . . . , x n ∈ g, ∀1 ≤ i < j ≤ n F , we introduce I n,p := x 1 ⊗ . . .⊗ x i ⊗ . . . ⊗ x j ⊗ . . . ⊗ x n | x i = x j ∀x 1 , . . . , x n ∈ g, ∀1 ≤ i < j ≤ p F .
We have then J n ⊂ (J n + I n,p ), and denoting Λ p g ∨ S n-p g := ⊗ n g / (I n,p + J n ), we have

F p CS n (g, M ) := Hom F (Λ p g ∨ S n-p g, M ) ⊂ CS n (g, M ) = Hom F (S n g, M ).
The successive inclusions (I n,p + J n ) ⊂ (I n,p+1 + J n ) lead to a finite decreasing filtration

F 1 C n rel,Λ,S (g, M ) = C n rel,Λ,S (g, M ) ⊃ F 2 C n rel,Λ (g, M ) ⊃ • • • ⊃ F n C n rel,Λ,S (g, M ) = {0}
, whose associated spectral sequence converges thus in the strong (i.e., finite) sense to H n rel,Λ,S (g, M ). Again like in Section 2 of [START_REF] Feldvoss | On Leibniz cohomology[END_REF], we have a product map

m 2 : Λ n g ∨ g → Λ n+1 g, x 1 ∧ . . . ∧ x n ∨ x → x 1 ∧ . . . ∧ x n ∧ x.
Here Λ n g ∨ g ⊂ S n+1 g is the quotient space of ⊗ n+1 g by the sum I n+1,n + J n+1 . The map m 2 induces a monomorphism

m • 2 : C • (g, F) ֒→ C • Λ,S (g, g *
). Thus we have a cochain complex

CR • Λ,S (g) := Coker(m • 2 )[-1]
, the cokernel of m 2 (up to a degree shift). We then have a long exact sequence in cohomology from the short exact sequence of complexes like in Proposition 2.1 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF].

For this filtration/spectral sequence associated to C • rel,Λ,S (g, M ), the arguments of Section 2 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF] still go through in order to show the following theorem: Theorem 5.2. Let g be a Lie algebra, and let M be a left g-module considered as a symmetric Leibniz g-bimodule M s . Then there is a spectral sequence converging to H • rel,Λ,S (g, M ) with second term E p,q 2 = HR p Λ,S (g) ⊗ HS q (g, M ) . 5.5. Comparison commutative-to Leibniz cohomology. In this subsection, g is now an arbitrary commutative Lie algebra (i.e. we have only [x, y] = [y, x] for all x, y ∈ g). For the comparison between commutative cohomology and Leibniz cohomology, we reason as follows. The version of J n which concerns only the first p tensor factors is

J n,p := x 1 ⊗ . . . ⊗ x i ⊗ x i+1 ⊗ . . . ⊗ x n + x 1 ⊗ . . . ⊗ x i+1 ⊗ x i ⊗ . . . ⊗ x n | x 1 , . . . , x n ∈ g and i ≤ p -1 F .
The sequence of inclusions

{0} ⊂ J n,2 ⊂ J n,3 ⊂ . . . ⊂ J n,n = J n induces a sequence of surjections ⊗ n g ։ S 2 g ⊗ ⊗ n-2 g ։ . . . ։ S n g,
where we identify S p g ⊗ ⊗ n-p g = ⊗ n g / J n,p , which induces in turn a sequence of monomorphisms

F n CL n (g, M ) = CS n (g, M ) ⊂ F n-1 CL n (g, M ) ⊂ • • • ⊂ F 1 CL n (g, M ) = CL n (g, M ),
where by definition F p CL n (g, M ) := Hom F (S p g ⊗ ⊗ n-p g, M ). We interpret this sequence of monomorphisms as a finite decreasing filtration of the quotient complex

F 1 C n rel,S (g, M ) = C n rel,S (g, M ) ⊃ F 2 C n rel,S (g, M ) ⊃ • • • ⊃ F n C n rel,S (
g, M ) = {0}, which arises from dividing the above sequence of monomorphisms by CS n (g, M ). We need to show that this filtration is compatible with the Chevalley-Eilenberg differential, i.e.

dF p C n rel,S (g, M ) ⊂ F p C n+1 rel,S (g, M ). Indeed, given a cochain f ∈ Hom F (S p g ⊗ ⊗ n-p g, M ), representing a class in F p C n rel,S (g, M ), we have to show that df , given by Equation (1.1), is still symmetric in the first p entries. But this is clear from the symmetry of the bracket of g together with the symmetry of the terms of (1.1). In conclusion, we have a spectral sequence converging finitely to H • rel,S (g, M ). Again like in Section 2 of [START_REF] Feldvoss | On Leibniz cohomology[END_REF], we have a product map

m 3 : S n g ⊗ g → S n+1 g, x 1 ∨ . . . ∨ x n ⊗ x → x 1 ∨ . . . ∨ x n ∨ x,
The map m 3 induces a monomorphism

m • 3 : CS • (g, F) ֒→ CS • (g, g * ).
Accordingly, we have a cochain complex

CR • S (g) := Coker(m • 3 )[-1] the cokernel of m •
3 (up to a degree shift). We then have a long exact sequence in cohomology from the short exact sequence of complexes like in Proposition 2.1 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF]. Now the reasoning of Section 2 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF] goes through (mutatis mutandis) in order to show the following theorem.

Theorem 5.3. Let g be a commutative Lie algebra, and let M be a left g-module considered as a symmetric Leibniz g-bimodule M s . Then there is a spectral sequence converging to H • rel,S (g, M ) with second term E p,q 2 = HR p S (g) ⊗ HL q (g, M ) .

Applications of the comparison spectral sequences

By general homological algebra, the spectral sequences of Theorems 5.1, 5.2, 5.3 imply (using the long exact sequences arising from the definitions of H • rel,Λ (g, M ), H • rel,Λ,S (g, M ) and H • rel,S (g, M ) resp.) lead (exactly as in the proof of Theorem 2.6 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF]) to the following three theorems. Theorem 6.1. Let g be a Lie algebra, let M be a left g-module considered as a symmetric Leibniz g-bimodule M s , and let n be a non-negative integer. If H k (g, M ) = 0 for every integer k with 0 ≤ k ≤ n, then HL k (g, M s ) = 0 for every integer k with 0 ≤ k ≤ n and HL n+1 (g, M s ) ∼ = H n+1 (g, M ) as well as HL n+2 (g, M s ) ∼ = H n+2 (g, M ). In particular, H • (g, M ) = 0 implies that HL • (g, M s ) = 0. This is exactly Theorem 2.6 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF]. It has a converse stated in the Remark after its proof in [START_REF] Feldvoss | On Leibniz cohomology[END_REF]. Theorem 6.2. Let g be a Lie algebra, let M be a left g-module considered as a symmetric Leibniz g-bimodule M s , and let n be a non-negative integer. If H k (g, M ) = 0 for every integer k with 0 ≤ k ≤ n, then HS k (g, M s ) = 0 for every integer k with 0 ≤ k ≤ n and HS n+1 (g, M s ) ∼ = H n+1 (g, M ) as well as HS n+2 (g, M s ) ∼ = H n+2 (g, M ). In particular, H • (g, M ) = 0 implies that HS • (g, M s ) = 0.

And for the more general case of commutative Lie algebras: Theorem 6.3. Let g be a commutative Lie algebra, let M be a left g-module considered as a symmetric Leibniz g-bimodule M s , and let n be a non-negative integer. If HS k (g, M ) = 0 for every integer k with 0 ≤ k ≤ n, then HL k (g, M s ) = 0 for every integer k with 0 ≤ k ≤ n and HL n+1 (g, M s ) ∼ = HS n+1 (g, M ) as well as HL n+2 (g, M s ) ∼ = HS n+2 (g, M ). In particular, HS • (g, M ) = 0 implies that HL • (g, M s ) = 0.

The analogous converse statements (like in the Remark after Theorem 2.6 in [START_REF] Feldvoss | On Leibniz cohomology[END_REF]) are obviously also true. Indeed, suppose for example that HL • (g, M s ) = 0. Then by the long exact sequence H p-2 rel,S (g, M ) = HS p (g, M ) for all p ≥ 2 (while HS 0 (g, M ) = HS 1 (g, M ) = 0). On the other hand, the spectral sequence in Theorem 5.3 shows that the E 2 -term, and thus the relative cohomology H p rel,S (g, M ) is zero for all p ≥ 0. Therefore all cohomology HS • (g, M ) = 0.

One may also formulate conditions relying on a vanishing of the tensor factor HR k Λ (g, M ) for 0 ≤ k ≤ n in order to obtain from the spectral sequence (together with the long exact sequence linking H • (g, M ) and HL • (g, M )) the isomorphy between H • (g, M ) and HL • (g, M ). For example, the isomorphy of H 2 (g, F) and HL 1 (g, g) on the one hand and H 3 (g, F) and HL 2 (g, g) on the other hand imply the vanishing of HR 0 Λ (g) which then can be reported into the spectral sequence of Theorem 5.1 to yield H 0 rel,Λ (g, M ) = 0 and thus (by the long exact sequence) H 2 (g, M ) ∼ = HL 2 (g, M ). Obviously, this can be done with a vanishing of the first n spaces HR k Λ (g). Similar methods can be applied to the situations in Theorems 5.2 and 5.3, too. We refrain form stating all the consequences as theorems.

As a concrete computation, we may consider similarly to Section 4a Lie algebra g which has a 1-dimensional ideal h, and consider cohomology with values in the 1-dimensional module F 1 . By the above theorems, we obtain directly with Theorem 4.1: Theorem 6.4. Let g be a Lie algebra which has a 1-dimensional ideal h. Suppose that h acts non-trivially on the module F 1 . Then HS • (g, F 1 ) = HL • (g, F 1 ) = H • (g, F 1 ) = 0.

Acknowledgments. The author thanks the organisors of the Porto conference on non-associative algebras in 2019 where this work was initiated, for inviting him. He thanks furthermore Pasha Zusmanovich for discussion and for sending him a copy of [START_REF] Lopatkin | Commutative Lie algebras and commutative cohomology in characteristic[END_REF].