J.-F Pommaret
email: pommaret@cermics.enpc.fr

BIANCHI IDENTITIES FOR THE RIEMANN AND WEYL TENSORS

Keywords: Riemann tensor, Weyl tensor, Bianchi identities, Spencer cohomology, Vessiot structure equations, Poincaré sequence, Differential sequence, Differential modules, Compatibility conditions, Lanczos tensor

The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential modules, duality). In particular, we prove that the n 2 (n 2 -1)(n -2)/24 generating Bianchi identities for the Riemann tensor are first order and can be easily described by means of the Spencer cohomology of the first order Killing symbol in arbitrary dimension n ≥ 2. Similarly, the n(n 2 -1)(n + 2)(n -4)/24 generating Bianchi identities for the Weyl tensor are first order and can be easily described by means of the Spencer cohomology of the first order conformal Killing symbol in arbitrary dimension n ≥ 5. As a most surprising result, the 9 generating Bianchi identities for the Weyl tensor are of second order in dimension n = 4 while the analogue of the Weyl tensor has 5 components of third order in the metric with 3 first order generating Bianchi identities in dimension n = 3. The above results, which could not be obtained otherwise, are valid for any non-degenerate metric of constant riemannian curvature and do not depend on any conformal factor. They are checked in an Appendix produced by Alban Quadrat (INRIA, Lille) by means of computer algebra. We finally explain why the work of Lanczos and followers is not coherent with these results and must therefore be also revisited.

1) INTRODUCTION

The language of differential modules has been recently introduced in applications as a way to understand the structural properties of systems of partial differential equations and the Poincaré duality between geometry and physics by using adjoint operators or variational calculus with differential constraints ( [2], [START_REF] Pommaret | Partial Differential Control Theory[END_REF], [START_REF] Schneiders | An Introduction to D-Modules[END_REF]). In order to explain briefly the ideas of Lanczos as a way to justify the title of this paper, let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook. If x = (x 1 , ..., x n ) is a point in space and ξ = (ξ 1 (x), ..., ξ n (x)) is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the "small" deformation tensor ǫ = (ǫ ij = ǫ ji = (1/2)(∂ i ξ j + ∂ j ξ i )) with n(n + 1)/2 (independent) components (ǫ i≤j ). If we study a part of a deformed body by means of a variational principle, we may introduce the local density of free energy ϕ(ǫ) = ϕ(ǫ ij |i ≤ j) and vary the total free energy Φ = ϕ(ǫ)dx with dx = dx 1 ∧ ... ∧ dx n by introducing σ ij = ∂ϕ/∂ǫ ij for i ≤ j and "deciding" to define the stress tensor σ by a symmetric matrix with σ ij = σ ji in a purely artificial way within such a variational principle. Indeed, the usual Cauchy Tetrahedron device (1828) assumes that each element of a boundary surface is acted on by a surface density of force σ with a linear dependence σ = (σ ir (x)n r ) on the outward normal unit vector n = (n r ) and does not make any assumption on the stress tensor. It is only by an equilibrium of forces and couples, namely the well known phenomenological static torsor equilibrium, that one can prove the symmetry of σ. However, if we assume this symmetry, we may now consider the summation δΦ = σ ij δǫ ij dx = σ ir ∂ r δξ i dx. An integration by parts and a change of sign produce the integral (∂ r σ ir )δξ i dx leading to the stress equations ∂ r σ ir = 0. This classical approach to elasticity theory, based on invariant theory with respect to the group of rigid motions, cannot therefore describe equilibrium of torsors by means of a variational principle where the proper torsor concept is totally lacking. It is however widely used through the technique of " finite elements " where it can also be applied to electromagnetism (EM) with similar quadratic (piezoelectricity) or cubic (photoelasticity) lagrangian integrals. In this situation, the 4-potential A of EM is used in place of ξ while the EM field dA = F = ( B, E) is used in place of ǫ.

However, there exists another equivalent procedure dealing with a variational calculus with constraint. Indeed, as we shall see later on, the deformation tensor is not any symmetric tensor as it must satisfy n 2 (n 2 -1)/12 Riemann compatibility conditions (CC), that is the only condition ∂ 22 ǫ 11 + ∂ 11 ǫ 22 -2∂ 12 ǫ 12 = 0 when n = 2. In this case, introducing the Lagrange multiplier λ, we have to vary the new integral [ϕ(ǫ) + λ(∂ 22 ǫ 11 + ∂ 11 ǫ 22 -2∂ 12 ǫ 12 )]dx for an arbitrary ǫ. Setting λ = -φ, a double integration by parts now provides the parametrization σ 11 = ∂ 22 φ, σ 12 = σ 21 = -∂ 12 φ, σ 22 = ∂ 11 φ of the stress equations by means of the Airy function φ and the formal adjoint of the Riemann CC ( [1], [START_REF] Pommaret | Parametrization of Cosserat Equations[END_REF]). The same variational calculus with constraint may thus also be used in order to avoid the introduction of the EM potential A by using the Maxwell equations dF = 0 in place of the Riemann CC for ǫ but, in all these situations, we have to eliminate the Lagrange multipliers or use them as potentials.

In arbitrary dimension, the above compatibility conditions are nothing else but the linearized Riemann tensor in Riemannian geometry, a crucial mathematical tool in the theory of general relativity and a good reason for studying the work of Cornelius Lanczos (1893Lanczos ( -1974) ) as it can be found in ( [START_REF] Lanczos | Lagrange Multiplier and Riemannian Spaces[END_REF], [START_REF] Lanczos | The Splitting of the Riemann Tensor[END_REF]) or in a few modern references ( [5], [START_REF] Edgar | The Lanczos potential for Weyl-Candidate Tensors Exists only in Four Dimension[END_REF], [START_REF] Edgar | A Local Potential for the Weyl tensor in all dimensions[END_REF], [START_REF] Donnell | A Brief Historical Review of the Important Developments in Lanczos Potential Theory[END_REF], [START_REF] Roberts | The Physical Interpretation of the Lanczos Tensor[END_REF]). The starting point of Lanczos has been to take EM as a model in order to introduce a Lagrangian that should be quadratic in the Riemann tensor (ρ k l,ij ⇒ ρ ij = ρ r i,rj = ρ ji ⇒ ρ = ω ij ρ ij ) while considering it independently of its expression through the second order derivatives of a metric (ω ij ) with inverse (ω ij ) or the first order derivatives of the corresponding Christoffel symbols (γ k ij ). According to the previous paragraph, the corresponding variational calculus must involve PD constraints made by the Bianchi identities and the new lagrangian to vary must therefore contain as many Lagrange multipliers as the number of generating Bianchi identities that can be written under the form: ∇ r ρ k l,ij + ∇ i ρ k l,jr + ∇ j ρ k l,ri = 0 ⇒ ∇ r ρ r l,ij = ∇ i ρ lj -∇ j ρ li Meanwhile, Lanczos and followers have been looking for a kind of "parametrization " by using the corresponding "Lanczos potential ", exactly like the Lagrange multiplier has been used as an Airy potential for the stress equations. However, we shall prove that the definition of a Riemann candidate cannot be done without the knowledge of the Spencer cohomology. Moreover, we have pointed out the existence of well known couplings between elasticity and electromagnetism, namely piezoelectricity and photoelasticity, which are showing that, in the respective Lagrangians, the EM field is on equal footing with the deformation tensor and not with the Riemann tensor. The shift by one step backwards that must be used in the physical interpretation of the differential sequences involved cannot therefore be avoided. Meanwhile, the ordinary derivatives ∂ i can be used in place of the covariant derivatives ∇ i when dealing with the linearized framework as the Christoffel symbols vanish when Euclidean or Minkowskian metrics are used.

The next tentative of Lanczos has been to extend his approach to the Weyl tensor:

τ k l,ij = ρ k l,ij - 1 (n -2) (δ k i ρ lj -δ k j ρ li + ω lj ω ks ρ si -ω li ω ks ρ sj ) + 1 (n -1)(n -2) (δ k i ω lj -δ k j ω li )ρ
The main problem is now that the Spencer cohomology of the symbols of the conformal Killing equations, in particular the 2-acyclicity, will be absolutely needed in order to provide the Weyl tensor and its relation with the Riemann tensor. It will follow that the CC for the Weyl tensor may not be first order contrary to the CC for the Riemann tensor made by the Bianchi identities, another reason for justifying the shift by one step already quoted. In order to provide an idea of the difficulty involved, let us define the following tensors:

Schouten = (σ ij = ρ ij - 1 2(n -1) ω ij ρ) ⇒ Cotton = (σ k,ij = ∇ i σ kj -∇ j σ ki )
An elementary but tedious computation allows to prove the formula:

∇ r τ r k,ij = (n -3) (n -2) σ k,ij
Then, of course, if Einstein equations in vacuum are valid, the Schouten and Cotton tensors vanish but the left member is by no way a differential identity for the Weyl tensor and great care must be taken when mixing up mathematics with physics.

The author thanks Prof. Lars Andersson (Einstein Institute, Postdam) for having suggested him to study the Lanczos potential within this new framework and Alban Quadrat (INRIA, Lille), a specialist of control theory and computer algebra, for having spent time checking directly in an Appendix the many striking results contained in this paper.

2) HOMOLOGICAL ALGEBRA

We now need a few definitions and results from homological algebra ([3], [START_REF] Hu | Introduction to Homological Algebra[END_REF], [START_REF] Northcott | An Introduction to Homological Algebra[END_REF], [START_REF] Pommaret | Partial Differential Control Theory[END_REF], [START_REF] Rotman | An Introduction to Homological Algebra[END_REF]). In the following two classical theorems, A, B, C, D, K, L, M, N, Q, R, S, T will be modules over a ring A or vector spaces over a field k and the linear maps are making the diagrams commutative. We start recalling the well known Cramer's rule for linear systems through the exactness of the ker/coker sequence for modules. When f : M → N is a linear map (homomorphism), we introduce the so-called ker/coker long exact sequence:

0 -→ ker(f ) -→ M f -→ N -→ coker(f ) -→ 0
In the case of vector spaces over a field k, we successively have rk(f

) = dim(im(f )), dim(ker(f )) = dim(M ) -rk(f ) and dim(coker(f )) = dim(N ) -rk(f )
is the proper number of compatibility conditions. We obtain by substraction:

dim(ker(f )) -dim(M ) + dim(N ) -dim(coker(f )) = 0
In the case of modules, we may replace the dimension by the rank with rk A (M ) = r when F ≃ A r is the greatest free submodule of M and obtain the same relations because of the additive property of the rank ( [START_REF] Pommaret | Partial Differential Control Theory[END_REF], [START_REF] Pommaret | Algebraic Analysis of Control Systems Defined by Partial Differential Equations[END_REF], [START_REF] Pommaret | Airy, Beltrami, Maxwell, Einstein and Lanczos potentials Revisited[END_REF]). The following theorems will be crucially used through the whole paper: SNAKE THEOREM 2.1: When one has the following commutative diagram resulting from the two central vertical short exact sequences by exhibiting the three corresponding horizontal ker/coker exact sequences:

0 0 0 ↓ ↓ ↓ 0 -→ K -→ A -→ A ′ -→ Q -→ 0 ↓ ↓ f ↓ f ′ ↓ 0 -→ L -→ B -→ B ′ -→ R -→ 0 ↓ ↓ g ↓ g ′ ↓ 0 -→ M -→ C -→ C ′ -→ S -→ 0 ↓ ↓ ↓ 0 0 0
then there exists a connecting map M -→ Q both with a long exact sequence:

0 -→ K -→ L -→ M -→ Q -→ R -→ S -→ 0.
Proof: We construct the connecting map by using the following succession of elements:

a • • • a ′ -→ q . . . ↓ b -→ b ′ ↓ . . . m -→ c • • • 0
Indeed, starting with m ∈ M , we may identify it with c ∈ C in the kernel of the next horizontal map. As g is an epimorphism, we may find b ∈ B such that c = g(b) and apply the next horizontal map to get b ′ ∈ B ′ in the kernel of g ′ by the commutativity of the lower square. Accordingly, there is a unique a ′ ∈ A ′ such that b ′ = f ′ (a ′ ) and we may finally project a ′ to q ∈ Q. The map is well defined because, if we take another lift for c in B, it will differ from b by the image under f of a certain a ∈ A having zero image in Q by composition. The remaining of the proof is similar. The above explicit procedure is called " chase " and will not be repeated. Q.E.D.

We may now introduce cohomology theory through the following definition:

DEFINITION 2.2: If one has a sequence L f -→ M g -→ N , that is if g • f = 0, then one may introduce the submodules coboundary = B = im(f ) ⊆ ker(g) = cocycle = Z ⊆ M
and define the cohomology at M to be the quotient H = Z/B. The sequence is said to be exact at M if im(f ) = ker(g).

COHOMOLOGY THEOREM 2.3:

The following commutative diagram where the two central vertical sequences are long exact sequences and the horizontal lines are ker/coker exact sequences:

0 0 0 ↓ ↓ ↓ 0 -→ K -→ A -→ A ′ -→ Q -→ 0 ↓ ↓ f ↓ f ′ ↓ 0 -→ L -→ B -→ B ′ -→ R -→ 0 • • • • • • ↓ • • • ↓ g • • • ↓ g ′ • • • ↓ • • • • • • • • • cut 0 -→ M -→ C -→ C ′ -→ S -→ 0 ↓ ↓ h ↓ h ′ ↓ 0 -→ N -→ D -→ D ′ -→ T -→ 0 ↓ ↓ ↓ 0 0 0
induces an isomorphism between the cohomology at M in the left vertical column and the kernel of the morphism Q → R in the right vertical column.

Proof: Let us "cut" the preceding diagram along the dotted line. We obtain the following two commutative and exact diagrams with im(g) = ker(h), im(g ′ ) = ker(h ′ ):

0 0 0 ↓ ↓ ↓ 0 -→ K -→ A -→ A ′ -→ Q -→ 0 ↓ ↓ f ↓ f ′ ↓ 0 -→ L -→ B -→ B ′ -→ R -→ 0 ↓ ↓ g ↓ g ′ 0 -→ cocycle -→ im(g) -→ im(g ′ ) ↓ ↓ 0 0 0 0 0 ↓ ↓ ↓ 0 -→ cocycle -→ ker(h) -→ ker(h ′ ) ↓ ↓ ↓ 0 -→ M -→ C -→ C ′ ↓ ↓ h ↓ h ′ 0 -→ N -→ D -→ D ′ ↓ ↓ 0 0
Using the snake theorem, we successively obtain the following long exact sequences:

=⇒ ∃ 0 -→ K -→ L g -→ cocycle -→ Q -→ R =⇒ ∃ 0 -→ coboundary -→ cocycle -→ ker (Q -→ R) -→ 0 =⇒ cohomology ≃ ker (Q -→ R) Q.E.D.
We finally quote for a later use:

PROPOSITION 2.4: If one has a short exact sequence:

0 -→ M ′ f -→ M g -→ M ′′ -→ 0
then the following conditions are equivalent:

• There exists an epimorphism u :

M → M ′ such that u • f = id M ′ (left inverse of f ).
• There exists a monomorphism v :

M ′′ → M such that g • v = id M ′′ (right inverse of g).
DEFINITION 2.5: In the above situation, we say that the short exact sequence splits. The relation

f • u + v • g = id M provides an isomorphism (u, g) : M → M ′ ⊕ M ′′ with inverse f + v : M ′ ⊕ M " → M . The short exact sequence 0 → Z → Q → Q/Z → 0 cannot split over Z.

3) DIFFERENTIAL SYSTEMS

If E is a vector bundle over the base manifold X with projection π and local coordinates (x, y) = (x i , y k ) projecting onto x = (x i ) for i = 1, ..., n and k = 1, ..., m, identifying a map with its graph, a (local) section f : U ⊂ X → E is such that π • f = id on U and we write y k = f k (x) or simply y = f (x). For any change of local coordinates (x, y) → (x = ϕ(x), ȳ = A(x)y) on E, the change of section is y

= f (x) → ȳ = f (x) such that f l (ϕ(x) ≡ A l k (x)f k (x)
. The new vector bundle E * obtained by changing the transition matrix A to its inverse A -1 is called the dual vector bundle of E. In particular, let T be the tangent vector bundle of vector fields on X, T * be the cotangent vector bundle of 1-forms on X and S q T * be the vector bundle of symmetric q-covariant tensors on X. Differentiating with respect to x i and using new coordinates

y k i in place of ∂ i f k (x), we obtain ȳl r ∂ i ϕ r (x) = A l k (x)y k i + ∂ i A l k (x)y k . Introducing a multi-index µ = (µ 1 , ..., µ n ) with length | µ |= µ 1 + ... + µ n
and prolonging the procedure up to order q, we may construct in this way, by patching coordinates, a vector bundle J q (E) over X, called the jet bundle of order q with local coordinates (x, y q ) = (x i , y k µ ) with 0 ≤| µ |≤ q and y k 0 = y k . We have therefore epimorphisms π q+r q : J q+r (E) → J q (E), ∀q, r ≥ 0 and the short exact sequences 0 → S q T * ⊗ E → J q (E)

π q q-1 -→ J q-1 (E) → 0.
For a later use, we shall set µ + 1 i = (µ 1 , ..., µ i-1 , µ i + 1, µ i+1 , ..., µ n ) and define the operator j q : E → J q (E) : f → j q (f ) on sections by the local formula j q (f ) :

(x) → (∂ µ f k (x) | 0 ≤| µ |≤ q, k = 1, ..., m). Moreover, a jet coordinate y k µ is said to be of class i if µ 1 = ... = µ i-1 = 0, µ i = 0. We finally inroduce the Spencer operator D : J q+1 (E) → T * ⊗J q (E) : f q+1 → j 1 (f q )-f q+1 with (Df q+1 ) k µ,i = ∂ i f k µ -f k µ+1i .
DEFINITION 3.1: A system of PD equations of order q on E is a vector subbundle R q ⊂ J q (E) locally defined by a constant rank system of linear equations for the jets of order q of the form a τ µ k (x)y k µ = 0. Its first prolongation R q+1 ⊂ J q+1 (E) will be defined by the equations a τ µ k (x)y k µ = 0, a τ µ k (x)y k µ+1i + ∂ i a τ µ k (x)y k µ = 0 which may not provide a system of constant rank as can easily be seen for xy x -y = 0 ⇒ xy xx = 0 where the rank drops at x = 0.

The next definition of formal integrability will be crucial for our purpose. DEFINITION 3.2: A system R q is said to be formally integrable if the R q+r are vector bundles ∀r ≥ 0 (regularity condition) and no new equation of order q + r can be obtained by prolonging the given PD equations more than r times, ∀r ≥ 0 or, equivalently, we have induced epimorphisms π q+r+1 q+r : R q+r+1 → R q+r , ∀r ≥ 0 allowing to compute " step by step " formal power series solutions.

A formal test first sketched by C. Riquier in 1910, has been improved by M. Janet in 1920 ( [START_REF] Janet | Sur les Systèmes aux Dérivées Partielles[END_REF], [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF]) and by E. Cartan in 1945 ([4]), finally rediscovered in 1965, totally independently, by B. Buchberger who introduced Gröbner bases, using the name of his thesis advisor ( [START_REF] Pommaret | Gröbner Bases in Algebraic Analysis: New perspectives for applications[END_REF]). However all these tentatives have been largely superseded and achieved in an intrinsic way, again totally independently of the previous approaches, by D.C. Spencer in 1965 ([19], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], [START_REF] Spencer | Overdetermined Systems of Partial Differential Equations[END_REF]). DEFINITION 3.3: The family g q+r of vector spaces over X defined by the purely linear equations a τ µ k (x)v k µ+ν = 0 for | µ |= q, | ν |= r is called the symbol at order q +r and only depends on g q .

The following procedure, where one may have to change linearly the independent variables if necessary, is the key towards the next definition which is intrinsic even though it must be checked in a particular coordinate system called δ-regular (See [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], [START_REF] Pommaret | Partial Differential Control Theory[END_REF] and [START_REF] Spencer | Overdetermined Systems of Partial Differential Equations[END_REF] for more details):

• Equations of class n: Solve the maximum number β n q of equations with respect to the jets of order q and class n. Then call (x 1 , ..., x n ) multiplicative variables.

----------------• Equations of class i: Solve the maximum number of remaining equations with respect to the jets of order q and class i. Then call (x 1 , ..., x i ) multiplicative variables and (x i+1 , ..., x n ) nonmultiplicative variables.

- ----------------• Remaining equations equations of order ≤ q -1: Call (x 1 , ..., x n ) non-multiplicative variables. DEFINITION 3.4: The above multiplicative and non-multiplicative variables can be visualized respectively by integers and dots in the corresponding Janet board. A system of PD equations is said to be involutive if its first prolongation can be achieved by prolonging its equations only with respect to the corresponding multiplicative variables. The following numbers are called characters:

α i q = m(q + n -i -1)!/((q -1)!(n -i)!) -β i q , ∀1 ≤ i ≤ n ⇒ α 1 q ≥ ... ≥ α n q
For an involutive system, (y β n q +1 , ..., y m ) can be given arbitrarily.

For an involutive system of order q in the above solved form, we shall use to denote by y pri the principal jet coordinates, namely the leading terms of the solved equations in the sense of involution, and any formal derivative of a principal jet coordinate is again a principal jet coordinate. The remaining jet coordinates will be called parametric jet coordinates and denoted by y par . PROPOSITION 3.5: Using the Janet board and the definition of involutivity, we get:

dim(g q+r ) = n i=1 (r + i -1)! r!(i -1)! α i q ⇒ dim(R q+r ) = dim(R q-1 ) + n i=1 (r + i)! r!i! α i q
Let now ∧ s T * be the vector bundle of s-forms on X with usual bases {dx I = dx i1 ∧ ... ∧ dx is } where we have set I = (i 1 < ... < i s ). . Moreover, if ξ, η ∈ T are two vector fields on X, we may

define their bracket [ξ, η] ∈ T by the local formula ([ξ, η]) i (x) = ξ r (x)∂ r η i (x)-η s (x)∂ s ξ i (x) leading to the Jacobi identity [ξ, [η, ζ]] + [η, [ζ, ξ]] + [ζ, [ξ, η]] = 0, ∀ξ, η, ζ ∈ T . We may finally introduce the exterior derivative d : ∧ r T * → ∧ r+1 T * : ω = ω I dx I → dω = ∂ i ω I dx i ∧ dx I with d 2 = d • d ≡ 0 in the Poincaré sequence: ∧ 0 T * d -→ ∧ 1 T * d -→ ∧ 2 T * d -→ ... d -→ ∧ n T * -→ 0
In a purely algebraic setting, one has ( [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], [START_REF] Pommaret | Partial Differential Control Theory[END_REF], [START_REF] Pommaret | Algebraic Analysis of Control Systems Defined by Partial Differential Equations[END_REF], [START_REF] Spencer | Overdetermined Systems of Partial Differential Equations[END_REF]):

PROPOSITION 3.6: There exists a map δ :

∧ s T * ⊗ S q+1 T * ⊗ E → ∧ s+1 T * ⊗ S q T * ⊗ E which restricts to δ : ∧ s T * ⊗ g q+1 → ∧ s+1 T * ⊗ g q and δ 2 = δ • δ = 0.
Proof: Let us introduce the family of s-forms ω = {ω k µ = v k µ,I dx I } and set (δω

) k µ = dx i ∧ ω k µ+1i . We obtain at once (δ 2 ω) k µ = dx i ∧ dx j ∧ ω k µ+1i+1j = 0. Q.E.D.
The kernel of each δ in the first case is equal to the image of the preceding δ but this may no longer be true in the restricted case and we set (See [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], p 85-88 for more details): DEFINITION 3.7: We denote by B s q+r (g q ) ⊆ Z s q+r (g q ) and H s q+r (g q ) = Z s q+r (g q )/B s q+r (g q ) respectively the coboundary space, cocycle space and cohomology space at ∧ s T * ⊗ g q+r of the restricted δ-sequence which only depend on g q and may not be vector bundles. The symbol g q is said to be s-acyclic if H 1 q+r = ... = H s q+r = 0, ∀r ≥ 0, involutive if it is n-acyclic and finite type if g q+r = 0 becomes trivially involutive for r large enough. For a later use, we notice that a symbol g q is involutive and of finite type if and only if g q = 0. Finally, S q T * ⊗ E is involutive ∀q ≥ 0 if we set S 0 T * ⊗ E = E. We shall prove later on that any symbol g q is 1-acyclc.

CRITERION THEOREM 3.8: If π q+1 q : R q+1 → R q is an epimorphism of vector bundles and g q is 2-acyclic (involutive), then R q is formally integrable (involutive). EXAMPLE 3.9: The system R 2 defined by the three PD equations Φ 3 ≡ y 33 = 0, Φ 2 ≡ y 23 -y 11 = 0, Φ 1 ≡ y 22 = 0 is homogeneous and thus automatically formally integrable but g 2 and g 3 are not involutive though finite type because g 4 = 0 and the sequence 0 → ∧ 3 T * ⊗ g 3 → 0 is not exact. Elementary computations of ranks of matrices shows that the δ-map:

0 → ∧ 2 T * ⊗ g 3 δ -→ ∧ 3 T * ⊗ g 2 → 0
is a 3 × 3 isomorphism and thus g 3 is 2-acyclic with dim(g 3 ) = 1, a crucial intrinsic property totally absent from any "old" work and quite more easy to handle than its Koszul dual. We invite the reader to treat similarly the system y 33 -y 11 = 0, y 23 = 0, y 22 -y 11 = 0 and compare.

The main use of involution is to construct differential sequences that are made up by successive compatibility conditions (CC) of order one. In particular, when R q is involutive, the differential operator D : E jq → J q (E) Φ → J q (E)/R q = F 0 of order q with space of solutions Θ ⊂ E is said to be involutive and one has the canonical linear Janet sequence ( [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], p 144):

0 -→ Θ -→ E D -→ F 0 D1 -→ F 1 D2 -→ ... Dn -→ F n -→ 0
where each other operator is first order involutive and generates the CC of the preceding one with the Janet bundles F r = ∧ r T * ⊗ J q (E)/(∧ r T * ⊗ R q + δ(∧ r-1 T * ⊗ S q+1 T * ⊗ E)). As the Janet sequence can be "cut at any place", that is can also be constructed anew from any intermediate operator, the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence for the exterior derivative, contrary to what many physicists still believe (n = 3 with D = div provides the simplest example). Moreover, the fiber dimension of the Janet bundles can be computed at once inductively from the board of multiplicative and non-multiplicative variables that can be exhibited for D by working out the board for D 1 and so on. For this, the number of rows of this new board is the number of dots appearing in the initial board while the number nb(i) of dots in the column i just indicates the number of CC of class i for i = 1, ..., n with nb(i) < nb(j), ∀i < j.

MAIN THEOREM 3.10: When R q ⊂ J q (E) is not involutive but formally integrable and its symbol g q becomes 2-acyclic after exactly s prolongations, the generating CC are of order s + 1 (See [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], Example 6, p 120 and previous Example).

Proof: We may introduce the canonical epimorphism Φ = J q (E) → J q (E)/R q = F 0 and denote by D = Φ • j q : E → F 0 the corresponding differential operator. As before, we may write formally

Φ τ (x, y q ) ≡ a τ µ,k (x)y k µ = z τ , obtain d i Φ τ ≡ a τ µ,k (x)y k µ+1i + ∂ i a τ µ,k (x)y k µ = z τ i for the first prolonga- tion ρ 1 (Φ) : J q+1 (E) → j 1 (F 0 ) and so on with ρ r (Φ) : J q+r (E) → J r (F 0 ) defined by d ν Φ τ = z τ ν with 0 ≤| ν |≤ r. Setting B r = im(ρ r (Φ)) ⊆ J r (F 0 ), we may introduce the canonical epimorphism Ψ : J s+1 (F 0 ) → J s+1 (F 0 )/B s+1 = F 1 .
Taking into account the formal integrability of R q (care), we obtain by composition of jets the following commutative prolongation diagrams ∀r ≥ 1:

0 0 0 0 ↓ ↓ ↓ ↓ 0 → g q+r+s+1 → S q+r+s+1 T * ⊗ E σr+s+1(Φ) -→ S r+s+1 T * ⊗ F 0 σr (Ψ) -→ S r T * ⊗ F 1 ↓ ↓ ↓ ↓ 0 → R q+r+s+1 → J q+r+s+1 (E) ρr+s+1(Φ) -→ J r+s+1 (F 0 ) ρr (Ψ) -→ J r (F 1 ) ↓ ↓ ↓ ↓ 0 → R q+r+s → J q+r+s (E) ρr+s(Φ) -→ J r+s (F 0 ) ρr-1(Ψ) -→ J r-1 (F 1 ) ↓ ↓ ↓ ↓ 0 0 0 0
and the only thing we know is that the bottom sequence is exact for r = 1 by construction and that the upper induced sequence is exact when r = 0 when σ 0 (Ψ) = σ(Ψ) is the restriction of Ψ to S s+1 T * ⊗ F 0 ⊂ J s+1 (F 0 ) after a chase in the following commutative diagram:

0 0 0 0 ↓ ↓ ↓ ↓ 0 → g q+s+1 → S q+s+1 T * ⊗ E σs+1(Φ) -→ S s+1 T * ⊗ F 0 σ(Ψ) -→ F 1 ↓ ↓ ↓ ↓ 0 → R q+s+1 → J q+s+1 (E) ρs+1(Φ) -→ J s+1 (F 0 ) Ψ -→ F 1 ↓ ↓ ↓ 0 → R q+s → J q+s (E) ρs(Φ) -→ J s (F 0 ) ↓ ↓ ↓ 0 0 0 because R q is formally integrable (care)
. Appying now the δ-maps to the upper row of the previous prolongation diagram and proceeding by induction, starting from r = 1, we shall prove that the upper row is exact. Indeed, setting h r+s = im(σ r+s (Φ)) ⊆ S r+s T * ⊗ F 0 , we may cut the full commutative diagram thus obtained as in the proof of the previous "cohomology theorem" into the following two commutative diagrams:

0 0 0 ↓ ↓ ↓ 0 → g q+r+s+1 → S q+r+s+1 T * ⊗ E → h r+s+1 → 0 ↓ ↓ ↓ 0 → T * ⊗ g q+r+s → T * ⊗ S q+r+s T * ⊗ E → T * ⊗ h r+s → 0 ↓ ↓ ↓ 0 → ∧ 2 T * ⊗ g q+r+s-1 → ∧ 2 T * ⊗ S q+r+s-1 T * ⊗ E → ∧ 2 T * ⊗ h r+s-1 → 0 ↓ ↓ 0 → ∧ 3 T * ⊗ S q+r+s-2 T * ⊗ E = ∧ 3 T * ⊗ S q+r+s-2 T * ⊗ E 0 0 0 ↓ ↓ ↓ 0 → h r+s+1 → S r+s+1 T * ⊗ F 0 → S r T * ⊗ F 1 ↓ ↓ ↓ 0 → T * ⊗ h r+s → T * ⊗ S r+s T * ⊗ F 0 → T * ⊗ S r-1 T * ⊗ F 1 ↓ ↓ 0 → ∧ 2 T * ⊗ h r+s-1 → ∧ 2 T * ⊗ S r+s-1 T * ⊗ F 0
An easy chase is showing that g q is always 1-acyclic and that we have an induced monomorphism 0 → h r+s+1 → T * ⊗h r+s . The crucial result that no classical approach could provide is that, whenever g q+s is 2-acyclic, then the full right column of the first diagram is also exact or, equivalently, h r+s+1 ⊆ S r+s+1 T * ⊗ F 0 is the r-prolongation of the symbol h s+1 ⊆ S s+1 T * ⊗ F 0 . Using finally the second diagram, it follows by induction and a chase that the upper row is exact whenever the central row is exact, a result achieving the first part of the proof. We may also use an inductive chase in the full diagram, showing directly that the cohomology at S r+s+1 T * ⊗ F 0 of the upper sequence:

0 → g q+r+s+1 → S q+r+s+1 T * ⊗ E σr+s+1(Φ) -→ S r+s+1 T * ⊗ F 0 σr (Ψ) -→ S r T * ⊗ F 1
is the same as the δ-cohomology of the left column at ∧ 2 T * ⊗g q+r+s-1 because all the other vertical δ-sequences are exact. Finally, starting from the ker/coker long exact sequence allowing to define Ψ and ending with F 1 while taking into account that the upper symbol row of the prolongation diagram is exact, we deduce by an inductive chase that the central row is also exact. It follows that B r+s+1 is the r-prolongation of B s+1 which is formally integrable because a chase shows that B r+s+1 projects onto B r+s , ∀r ≥ 1.The case of an involutive symbol can be studied similarly by choosing s = 0 and explains why all the CC operators met in the Janet sequence are first order involutive operators.

Q.E.D.

As we shall see through explicit examples, in particular the conformal Killing system, there is no rule in general in order to decide about the minimum number s ′ ≥ 0 such that h s+s ′ +1 becomes 2-acyclic in order to repeat the above procedure. However, replacing r by s ′ + 2 and chasing in the first of the last two diagrams, we have: COROLLARY 3.11: The symbol h s+s ′ +1 becomes 2-acyclic whenever the symbol g q+s+s ′ becomes 3-acyclic. DEFINITION 3.12: More generally, a differential sequence is said to be formally exact if each operator generates the CC of the operator preceding it. EXAMPLE 3.13: ([16], §38, p 40) The second order system y 11 = 0, y 13 -y 2 = 0 is neither formally integrable nor involutive. Indeed, we get d 3 y 11 -d 1 (y 13 -y 2 ) = y 12 and d 3 y 12 -d 2 (y 13 -y 2 ) = y 22 , that is to say each first and second prolongation does bring a new second order PD equation. Considering the new system y 22 = 0, y 12 = 0, y 13 -y 2 = 0, y 11 = 0, the (evident !) permutation of coordinates (1, 2, 3) → (3, 2, 1) provides the following involutive second order system with one equation of class 3, 2 equations of class 2 and 1 equation of clas 1:

       Φ 4 ≡ y 33 = 0 Φ 3 ≡ y 23 = 0 Φ 2 ≡ y 22 = 0 Φ 1 ≡ y 13 -y 2 = 0 1 2 3 1 2 • 1 2 • 1 • • We have α 3 2 = 0, α 2 2 = 0, α 1 2 = 2
and we get therefore the (formally exact) Janet sequence:

0 -→ Θ -→ 1 -→ 4 -→ 4 -→ 1 -→ 0
However, keeping only Φ 1 and Φ 4 while using the fact that d 33 commutes with d 13 -d 2 , we get the formally exact sequence 0 → Θ → 1 → 2 → 1 → 0 which is not a Janet sequence though the Euler-Poincaré characteristics vanishes in both cases with 1-4+4-1 = 1-2+1 = 0 ( [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], p 159 and [START_REF] Pommaret | Partial Differential Control Theory[END_REF]).

EXAMPLE 3.14: Coming back to Example 3.9 while intoducing the three second order operators P = d 22 , Q = d 23 -d 11 , R = d 33 which are commuting between themselves, we have now q = 2, s = 1 and we obtain the second order CC:

Ψ 3 ≡ P Φ 2 -QΦ 1 = 0, Ψ 2 ≡ RΦ 1 -P Φ 3 = 0, Ψ 1 = QΦ 3 -RΦ 2 = 0
Exactly like in the Poincaré sequence, we finally get the new second order CC:

P Ψ 1 + QΨ 2 + RΨ 3 = 0
Writing out only the number of respective equations, we obtain the formally exact differential sequence with vanishing Euler-Poincaré characteristics:

0 → Θ → 1 → 3 → 3 → 1 → 0
which is not a Janet sequence. We let the reader check that h 2 is not 2-acyclic but that h 3 is 2-acyclic and thus s ′ = 1 because 2 + 1 + 1 = 4 and g 4 = 0 is trivially involutive. A similar situation will be met with the conformal Killing equations.

We may finally extend the restriction D : R q+1 → T * ⊗ R q of the Spencer operator to:

D : ∧ r T * ⊗ R q+1 → ∧ r+1 T * ⊗ R q : α ⊗ f q+1 → dα ⊗ f q + (-1) r α ∧ Df q+1 ⇒ D 2 = D • D ≡ 0
in order to construct the first Spencer sequence which is another resolution of Θ because the kernel of the first D is such that f q+1 ∈ R q+1 , Df q+1 = 0 ⇔ f q+1 = j q+1 (f ), f ∈ Θ when q is large enough. This standard notation for the Spencer operator must not be confused with the same notation used in the next section for the ring of differential operators but the distinction will always be pointed out whenever a confusion could exist.

4) DIFFERENTIAL MODULES

Let K be a differential field, that is a field containing Q with n commuting derivations

{∂ 1 , ..., ∂ n } with ∂ i ∂ j = ∂ j ∂ i = ∂ ij , ∀i, j = 1, ..., n such that ∂ i (a + b) = ∂ i a + ∂ i b, ∂ i (ab) = (∂ i a)b + a∂ i b, ∀a, b ∈ K and ∂ i (1/a) = -(1/a 2 )∂ i a, ∀a ∈ K.
Using an implicit summation on multi-indices, we may introduce the (noncommutative) ring of differential operators

D = K[d 1 , ..., d n ] = K[d] with elements P = a µ d µ such that | µ |< ∞ and d i a = ad i + ∂ i a.
The highest value of |µ| with a µ = 0 is called the order of the operator P and the ring D with multiplication (P, Q) -→ P • Q = P Q is filtred by the order q of the operators. We have the filtration If we introduce differential indeterminates y = (y 1 , ..., y m ), we may extend µ by ȳk µ when there can be a confusion. Introducing the two free differential modules F 0 ≃ D m0 , F 1 ≃ D m1 , we obtain equivalently the free presentation F 1 d1 -→ F 0 → M → 0 of order q when m 0 = m, m 1 = p and d 1 = D = Φ • j q with (P 1 , ..., P p ) → (P 1 , ..., P p ) • D = (Q 1 , ..., Q m ). We shall moreover assume that D provides a strict morphism, namely that the corresponding system R q is formally integrable. It follows that M can be endowed with a quotient filtration obtained from that of D m which is defined by the order of the jet coordinates y q in D q y. We have therefore the inductive

0 ⊂ K = D 0 ⊂ D 1 ⊂ ... ⊂ D q ⊂ ... ⊂ D ∞ = D. Moreover, it is clear that D, as an algebra, is generated by K = D 0 and T = D 1 /D 0 with D 1 = K ⊕ T if we identify an element ξ = ξ i d i ∈ T with the vector field ξ = ξ i (x)∂ i of differential geometry, but with ξ i ∈ K now. It follows that D = D D D is a bimodule over itself,
d i y k µ = y k µ+1i to Φ τ ≡ a τ µ k y k µ di -→ d i Φ τ ≡ a τ µ k y k µ+1i + ∂ i a τ µ k y k µ for τ =
limit 0 = M -1 ⊆ M 0 ⊆ M 1 ⊆ ... ⊆ M q ⊆ ... ⊆ M ∞ = M with d i M q ⊆ M q+1 but it is important to notice that D r D q = D q+r ⇒ D r M q = M q+r , ∀q, r ≥ 0 ⇒ M = DM q , ∀q
≥ 0 in this particular case. It also follows from noetherian arguments and involution that D r I q = I q+r , ∀r ≥ 0 though we have in general only D r I s ⊆ I r+s , ∀r ≥ 0, ∀s < q. As K ⊂ D, we may introduce the forgetful functor f or : mod(D) → mod(K)

: D M → K M .
More generally, introducing the successive CC as in the preceding section while changing slightly the numbering of the respective operators, we may finally obtain the free resolution of M , namely the exact sequence ...

d3 -→ F 2 d2 -→ F 1 d1 -→ F 0 -→ M -→ 0.
In actual practice, one must never forget that D = Φ • j q acts on the left on column vectors in the operator case and on the right on row vectors in the module case. Also, with a slight abuse of language, when D = Φ • j q is involutive as in section 3 and thus R q = ker(Φ) is involutive, one should say that M has an involutive presentation of order q or that M q is involutive.

REMARK 4.2:

As can be seen from the last two examples of Section 3, when D is involutive, then ad(D) may not be involutive. In the differential framework, we may set rk D (D) = m -α n q = β n q . Comparing to similar concepts used in differential algebra, this number is just the maximum number of differentially independent equations to be found in the differential module I of equations. Indeed, pointing out that differential indeterminates in differential algebra are nothing else than jet coordinates in differential geometry and using standard notations, we have K{y} = lim q→∞ K[y q ]. In that case, the differential ideal I automatically generates a prime differential ideal p ⊂ K{y} providing a differential extension L/K with L = Q(K{y}/p) and differential transcendence degree dif f trd(L/K) = α n q , a result explaining the notations ( [START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF]). Now, from the dimension formulas of R q+r , we obtain at once rk D (M ) = α n q and thus rk D (D) = m -rk D (M ) in a coherent way with any free presentation of M starting with D. However, D acts on the left in differential geometry but on the right in the theory of differential modules. For an operator of order zero, we recognize the fact that the rank of a matrix is eqal to the rank of the transposed matrix. PROPOSITION 4.3: If f ∈ aut(X) is a local diffeomorphisms on X, we may set x = f -1 (y) = g(y) and we have the identity:

∂ ∂y k ( 1 ∆(g(y)) ∂ i f k (g(y))) ≡ 0 ⇒ ∂ ∂y k ( 1 ∆ ∂f k ∂x i A i ) = 1 ∆ ∂f k ∂x i ∂A i ∂y k = 1 ∆ ∂ i A i
and the adjoint of the well defined intrinsic operator

∧ 0 T * d -→ ∧ 1 T * = T * : A -→ ∂ i A is (mi- nus) the well defined intrinsic operator ∧ n T * d ←-∧ n T * ⊗ T ≃ ∧ n-1 T * : ∂ i A i ←-A i . Hence, if we have an operator E D -→ F , we obtain the formal adjoint operator ∧ n T * ⊗E * ad(D) ←-∧ n T * ⊗F * .
Having in mind that D is a K-algebra, that K is a left D-module with the standard action (D, K) -→ K : (P, a) -→ P (a) : (d i , a) -→ ∂ i a and that D is a bimodule over itself, we have only two possible constructions leading to the following two definitions: DEFINITION 4.4: We may define the inverse system R = hom K (M, K) of M and introduce R q = hom K (M q , K) as the inverse system of order q. Proof: For any f ∈ hom K (M, N ), let us define:

(af )(m) = af (m) = f (am) ∀a ∈ K, ∀m ∈ M (ξf )(m) = ξf (m) -f (ξm) ∀ξ = ξ i d i ∈ T, ∀m ∈ M
It is easy to check that ξa = aξ + ξ(a) in the operator sense and that ξη -ηξ = [ξ, η] is the standard bracket of vector fields. We have in particular with d in place of any d i :

((da)f )(m) = (d(af ))(m) = d(af (m)) -af (dm) = (∂a)f (m) + ad(f (m)) -af (dm) = (a(df ))(m) + (∂a)f (m) = ((ad + ∂a)f )(m)
For any m ⊗ n ∈ M ⊗ K N with arbitrary m ∈ M and n ∈ N , we may then define:

a(m ⊗ n) = am ⊗ n = m ⊗ an ∈ M ⊗ K N ξ(m ⊗ n) = ξm ⊗ n + m ⊗ ξn ∈ M ⊗ K N
and conclude similarly with:

(da)(m ⊗ n) = d(a(m ⊗ n)) = d(am ⊗ n) = d(am) ⊗ n + am ⊗ dn = (∂a)m ⊗ n + a(dm) ⊗ n + am ⊗ dn = (ad + ∂a)(m ⊗ n) Using K in place of N , we finally get (d i f ) k µ = (d i f )(y k µ ) = ∂ i f k µ -f k µ+1i
that is we recognize exactly the Spencer operator with now Df = dx i ⊗ d i f and thus:

(d i (d j f )) k µ = ∂ ij f k µ -(∂ i f k µ+1j + ∂ j f k µ+1i ) + f k µ+1i+1j ⇒ d i (d j f ) = d j (d i f ) = d ij f
In fact, R is the projective limit of π q+r q : R q+r → R q in a coherent way with jet theory ( [2], [START_REF] Pommaret | Macaulay Inverse Systems Revisited[END_REF], [START_REF] Schneiders | An Introduction to D-Modules[END_REF]).

Q.E.D.

N D ←-F * 1 D * ←-F * 0 ←-M * ←-0.
COROLLARY 4.9: We have the side changing procedure

N D → N = D N = hom K (∧ n T * , N D ) with inverse M = D M → M D = ∧ n T * ⊗ K M whenever M, N ∈ mod(D).
Proof: According to the above Theorem, we just need to prove that ∧ n T * has a natural right module structure over D. For this, if α = adx 1 ∧ ... ∧ dx n ∈ T * is a volume form with coefficient a ∈ K, we may set α.P = ad(P )(a)dx 1 ∧ ... ∧ dx n when P ∈ D. As D is generated by K and T , we just need to check that the above formula has an intrinsic meaning for any ξ = ξ i d i ∈ T . In that case, we check at once: REMARK 4.13: Of course, solving the direct problem (Janet, Spencer) is necessary for solving the inverse problem. However, though the direct problem always has a solution, the inverse problem may not have a solution at all and the case of the Einstein operator is one of the best non-trivial PD counterexamples ( [START_REF] Pommaret | Algebraic Analysis of Control Systems Defined by Partial Differential Equations[END_REF], [START_REF] Pommaret | The Mathematical Foundations of General Relativity Revisited[END_REF]). It is rather striking to discover that, in the case of OD operators, it took almost 50 years to understand that the possibility to solve the inverse problem was equivalent to the controllability of the corresponding control system ( [START_REF] Pommaret | Algebraic Analysis of Control Systems Defined by Partial Differential Equations[END_REF], [START_REF] Pommaret | QUADRAT: Algebraic Analysis of Linear Multidimensional Control Systems[END_REF]). EXAMPLE 4.14: (contact transformations) With n = 3, K = Q(x 1 , x 2 , x 3 ), let us consider the Lie pseudogroup of transfomations preserving the first order geometric object ω like a 1-form but up to the square root of ∆. The infinitesimal transformations are among the solutions Θ of the general system:

α.ξ = -∂ i (aξ i )dx 1 ∧ ... ∧ dx n = -L(ξ)
Ω i ≡ (L(ξ)ω) i ≡ ω r (x)∂ i ξ r -(1/2)ω i (x)∂ r ξ r + ξ r ∂ r ω i (x) = 0
When ω = (1, -x 3 , 0), we obtain the special involutive system:

∂ 3 ξ 3 + ∂ 2 ξ 2 + 2x 3 ∂ 1 ξ 2 -∂ 1 ξ 1 = 0, ∂ 3 ξ 1 -x 3 ∂ 3 ξ 2 = 0, ∂ 2 ξ 1 -x 3 ∂ 2 ξ 2 + x 3 ∂ 1 ξ 1 -(x 3 ) 2 ∂ 1 ξ 2 -ξ 3 = 0
with 2 equations of class 3, 1 equation of class 2 and thus only 1 first order CC for the second members coming from the linearization of the Vessiot structure equation:

ω 1 (∂ 2 ω 3 -∂ 3 ω 2 ) + ω 2 (∂ 3 ω 1 -∂ 1 ω 3 ) + ω 3 (∂ 1 ω 2 -∂ 2 ω 1 ) = c
involving the only structure constant c. This system can be parametrized by a single potential θ:

-x 3 ∂ 3 θ + θ = ξ 1 , -∂ 3 θ = ξ 2 , ∂ 2 θ -x 3 ∂ 1 θ = ξ 3 ⇒ ξ 1 -x 3 ξ 2 = θ
and we have the formally exact differential sequence 0 → 1

D-1 -→ 3 D -→ 3 D1 -→ 1 → 0.
However, we have yet not proved the most difficult result that could not be obtained without homological algebra and the next example will explain ths additional difficulty.

EXAMPLE 4.15: With ∂ 22 ξ = η 2 , ∂ 12 ξ = η 1 for D, we get ∂ 1 η 2 -∂ 2 η 1 = ζ for D 1 . Then ad(D 1 ) is defined by µ 2 = -∂ 1 λ, µ 1 = ∂ 2 λ while ad(D) is defined by ν = ∂ 12 µ 1 + ∂ 22 µ 2 but the CC of ad(D 1 ) are generated by ν ′ = ∂ 1 µ 1 + ∂ 2 µ 2 .
In the operator framework, we have the differential sequences:

ξ D -→ η D1 -→ ζ → 0 0 ← ν ad(D) ←-µ ad(D1) ←-λ
where the upper sequence is formally exact at η but the lower sequence is not formally exact at µ. Passing to the module framework, we obtain the sequences:

0 → D D1 -→ D 2 D -→ D → M → 0 D ad(D1) ←-D 2 ad(D) ←-D ← 0
where the lower sequence is not exact at D 2 .

Therefore, we have to find out situations in which ad(D) generates the CC of ad(D 1 ) whenever D 1 generates the CC of D and conversely. This problem will be studied in Section 5, Part C.

5) APPLICATIONS

Though the next pages will only be concerned with a study of the Lie pseudogroups of isometries (A) and conformal isometries (B), the reader must never forget that they can be used similarly for any arbitrary transitive Lie pseudogroup of transformations ( [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Lie Pseudogroups and Mechanics, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], [START_REF] Pommaret | Deformation Cohomology of Algebraic and Geometric Structures[END_REF]).

A) RIEMANN TENSOR

Let ω = (ω ij = ω ji ) ∈ S 2 T * be a non-degenerate metric with det(ω) = 0. We shall apply the Main Theorem to the first order Killing system R 1 ⊂ J 1 (T ) defined by the n(n + 1)/2 linear equations Ω ij ≡ ω rj ξ r i + ω ir ξ r j + ξ r ∂ r ω ij = 0 for any section ξ 1 ∈ R 1 . Its symbol g 1 ⊂ T * ⊗ T is defined by the n(n + 1)/2 linear equations ω rj ξ r i + ω ir ξ r j = 0 and we obtain at once isomorphisms g 1 ≃ ∧ 2 T ≃ ∧ 2 T * by lowering or raising the indices by means of the metric, obtaining for example ξ i,j + ξ j,i = 0. As det(ω) = 0, we may introduce the well known Chrisoffel symbols γ = (γ k ij = γ k ji ) through the standard Ricci/Levi-Civita isomorphism j 1 (ω) ≃ (ω, γ) and obtain by one prolongation the linear second order equations for any section ξ 2 ∈ R 2 :

Γ k ij ≡ ξ k ij + γ k rj ξ r i + γ k ir -γ r ij ξ k r + ξ r ∂ r γ k ij = 0 and we have Ω ∈ S 2 T * ⇒ Γ ∈ S 2 T * ⊗
T for the respective linearization/variation of ω and γ. As we shall see that g 1 is not 2-acyclic and g 2 = 0 is defined by the n 2 (n + 1)/2 linear equations ξ k ij = 0, we may apply the Main Theorem with q = 1, s = 1, E = T , on the condition that R 1 should be formally integrable as it is finite type and cannot therefore be involutive. First of all, we have the following commutative and exact diagram allowing to define

F 0 = S 2 T * : 0 0 0 ↓ ↓ ↓ 0 → g 1 -→ T * ⊗ T σ(Φ) -→ F 0 → 0 ↓ ↓ 0 → R 1 -→ J 1 (T ) Φ -→ F 0 → 0 ↓ ↓ ↓ 0 → T = T -→ 0 ↓ ↓ 0 0 Now, R 2 π 2 1 -→ R 1 is an isomorphism because g 2 = 0 and dim(R 2 ) = dim(R 1 ) = n(n + 1)/2. Hence, R 2 ⊂ J 2 (T ) is involutive if and only if R 3 π 3 2
-→ R 2 is also an isomorphism too because g 2 = 0 ⇒ g 2+r = 0, ∀r ≥ 0. Such a differential condition for ω has been shown by L.P. Eisenhart in ( [START_REF] Eisenhart | Riemannian Geometry[END_REF]) to be equivalent to the Vessiot structure equation with one constant called constant riemannian curvature ρ k l,ij = c(δ k i ω lj -δ k j ω li ) (See [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF] and [START_REF] Pommaret | Deformation Cohomology of Algebraic and Geometric Structures[END_REF] for effective calculations still not acknowledged today). In this formula, c is an arbitrary constant and the Riemann tensor (ρ k l,ij ) ∈ ∧ 2 T * ⊗ T * ⊗ T satisfies the two types of purely algebraic relations:

ω rl ρ r k,ij + ω kr ρ r l,ij = 0, ρ k l,ij + ρ k i,jl + ρ k j,li = 0
We shall suppose that ω is the Euclidean metric if n = 2, 3 and the Minkowskian metric if n = 4 but any other compatible choice should be convenient. As a next step, we know from the Main Theorem that the generating CC for the operator Killing = Φ • j 1 : T → F 0 are made by an operator Riemann = Ψ•j 2 : F 0 → F 1 of order s+1 = 2. We shall define F 1 by setting q = 1, r = 0, s = 1 in the corresponding diagram in order to get the following commutative diagram:

0 0 0 ↓ ↓ ↓ 0 → g 3 → S 3 T * ⊗ T → S 2 T * ⊗ F 0 → F 1 → 0 ↓ δ ↓ δ ↓ δ 0 → T * ⊗ g 2 → T * ⊗ S 2 T * ⊗ T → T * ⊗ T * ⊗ F 0 → 0 ↓ δ ↓ δ ↓ δ 0 → ∧ 2 T * ⊗ g 1 → ∧ 2 T * ⊗ T * ⊗ T → ∧ 2 T * ⊗ F 0 → 0 ↓ δ ↓ δ ↓ 0 → ∧ 3 T * ⊗ T = ∧ 3 T * ⊗ T → 0 ↓ ↓ 0 0
where all the rows are exact and all the columns are also exact but the first at ∧ 2 T * ⊗ g 1 with g 2 = 0 ⇒ g 3 = 0. We shall denote by B 2 (g 1 ) the coboundary as the image of the central δ, by Z 2 (g 1 ) the cocycle as the kernel of the lower δ and by H 2 (g 1 ) = Z 2 (g 1 )/B 2 (g 1 ) the Spencer δ-cohomology at ∧ 2 T * ⊗ g 1 as the quotient. Chasing in the previous diagram, we discover that the Riemann tensor is a section of the bundle

F 1 = H 2 (g 1 ) = Z 2 (g 1 ) with dim(F 1 ) = (n 2 (n + 1) 2 /4) -(n 2 (n + 1)(n + 2)/6) = (n 2 (n -1) 2 /4) -(n 2 (n -1)(n -2)/6) = n 2 (n 2 -1)/12
by using the top row or the left column. We discover at once the two properties of the (linearized) Riemann tensor through the chase involved, namely (R k l,ij ) ∈ ∧ 2 T * ⊗ T * ⊗ T is killed by both δ and σ 0 (Φ). Similarly, going one step further, we get the (linearized) Bianchi identities by defining the first order operator Bianchi :

F 1 → F 2 where F 2 = H 3 (g 1 ) = Z 3 (g 1 ) with dim(F 2 ) = dim(∧ 3 T * ⊗ g 1 ) -dim(∧ 4 T * ⊗ T ) = n 2 (n -1) 2 (n -2)/12 -n 2 (n -1)(n -2)(n -3)/24 = n 2 (n 2 -1)(n -2)/
24 may be defined by the following commutative diagram:

0 0 0 0 ↓ ↓ ↓ ↓ 0 → g 4 → S 4 T * ⊗ T → S 3 T * ⊗ F 0 → T * ⊗ F 1 → F 2 → 0 ↓ ↓ ↓ 0 → T * ⊗ g 3 → T * ⊗ S 3 T * ⊗ T → T * ⊗ S 2 T * ⊗ F 0 → T * ⊗ F 1 → 0 ↓ ↓ ↓ ↓ 0 → ∧ 2 T * ⊗ g 2 → ∧ 2 T * ⊗ S 2 T * ⊗ T → ∧ 2 T * ⊗ T * ⊗ F 0 → 0 ↓ ↓ ↓ 0 → ∧ 3 T * ⊗ g 1 → ∧ 3 T * ⊗ T * ⊗ T → ∧ 3 T * ⊗ F 0 → 0 ↓ ↓ ↓ 0 → ∧ 4 T * ⊗ T = ∧ 4 T * ⊗ T → 0 ↓ ↓ 0 0
This approach is relating for the first time the concept of Riemann tensor candidate, introduced by Lanczos and others, to the Spencer δ-cohomology of the Killing symbols. We obtain therefore the formally exact sequence:

0 → Θ → n Killing -→ n(n + 1)/2 Riemann -→ n 2 (n 2 -1)/12 Bianchi -→ n 2 (n 2 -1)(n -2)/24 → ...
with operators of successive orders 1, 2, 1, ... and so on.

In the present situation, we have the (split) short exact sequences:

0 → F 1 → ∧ 2 T * ⊗ g 1 δ -→ ∧ 3 T * ⊗ T → 0, 0 → F 2 → ∧ 3 T * ⊗ g 1 δ → ∧ 4 T * ⊗ T → 0
and obtain the operator ad(Bianchi) :

∧ n T * ⊗ F * 2 → ∧ n T * ⊗ F * 1 with the short exact sequence: 0 ← ∧ n T * ⊗ F * 2 ← ∧ n-3 T * ⊗ g * 1 ← ∧ n-2 T * ⊗ T * ← 0 explaining at once why the Lagrange multipliers λ ∈ ∧ n T * ⊗ F *
2 can be represented by a section of ∧ n-3 T * ⊗ ∧ 2 T * , that is by a Lanczos potential in T ⊗ ∧ 2 T * when n = 4. We shall see in part C that ad(Bianchi) is parametrizing ad(Riemann) contrary to the claims of Lanczos. Moreover, we have already pointed out in many books ( [START_REF] Pommaret | Lie Pseudogroups and Mechanics, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Control Theory[END_REF]) or papers ( [START_REF] Pommaret | Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics[END_REF], [START_REF] Pommaret | From Thermodynamics to Gauge Theory: the Virial Theorem Revisited[END_REF]) that continuum mechanics may be presented through a variational problem with a differential constraints which is shifted by one step backwards in the previous differential sequence because the infinitesimal deformation tensor ǫ = 1 2 Ω ∈ S 2 T * must be now killed by the operator Riemann and the corresponding Lagrange multipliers λ ∈ ∧ n T * ⊗ F * 1 must be used because ad(Riemann) is parametrizing ad(Killing) = Cauchy. Anybody using computations with finite elements also knows that a similar situation is held by electromagnetism too because the EM field is killed by d : ∧ 2 T * → ∧ 3 T * ⇒ ad(d) : ∧ 1 T * → ∧ 2 T * , another fact contradicting Lanczos claims. Finally, the passage to differential modules can be achieved easily by using K = Q as will be done in the Appendix or K = Q < ω > with standard notations because the Lie pseudogroup of isometries is an algebraic Lie pseudogroup as it can be defined by differential polynomials in the jets of order ≥ 1 (See [START_REF] Kolchin | Differential Algebra and Algebraic Groups[END_REF], [START_REF] Pommaret | Differential Galois Theory[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF] for details).

B) WEYL TENSOR

If the study of the Riemann tensor/operator has been related to many classical results, the study of the Weyl tensor/operator in this new framework is quite different because these new mathematical tools have not been available before 1975 and are still not acknowledged today by mathematical physicists. In particular, we may quote the link existing between acyclicity and formal integrability both with the possibility to use the Vessiot structure equations in order to combine in a unique framework the constant riemannian curvature condition needed for the Killing system, which only depends on one arbitrary constant, with the zero Weyl tensor condition needed for the conformal Killing system, which does not depend on any constant. For this reason, we shall follow as closely as possible the previous part A, putting a "hat" on the corresponding concepts. The conformal Killing system R1 ⊂ J 1 (T ) is defined by eliminating the function A(x) in the system L(ξ)ω = A(x)ω. It is also a Lie operator D with solutions Θ ⊂ T satisfying [ Θ, Θ] ⊂ Θ. Its symbol ĝ1 is defined by the linear equations ω rj ξ r i + ω ir ξ r j -2 n ω ij ξ r r = 0 which do not depend on any conformal factor and is finite type because ĝ3 = 0 when n ≥ 3. We have ( [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Differential Galois Theory[END_REF], [START_REF] Pommaret | From Thermodynamics to Gauge Theory: the Virial Theorem Revisited[END_REF]): LEMMA 5.1: ĝ2 ⊂ S 2 T * ⊗ T is now 2-acyclic only when n ≥ 4 and 3-acyclic only when n ≥ 5.

It is known that R2 and thus R1 too (by a chase) are formally integrable if and only if ω has zero Weyl tensor:

τ k l,ij ≡ ρ k l,ij - 1 (n -2) (δ k i ρ lj -δ k j ρ li + ω lj ω ks ρ si -ω li ω ks ρ sj ) + 1 (n -1)(n -2) (δ k i ω lj -δ k j ω li )ρ = 0 If we use the formula id M -f • u = v • g of Proposition 2
.4 in the split short exact sequence induced by the inclusions g 1 ⊂ ĝ1 , 0 = g 2 ⊂ ĝ2 , g 3 = ĝ3 = 0 ( [START_REF] Pommaret | Lie Pseudogroups and Mechanics, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], [START_REF] Pommaret | Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics[END_REF]):

0 -→ Ricci -→ Riemann -→ W eyl -→ 0
according to the Vessiot structure equations, in particular if ω has constant Riemannian curvature and thus [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Lie Pseudogroups and Mechanics, Gordon and Breach[END_REF], [START_REF] Pommaret | The Mathematical Foundations of General Relativity Revisited[END_REF], [START_REF] Pommaret | The Mathematical Foundations of Gauge Theory Revisited[END_REF]). Using the same diagrams as before, we get F0 = T * ⊗ T /ĝ 1 with dim( F0 ) = (n -1)(n + 2)/2 and F1 = H 2 (ĝ 1 ) = Z 2 (ĝ 1 ) for defining any Weyl tensor candidate. As a byproduct, we could believe that the linearized operator W eyl : F0 → F1 is of order 2 with a symbol ĥ2 ⊂ S 2 T * ⊗ F0 which is not 2-acyclic by applying the δ-map to the short exact sequence:

ρ ij = ρ r i,rj = c(n -1)ω ij ⇒ ρ = ω ij ρ ij = cn(n -1) ([
0 → ĝ3+r -→ S 3+r T * ⊗ T σ2+r (Φ)
-→ ĥ2+r → 0 and chasing through the commutative diagram thus obtained with r = 0, 1, 2. As ĥ3 becomes 2-acyclic after one prolongation of ĥ2 only, it follows that the generating CC for the Weyl operator are of order 2, a result showing that the so-called Bianchi identities for the Weyl tensor are not CC in the strict sense of the definition as they do not involve only the Weyl tensor.

In fact, things are quite different and we have to distinguish three different cases:

• n = 3: According to the last Lemma, ĝ2 is not 2-acyclic but ĝ3 = 0 becomes trivially 2-ayclic and even involutive, that is s = 2. According to the Main Theorem, the operator W eyl : F0 → F1 is third order because s + 1 = 3 (See Appendix) and F1 is defined by the short exact sequences:

0 → S 4 T * ⊗ T → S 3 T * ⊗ F0 → F1 → 0, 0 → F1 → ∧ 2 T * ⊗ ĝ2 δ -→ ∧ 3 T * ⊗ ĝ1 → 0
with dim( F1 ) = 50 -45 = 9 -4 = 5. As now ĥ3 ⊂ S 3 T * ⊗ F0 , applying the δ-map to the short exact sequence:

0 → ĝ6 → S 6 T * ⊗ T → ĥ5 → 0
and chasing, we discover that ĥ3 is 2-acyclic because ĝ3 = 0. Accordingly, the operator Bianchi : F1 → F2 is first order and F2 is defined by the long exact sequence:

0 → S 5 T * ⊗ T → S 4 T * ⊗ F0 → T * ⊗ F1 → F2 → 0 or by the isomorphism 0 → F2 → ∧ 3 T * ⊗ ĝ2 → 0 giving dim( F2 ) = 63 -75 + 15 = 1 × 3 = 3.
Recapitulating, when n = 3 we have the formally exact differential sequence with 3 -5 + 5 -3 = 0:

0 → Θ → 3 CKilling -→ 5 W eyl -→ 5 Bianchi -→ 3 → 0
In actual practice, introducing the new geometric objects γk

ij = γ k ij -1 n (δ k i γ r rj + δ k j γ r ri -ω ij ω ks γ r rs
), linearizing and using the cyclic summation C(1, 2, 3), we get for example:

C(1, 2, 3)[d 23 ( Γ2 12 + Γ1 33 -Γ1 11 -Γ1 22 )] = 0
• n = 4: This situation is even more striking because ĝ2 is 2 acyclic but not 3-acyclic and thus s = 1. As before, we have dim( F0 ) = (n -1)(n + 2)/2 = 9 but, according to the Main Theorem, the operator W eyl : F0 → F1 is of order s+1 = 2 and F1 is defined by the short exact sequence:

0 → S 3 T * ⊗ T → S 2 T * ⊗ F0 → F1 → 0 ⇒ dim( F1 ) = 90 -80 = 10
or by F1 = H 2 (ĝ 1 ) = Z 2 (ĝ 1 )/B 2 (ĝ 1 ) with exact sequences:

0 → T * ⊗ ĝ2 δ -→ B 2 (ĝ 1 ) → 0, 0 → Z 2 (ĝ 1 ) → ∧ 2 T * ⊗ ĝ1 δ -→ ∧ 3 T * ⊗ T → 0
providing again dim( F1 ) = 26 -16 = 10. The main problem is that,now, ĝ2 is not 3-acyclic and thus ĥ2 is not 2-acyclic according to a chase in the commutative diagram:

0 0 0 ↓ ↓ ↓ 0 → ĝ5 → S 5 T * ⊗ T → ĥ4 → 0 ↓ ↓ ↓ 0 → T * ⊗ ĝ4 → T * ⊗ S 4 T * ⊗ T → T * ⊗ ĥ3 → 0 ↓ ↓ ↓ 0 → ∧ 2 T * ⊗ ĝ3 → ∧ 2 T * ⊗ S 3 T * ⊗ T → ∧ 2 T * ⊗ ĥ2 → 0 ↓ ↓ ↓ 0 → ∧ 3 T * ⊗ ĝ2 → ∧ 3 T * ⊗ S 2 T * ⊗ T → ∧ 3 T * ⊗ T * ⊗ F0 → 0 ↓ ↓ ↓ 0 → ∧ 4 T * ⊗ ĝ1 → ∧ 4 T * ⊗ T * ⊗ T → ∧ 4 T * ⊗ F0 → 0 ↓ ↓ ↓ 0 0 0 but ĥ3 becomes 2-acyclic by chasing in the next diagram: 0 0 0 ↓ ↓ ↓ 0 → ĝ6 → S 6 T * ⊗ T → ĥ5 → 0 ↓ ↓ ↓ 0 → T * ⊗ ĝ5 → T * ⊗ S 5 T * ⊗ T → T * ⊗ ĥ4 → 0 ↓ ↓ ↓ 0 → ∧ 2 T * ⊗ ĝ4 → ∧ 2 T * ⊗ S 4 T * ⊗ T → ∧ 2 T * ⊗ ĥ3 → 0 ↓ ↓ ↓ 0 → ∧ 3 T * ⊗ ĝ3 → ∧ 3 T * ⊗ S 3 T * ⊗ T → ∧ 3 T * ⊗ ĥ2 → 0 ↓ ↓ ↓ 0 → ∧ 4 T * ⊗ ĝ2 → ∧ 4 T * ⊗ S 2 T * ⊗ T → ∧ 4 T * ⊗ T * ⊗ F0 → 0 ↓ ↓ ↓ 0 0 0
and we have s ′ = 1. Accordingly, the operator Bianchi : F1 → F2 is of order s ′ + 1 = 2 and F2 is defined by following commutative diagram where all the rows are exact and all the columns are exact but the first:

0 0 0 0 ↓ ↓ ↓ ↓ 0 → ĝ5 → S 5 T * ⊗ T → S 4 T * ⊗ F0 → S 2 T * ⊗ F1 → F2 → 0 ↓ ↓ ↓ ↓ 0 → T * ⊗ ĝ4 → T * ⊗ S 4 T * ⊗ T → T * ⊗ S 3 T * ⊗ F0 → T * ⊗ T * ⊗ F1 → 0 ↓ ↓ ↓ ↓ 0 → ∧ 2 T * ⊗ ĝ3 → ∧ 2 T * ⊗ S 3 T * ⊗ T → ∧ 2 T * ⊗ S 2 T * ⊗ F0 → ∧ 2 T * ⊗ F1 → 0 ↓ ↓ ↓ ↓ 0 → ∧ 3 T * ⊗ ĝ2 → ∧ 3 T * ⊗ S 2 T * ⊗ T → ∧ 3 T * ⊗ T * ⊗ F0 → 0 ↓ ↓ ↓ 0 → ∧ 4 T * ⊗ ĝ1 → ∧ 4 T * ⊗ T * ⊗ T → ∧ 4 T * ⊗ F0 → 0 ↓ ↓ ↓ 0 0 0 0 → ĝ5 → S 5 T * ⊗ T → S 4 T * ⊗ F0 → S 2 T * ⊗ F1 → F2 → 0 0 → 224 → 315 → 100 → 9 → 0 0 → ĝ4 → S 4 T * ⊗ T → S 3 T * ⊗ F0 → T * ⊗ F1 → 0 0 → 140 → 180 → 40 → 0
We could define similarly the first order generating CC F2 → F3 where F3 is defined by the following long exact sequence:

0 → ĝ6 → S 6 T * ⊗ T → S 5 T * ⊗ F0 → S 3 T * ⊗ F1 → T * ⊗ F2 → F3 → 0 0 → 336 → 504 → 200 → 36 → 4 → 0
and obtain finally the formally exact differential sequence:

0 → Θ → 4 CKilling -→ 9 W eyl -→ 10 Bianchi -→ 9 -→ 4 → 0
with vanishing Euler-Poincaré characteristic 4 -9 + 10 -9 + 4 = 0. We conclude the study of n = 4 by exhibiting the short exact sequence:

0 → ∧ 2 T * ⊗ ĝ2 → ∧ 3 T * ⊗ ĝ1 → ∧ 4 T * ⊗ T → 0 0 → 24 → 28 → 4 → 0
a result showing that H 3 (ĝ 1 ) = 0 for n = 4, contrary to what will happen when n ≥ 5.

• n ≥ 5: We still have dim( F0 ) = (n -1)(n + 2)/2 but now ĝ2 is 2-acyclic and 3-acyclic with s = 1 again but now with s ′ = 0. Hence, according to the Main Theorem and its Corollary, the operator W eyl is second order while the operator Bianchi is first order. We may therefore use the same diagrams already introduced in part A but with a "hat " symbol and n ≥ 4. In particular we get:

dim(Z 3 (ĝ 1 )) = dim(∧ 3 T * ⊗ ĝ1 ) -dim(∧ 4 T * ⊗ T ) = n(n -1)(n -2)(n 2 + n + 4)/24 dim(B 3 (ĝ 1 )) = n 2 (n -1)/2 ⇒ dim(H 3 (ĝ 1 )) = n(n 2 -1)(n + 2)(n -4)/24
We find again H 3 (ĝ 1 ) = 0 when n = 4 but H 3 (ĝ 1 ) = 0 when n ≥ 5, a key step that no classical technique can even imagine. We have therefore a first order operator Bianchi : F1 → F2 with F1 = H 2 (ĝ 1 ) and F2 = H 3 (ĝ 1 ) when n ≥ 5. We obtain therefore a formally exact differential sequence:

0 → Θ → n CKilling -→ (n-1)(n+2)/2 W eyl -→ n(n+1)(n+2)(n-3)/12 Bianchi -→ n(n 2 -1)(n+2)(n-4)/24
In order to convince the reader about the powerfulness of the previous methods, let us prove that the generating CC of Bianchi is a second order operator with 14 equations when n = 5. First, we ask the reader to prove, as an exercise, that H 4 (ĝ 1 ) = 0 through the exact δ-sequence 0 → ∧ 3 T * ⊗ ĝ2 → ∧ 4 T * ⊗ ĝ1 → ∧ 5 T * ⊗ T → 0 (Hint: 50 = 55 -5). Conclude that the generating CC cannot be of first order. Then, prove that dim(H 4 (ĝ 2 )) = 25 -11 = 14. Finally, prove that the generating CC of these CC is first order with 5 equations (Hint: Use the vanishing of the Euler-Poincaré characteristic with 5-14+35-35+14-5 = 0) (See Appendix for confirmation).

C) PARAMETRIZATIONS

Let E and F be two vector bundles with respective fiber dimensions dim(E) = m and dim(F ) = p. Starting with a differential operator D : E → F of order q with solutions Θ ⊂ E and such that the corresponding system R q ⊂ J q (E) is formally integrable, we have explained in Section 3 how to construct a formally exact differential sequence:

0 → Θ → E D -→ F 0 D1 -→ F 1 D2 -→ F 2 D3 -→ ...
where F 0 = F and each operator D i of order q i generates the CC of the previous one. In particular, if the starting operator D is involutive, then q 1 = ... = q n = 1 and each D i is involutive in the resulting Janet sequence finishing at F n in the sense that D n is formally surjective. Equivalently, it is possible to pass to the framework of differential module and look for a free resolution of a differential module M starting with a free finite presentation D p D -→ D m → M → 0 with an operator acting on the right. In general, we have proved in the last two parts A and B that the succession of the orders q, q 1 , q 2 , q 3 , ... can be nevertheless quite strange. Meanwhile, we have proved through examples that many possible finite length such sequences can be exhibited and the purpose of Homological Algebra is to study formal properties that should not depend on the sequence used. At the end od Section 4, we have pointed out the fact that, whenever an operator D 1 generates all the CC of an operator D, this does not imply in general that the operator ad(D) generates all the CC of ad(D 1 ). The following (quite difficult) theorem is a main result of homological algebra, adapted to differential systems and differential modules ( [2], [3], [START_REF] Hu | Introduction to Homological Algebra[END_REF], [START_REF] Northcott | An Introduction to Homological Algebra[END_REF], [START_REF] Pommaret | Partial Differential Control Theory[END_REF], [START_REF] Rotman | An Introduction to Homological Algebra[END_REF]): THEOREM 5.2: The fact that a formally exact differential sequence considered as a resolution of Θ has the property that the adjoint sequence is also formally exact does not depend on the sequence but only on Θ.

COROLLARY 5.3:

The fact that a free resolution of a differential module M has the property that the adjoint sequence is also a free resolution does not depend on the sequence but only on M .

Our problem in this part C is to describe a sufficiently general situation in such a way that all the results of the parts A and B can fit together, in the sense that we shall no longer need to use a hat in order to distinguish them. For this, with E = T , let us say that an operator

D = Φ • j q : T → F is a Lie operator if Dξ = 0, Dη = 0 ⇒ D[ξ, η] = 0 that is to say [Θ, Θ] ⊆ Θ.
The corresponding system R q = ker(Φ) is called a system of infinitesimal Lie equations and one can define a "bracket on sections" satisfying [R q , R q ] ⊆ R q in order to check formally the previous definition ( [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Deformation Cohomology of Algebraic and Geometric Structures[END_REF], [START_REF] Pommaret | The Mathematical Foundations of General Relativity Revisited[END_REF]). It has been found by E. Vessiot, as early as in ... 1903 ( [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Vessiot | Sur la Théorie des Groupes Continus[END_REF]), that the condition of formal integrability of R q can be described by the Vessiot structure equations, a set of (non-linear in general) differential conditions depending on a certain number of constants, for one or a family of geometric objects that can be vectors, forms, tensors or even higher order objects. The idea has been to look for "general" systems or symbols having the same dimensions as for a model object called "special", for example the euclidean metric when n = 2, 3 or the minkowskian metric when n = 4. The case of part A has been a metric ω with det(ω) = 0 and constant riemannian curvature with one constant while the case of part B has been a metric density ω = ω/(| det(ω) | 1 n ) with zero Weyl tensor and no constant involved. The following results will be local.

Let us suppose that we have a Lie group of transformations of X, namely a Lie group G and an action X × G → X : (x, a) → y = ax = f (x, a) or, better, its graph X × G → X × X : (x, a) → (x, y = ax = f (x, a). Differentiating enough times, we may eliminate the parameters a among the equations y q = j q (f )(x, a) for q large enough and get a (non-linear in general) system of finite Lie equations. Linearising this system for a close to the identity e ∈ G, that is for y close to x, provides the system R q ⊂ J q (T ) and the corresponding Lie operator of finite type. Equivalently, the three theorems of Sophus Lie assert that there exists a finite number of infinitesimal generators {θ τ } of the action that should be linearly independent over the constants and satisfy [θ ρ , θ σ ] = c τ ρσ θ τ where the structure constants c define a Lie algebra G = T e (G). We have therefore ξ ∈ Θ ⇔ ξ = λ τ θ τ with λ τ = cst. Hence, we may replace locally the system of infinitesimal Lie equations by the system ∂ i λ τ = 0, getting therefore the differential sequence:

0 → Θ → ∧ 0 T * ⊗ G d -→ ∧ 1 T * ⊗ G d -→ ... d -→ ∧ n T * ⊗ G → 0
which is the tensor product of the Poincaré sequence by G. Finally, we are in a position to apply the previous Theorem and Corollary because the Poincaré sequence is self adjoint (up to sign), that is ad(d) generates the CC of ad(d) at any position, exactly like d generates the CC of d at any position. We invite the reader to compare with the situation of the Maxwell equations in electromagnetisme. However, we have proved in ( [START_REF] Pommaret | Lie Pseudogroups and Mechanics, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF], [START_REF] Pommaret | The Mathematical Foundations of General Relativity Revisited[END_REF], [START_REF] Pommaret | The Mathematical Foundations of Gauge Theory Revisited[END_REF], [START_REF] Pommaret | From Thermodynamics to Gauge Theory: the Virial Theorem Revisited[END_REF]) why neither the Janet sequence nor the Poincaré sequence can be used in physics and must be replaced by the Spencer sequence which is another resolution of Θ. We provide a few additional details on the motivations for such a procedure.

For this, if q is large enough in such a way that g q = 0 and thus R q+1 ≃ R q , let us define locally a section ξ q+1 ∈ R q+1 by the formula ξ k µ+1i = λ τ (x)∂ µ+1i θ k τ (x) and apply the Spencer operator D. We obtain at once (Dξ q+1 )

k µ,i = ∂ i ξ k µ -ξ k µ+1i = ∂ i λ τ ∂ µ θ k τ
, a result proving that the previous sequence is (locally) isomorphic to the Spencer sequence:

0 → Θ → ∧ 0 T * ⊗ R q D -→ ∧ 1 T * ⊗ R q D -→ ... D -→ ∧ n T * ⊗ R q → 0
In the present paper, we had dim(G) = n(n + 1)/2 in part A and dim( Ĝ) = (n + 1)(n + 2)/2 in part B. Moreover, whatever is the part concerned, ad(D) generates the CC of ad(D 1 ) because D 1 generates the CC of D while, similarly, ad(D 1 ) generates the CC of ad(D 2 ) because D 2 generates the CC of D 1 and so on. We conclude with the following comments.

1) Coming back to the first differential sequence constructed in part A, the Riemann tensor is a section of F 1 which is in the image of the operator Riemann or in the kernel of the operator Bianchi. Indeed, Lanczos has been considering the Riemann tensor as a section of F 1 killed by the operator Bianchi considered as a differential constraint. Accordingly, he has used the action of ad(Bianchi) on the corresponding Lagrange multipliers. However, this operator parametrizes ad(Riemann) as we saw and cannot be used in order to parametrize the Riemann tensor by means of the Lanczos potential.

2) Coming back to the second differential sequence constructed (independently) in part B, the Weyl tensor is a section of F1 which is in the image of the operator W eyl or in the kernel of the operator Bianchi. As most of the results presented are unknown, in particular the fact that both operators W eyl and Bianchi are second order in dimension n = 4, we believe that even the proper concept of a Weyl tensor candidate must be revisited within this new framework.

3) In parts A and B, only linear differential operators have been used. However, it is known from the formal theory of Lie pseudogroups that non-linear differential sequences can be similarly constructed ( [START_REF] Kumpera | Lie Equations[END_REF], [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], [START_REF] Pommaret | Partial Differential Equations and Group Theory,New Perspectives for Applications[END_REF]). As a matter of fact, if non-linear analogues of D and D 1 may be exhibited, this is not possible for D 2 . Moreover, the only important problem is to compare the image of D with the kernel of D 1 in the finite/infinitesimal equivalence problem (See [START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach[END_REF], p 333 for a nice counterexample). We believe that this shift by one step backwards in the interpretation of a differential sequence will become important for future physics. It is commonly done in the variational calculus using finite elements where the free energy brings together the deformation tensor and the EM field on equal footing, quite contrary to the ideas of Lanczos.

6) CONCLUSION

In most textbooks, the Weyl tensor is always presented today by comparison with the Riemann tensor after eliminating a conformal factor and its derivatives. We have exhibited new methods in order to introduce both the Riemann and the Weyl tensor independently by using the formal theory of systems of partial differential equations (Spencer cohomology) in the study of arbitrary Lie pseudogroups while using the Vessiot structure equations for the Killing and conformal Killing systems separately. In particular, we have revisited, in both cases, the proper concept of Bianchi identities by means of homological algebra and diagram chasing, obtaining explicit numbers and orders for each dimension. These striking results are confirmed by means of computer algebra in the Appendix. They prove that the work of Lanczos and followers must be revisited within this new framework.

7) APPENDIX by Alban QUADRAT (INRIA, Lille; alban.quadrat@inria.fr):

7.1) Conformal Killing system

Using an Euclidean or Minkowskian metric, the system of conformal Killing equations is:

ω rj ξ r i + ω ir ξ r j - 2 n ω ij ξ r r = 0. ( 1
)
but any other choice could be convenient too as we have explained in the paper.

7.2) Weyl tensor in dimension 2

Let us consider n = 2 and the euclidean metric

ω ij = δ i j for 1 ≤ i < j ≤ 2. If D = Q[d 1 , d 2 ]
denotes the commutative ring of PD operators in d 1 = ∂ ∂x1 and d 2 = ∂ ∂x2 , then the system can be rewritten as W 2 η = 0, where η = (ξ 1 ξ 2 ) T and:

W 2 =     d 1 -d 2 d 2 d 1 -d 1 d 2    
If we note λ 2 = 1 0 1 , then we have λ 2 W 2 = 0, which shows that the last row of W 2 is minus the first one. Hence, the D-module im D (W 2 ) = {λ W 2 | λ ∈ D 1×2 } generated by the rows of W 2 can be generated by the first two rows of W 2 , i.e., we have im D (W 2 ) = im D (R 2 ), where R 2 is the following matrix:

R 2 = d 1 -d 2 d 2 d 1
We can check that the D-module ker

D (R 2 ) = {µ ∈ D 1×2 | µ R 2 = 0}, called the second syzygy module ([4]) of the D-module M = D 2 /(D 2 R 2 )
, is reduced to 0, i.e., R 2 has full row rank. Hence, we obtain the following finite free resolution of M :

0 → D 2 R2
-→ D 2 → M → 0

7.3) Weyl tensor in dimension 3

Let us consider n = 3 and the euclidean metric

ω ij = δ i j for 1 ≤ i < j ≤ 3. If D = Q[d 1 , d 2 , d 3 ]
denotes the commutative ring of PD operators in d i = ∂ ∂xi for i = 1, 2, 3, then the system can be rewritten as W 3 η = 0, where η = (ξ 1 ξ 2 ξ 3 ) T and:

W 3 =              4 3 d 1 -2 3 d 2 -2 3 d 3 d 2 d 1 0 d 3 0 d 1 -2 3 d 1 4 3 d 2 -2 3 d 3 0 d 3 d 2 -2 3 d 1 -2 3 d 2 4 3 d 3             
If we note λ 3 = 1 0 0 1 0 1 ∈ D 1×6 , then we have λ 3 W 3 = 0 and shows that the last row of W 3 is a linear combination of the first and fourth rows of W 3 . Hence, the D-module im D (W 3 ), generated by the rows of W 3 , can be generated by the first five rows of W 3 , i.e., we have im D (W 3 ) = im D (R 3 ), where R 3 is the following matrix:

R 3 =           4 3 d 1 -2 3 d 2 -2 3 d 3 d 2 d 1 0 d 3 0 d 1 -2 3 d 1 4 3 d 2 -2 3 d 3 0 d 3 d 2          
Using the OreModules package, we can show that the left kernel of R 3 , i.e., the D-module ker D (R 3 ) = {λ ∈ D 1×3 | λ R 3 = 0}, is generated by the rows of the following matrix S 3 :

          -d 2 d 3 d 1 d 2 2 d 3 + d 3 3 -d 3 2 -d 2 d 2 3 -2 d 2 d 3 d 1 d 1 d 2 2 -d 1 d 2 3 -d 2 1 d 3 + 2 d 2 2 d 3 + d 3 3 0 -2 d 1 d 2 2 -2 d 1 d 2 3 -2 d 2 1 d 3 + d 2 2 d 3 -d 3 3 2 d 2 1 d 2 + 2 d 2 d 2 3 d 2 1 d 2 + 2 d 3 2 + d 2 d 2 3 -2 d 1 d 2 2 -2 d 1 d 2 3 0 2 d 2 1 d 2 + d 3 2 -d 2 d 2 3 2 d 2 1 d 3 + 2 d 2 2 d 3 2 d 2 d 3 d 1 -d 2 1 d 3 -d 3 3 -d 2 1 d 2 + d 2 d 2 3 d 2 d 3 d 1 d 3 1 + d 1 d 2 3 d 3 1 + 2 d 1 d 2 2 -d 1 d 2 3 -2 d 2 1 d 2 -2 d 2 d 2 3 2 d 2 1 d 3 + 2 d 2 2 d 3 2 d 3 1 + d 1 d 2 2 + d 1 d 2 3 0          
i.e., we have ker ) (see also [3]). Similarly, the left kernel of T 3 is generated by the rows of the following matrix

D (R 3 ) = im D (S 3 ) = {µ ∈ D 1×5 | µ S 3 }.
T 3 =     -2 d 3 0 d 1 -2 d 3 -d 2 2 d 1 -d 2 d 3 0 0 0 d 1 0 -2 d 2 d 3    
i.e., we have ker D (S 3 ) = im D (T 3 ). We can check that that the matrix T 3 has full row rank, i.e., ker D (T 3 ) = 0, or equivalently that the rows of T 3 are D-linearly independent. Finally, if we denote by M = D 3 /(D 5 R 3 ) = D 3 /(D 6 W 3 ) the D-module finitely presented by R 3 , associated with the system, then we obtain the following finite free resolution:

0 → D 3 T3 -→ D 5 S3 -→ D 5 R3 -→ D 3 → M → 0 7.

4) Weyl tensor in dimension 4

Let us consider n = 4 and the Minkowski metric ω = (1, 1, 1, -1).

If D = Q[d 1 , d 2 , d 3 , d 4
] denotes the commutative ring of PD operators in d i = ∂ ∂xi for i = 1, . . . , 4, then the system can be rewritten as W 4 η = 0, where η = (ξ 1 ξ 2 ξ 3 ξ 4 ) T and:

W 4 =                          3 2 d 1 -1 2 d 2 -1 2 d 3 -1 2 d 4 d 2 d 1 0 0 d 3 0 d 1 0 d 4 0 0 -d 1 -1 2 d 1 3 2 d 2 -1 2 d 3 -1 2 d 4 0 d 3 d 2 0 0 d 4 0 -d 2 -1 2 d 1 -1 2 d 2 3 2 d 3 -1 2 d 4 0 0 d 4 -d 3 1 2 d 1 1 2 d 2 1 2 d 3 -3 2 d 4                         
If we note λ 4 = 1 0 0 0 1 0 0 1 0 -1 , then we have λ 4 W 4 = 0, which shows that the last row of W 4 is a linear combination of the first, fourth, and eighth rows. Hence, the D-module im D (W 4 ), generated by the rows of W 4 , can be generated by the first nine rows of W 4 , i.e., we have im D (W 4 ) = im D (R 4 ), where R 4 is the following matrix:

R 4 =                       3 2 d 1 -1 2 d 2 -1 2 d 3 -1 2 d 4 d 2 d 1 0 0 d 3 0 d 1 0 d 4 0 0 -d 1 -1 2 d 1 3 2 d 2 -1 2 d 3 -1 2 d 4 0 d 3 d 2 0 0 d 4 0 -d 2 -1 2 d 1 -1 2 d 2 d 3 d 4 -2 d 1 d 3 d 1 d 2 0 -d 1 d 3 d 4 d 1 d 1 d 2 d 2 3 + d 2 4 -d 2 d 3 -d 4 d 2 d 1 d 2 -d 1 d 3 -d 4 d 1 2 d 1 d 2 0 d 3 d 4 0 -d 4 d 1 -d 1 d 3 -d 3 d 4 d 4 d 2 d 2 d 3 0 d 2 1 -d 2 2 d 4 d 2 -d 4 d 1 0 -d 1 d 2 0 d 3 d 4 d 2 1 -d 2 3 -d 4 d 2 d 2 d 3 2 d 2 d 3 -d 1 d 3 -d 1 d 2 0 d 2 d 3 d 2 1 + d 2 4 -d 3 d 4 d 2 d 3 -d 4 d 2 2 d 2 2 -2 d 2 3 -2 d 1 d 2 2 d 1 d 3 0 d 2 1 + d 2 2 -d 2 3 + d 2 4 0 -2 d 4 d 2 -d 2 1 + d 2 2 -d 2 3 -d 2 4 2 d 3 d 4 d 2 1 -3 d 2 1 + 3 d 2 3 + d 2 4 2 d 1 d 2 -4 d 1 d 3 -2 d 4 d 1 -2 d 2 2 -2 d 2 4 2 d 2 d 3 4 d 4 d 2 3 d 2 1 -3 d 2 2 + d 2 3 + d 2 4 -2 d 3 d 4 0 d 3 d 4 0 -d 2 d 3 0 -d 4 d 1 0 0 d 1 d 2 0 d 4 d 2 -d 3 d 4 -d 2 2 + d 2 3 -d 4 d 1 0 d 1 d 2 d 4 d 1 -d 1 d 3 0 0 d 4 d 2 -d 2 d 3 0 -d 4 d 1 d 1 d 3 0 0                           
i.e., we have ker D (R 4 ) = im D (S 4 ). In fact, using the OreModules package, we can prove that ker D (R 4 ) = im D (S 4 ), where S 4 = (S T 4 S ′ 4 T ) T , where the matrix S ′ 4 ∈ D 6×9 is defined by

S ′ 4 ∈ D 6×9 :                     -d 2 d 3 d 4 0 0 0 -d 2 d 3 d 4 d 2 3 d 4 -d 3 4 -d 3 3 + d 3 d 2 4 -2 d 2 d 3 d 4 d 2 d 2 3 + d 2 d 2 4 -d 1 d 3 d 4 0 d 2 3 d 4 -d 3 4 -d 3 3 + d 3 d 2 4 -d 1 d 3 d 4 0 0 -2 d 1 d 3 d 4 d 1 d 2 3 + d 1 d 2 4 -d 2 2 d 4 + d 2 3 d 4 0 0 0 -d 2 2 d 4 + 2 d 2 3 d 4 -d 3 4 0 -2 d 2 d 2 3 + 2 d 2 d 2 4 -2 d 2 2 d 4 + d 2 3 d 4 + d 3 4 2 d 2 2 d 3 -2 d 3 d 2 4 d 2 2 d 3 + d 3 3 0 0 0 d 2 2 d 3 + 2 d 3 3 -d 3 d 2 4 -2 d 2 d 2 3 + 2 d 2 d 2 4 0 2 d 2 2 d 3 + d 3 3 + d 3 d 2 4 -2 d 2 2 d 4 -2 d 2 3 d 4 d 2 d 3 d 4 0 0 0 2 d 2 d 3 d 4 -d 2 2 d 4 + d 3 4 -d 2 2 d 3 -d 3 d 2 4 d 2 d 3 d 4 d 3 2 -d 2 d 2 4 d 3 2 + d 2 d 2 3 0 0 0 d 3 2 + 2 d 2 d 2 3 + d 2 d 2 4 -2 d 2 2 d 3 + 2 d 3 d 2 4 -2 d 2 2 d 4 -2 d 2 3 d 4 2 d 3 2 + d 2 d 2 3 -d 2 d 2 4 0                    
.

But, we have S ′ 4 = F 4 S 4 , where

F 4 =              0 0 0 d 3 -d 4 0 0 0 0 -d 1 d 4 0 0 0 0 0 0 0 -d 3 d 2 0 0 -d 3 d 2 0 -d 4 0 0 -d 1 0 -d 1 0 d 4 0 d 2 -2 d 3 -d 3 0 0 0 0 0 -d 2 0 d 4 0 0 d 1 0 0 0 d 1 0 d 4 d 3 -d 2 -d 2 0 0 0              ∈ D 6×10 ,
which shows that im D (S 4 ) = im D (S 4 ). Similarly, we get that ker D (S 4 ) = im D (T 4 ), where the matrix T 4 ∈ D 9×10 is defined by

                       d 4 d 2 d 3 d 4 0 0 0 0 0 -d 2 3 -d 2 4 -d 2 d 3 d 2 2 + d 2 4 d 4 d 1 0 -d 2 3 -d 2 4 -d 2 d 3 d 4 d 2 d 3 d 4 d 3 d 4 0 -d 1 d 3 2 d 1 d 2 d 1 d 2 d 1 d 3 d 4 d 2 -d 3 d 4 -d 2 2 + d 2 3 d 2 d 3 0 -2 d 4 d 1 0 2 d 4 d 1 -d 3 d 4 -d 4 d 2 d 1 d 3 -d 1 d 2 0 d 4 d 1 0 2 d 2 d 3 d 2 1 + d 2 4 -2 d 2 d 3 d 2 d 3 d 2 1 -d 2 2 0 d 4 d 1 d 1 d 3 -d 1 d 2 -d 1 d 2 0 d 4 d 2 2 d 3 d 4 d 2 1 -d 2 3 d 2 d 3 -d 4 d 1 0 -d 1 d 2 2 d 1 d 3 d 1 d 3 -2 d 4 d 2 d 3 d 4 0 0 -d 4 d 1 -d 2 d 3 -d 2 2 -d 2 4 d 3 d 4 2 d 4 d 2 d 4 d 2 2 d 1 d 3 d 1 d 2 0 0 -d 3 d 4 0 -d 1 d 3 d 4 d 1 0 0 d 2 3 + d 2 4 0 d 2 1 -d 2 3 0 -d 3 d 4 -d 1 d 2 0 d 4 d 1 0 0 d 2 1 -d 2 2 + d 2 3 + d 2       -2 d 3 0 0 -d 2 d 4 0 d 1 0 -2 d 3 2 d 1 -d 2 -d 4 0 0 0 d 3 0 0 0 d 1 0 d 3 0 -d 4 0 -2 d 2 0 0 0 d 1 0 -d 3 -d 2 0 -2 d 4 2 d 4       
.

Finally, we can check that U 4 has full row rank, i.e., ker D (U 4 ) = 0, which shows that the D-module M = D 4 /(D 9 R 4 ) = D 4 /(D 10 W 4 ), associated with the system, admits the following finite free resolution:

0 → D 4 U4 -→ D 9 T4 -→ D 10 S4 -→ D 9 R4 -→ D 4 → M → 0
Finally, we note that a free resolution of M can be computed in 0.616 CPU seconds when n = 4.

7.5) Weyl tensor in dimension 5

Let us consider n = 5 and the euclidean metric

ω ij = δ i j for 1 ≤ i < j ≤ 5. If D = Q[d 1 , d 2 , d 3 , d 4 , d 5 ]
denotes the commutative ring of PD operators in d i = ∂ ∂xi for i = 1, . . . , 5, then the system can be rewritten as W 5 η = 0, where η = (ξ 1 . . . ξ 5 ) T and: 

    8 5 d 1 -2 5 d 2 -2 5 d 3 -2 5 d 4 -2 5 d 5 d 2 d 1 0 0 0 d 3 0 d 1 0 0 d 4 0 0 d 1 0 d 5 0 0 0 d 1 -2 5 d 1 8 5 d 2 -2 5 d 3 -2 5 d 4 -2 5 d 5 0 d 3 d 2 0 0 0 d 4 0 d 2 0 0 d 5 0 0 d 2 -2 5 d 1 -2 5 d 2 8 5 d 3 -2 5 d 4 -2 5 d 5 0 0 d 4 d 3 0 0 0 d 5 0 d 3 -2/5 d 1 -2 5 d 2 -2 5 d 3 8 5 d 4 -2 5 d 5 0 0 0 d 5 d 4 -2 5 d 1 -2 5 d 2 -2 5 d 3 -2 5 d 4
If we note λ 5 = 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 , then we have λ 5 W 5 = 0, which shows that the D-module im D (W 5 ), generated by the rows of W 5 , can be generated by the first fourteen rows of W 5 , i.e., we have im D (W 5 ) = im D (R 5 ), where R 5 is the following matrix:

  being at the same time a left D-module D D by the composition P -→ QP and a right D-module D D by the composition P -→ P Q with D r D s = D r+s , ∀r, s ≥ 0.

  1, ..., p. Therefore, setting Dy 1 + ... + Dy m = Dy ≃ D m and calling I = DΦ ⊂ Dy the differential module of equations, we obtain by residue the differential module or D-module M = Dy/DΦ, denoting the residue of y k

DEFINITION 4 . 1 :

 41 Setting P = a µ d µ ∈ D ad ←→ ad(P ) = (-1) |µ| d µ a µ ∈ D, we have ad(ad(P )) = P and ad(P Q) = ad(Q)ad(P ), ∀P, Q ∈ D. Such a definition can be extended to any matrix of operators by using the transposed matrix of adjoint operators and we get: < λ, Dξ >=< ad(D)λ, ξ > + div (...) from integration by part, where λ is a row vector of test functions and <> the usual contraction. We quote the useful formulas [ad(ξ), ad(η)] = ad(ξ)ad(η) -ad(η)ad(ξ) = -ad([ξ, η]), ∀ξ, η ∈ T (care about the minus sign) and rk D (D) = rk D (ad(D)) as in ([23], p 610-612).

DEFINITION 4 . 5 :

 45 We may define the right differential module M * = hom D (M, D) by using the bimodule structure of D = D D D .

THEOREM 4 . 6 :

 46 When M and N are left D-modules, then hom K (M, N ) and M ⊗ K N are left D-modules. In particular R = hom K (M, K) is also a left D-module for the Spencer operator.

COROLLARY 4 . 7 :REMARK 4 . 8 :

 4748 If M and N are right D-modules, then hom K (M, N ) is a left D-module. Moreover, if M is a left D-module and N is a right D-module, then M ⊗ K N is a right D-module. Proof: If M and N are right D-modules, we just need to set (ξf )(m) = f (mξ) -f (m)ξ, ∀ξ ∈ T, ∀m ∈ M and conclude as before. Similarly, if M is a left D-module and N is a right D-module, we just need to set (m ⊗ n)ξ = m ⊗ nξ -ξm ⊗ n. Q.E.D. When M = D M ∈ mod(D) and N = N D , , then hom K (N, M ) cannot be endowed with any left or right differential structure. When M = M D and N = N D , then M ⊗ K N cannot be endowed with any left or right differential structure (See [2], p 24 for more details). As M * = hom D (M, D) is a right D-module, let us define the right D-module N D by the ker/coker long exact sequence 0 ←-

1 D1- 1 D1-→ F 0 D

 110 α by introducing the Lie derivative of α with respect to ξ, along the intrinsic formula L(ξ) = i(ξ)d + di(ξ) where i() is the interior multiplication and d is the exterior derivative of exterior forms. According to well known properties of the Lie derivative, we get :α.(aξ) = (α.ξ).a -α.ξ(a), α.(ξη -ηξ) = -[L(ξ), L(η)]α = -L([ξ, η])α = α.[ξ, η]. Q.E.D.Collecting the previous results, if a differential operator D is given in the framework of differential geometry, we may keep the same notation D in the framework of differential modules which are left modules over the ring D of linear differential operators and apply duality, provided we use the notation D * and deal with right differential modules or use the notation ad(D) and deal again with left differential modules by using the lef t ↔ right conversion procedure. DEFINITION 4.10: If an operator ξ D -→ η is given, a direct problem is to look for (generating) compatibility conditions (CC) as an operator η D1 -→ ζ such that Dξ = η ⇒ D 1 η = 0. Conversely, given η D1 -→ ζ, the inverse problem will be to look for ξ D -→ η such that D 1 generates the CC of D and we shall say that D 1 is parametrized by D if such an operator D is existing. As ad(ad(P )) = P, ∀P ∈ D, any operator is the adjoint of a certain operator and we get: DOUBLE DUALITY CRITERION 4.11: An operator D 1 can be parametrized by an operator D if, whenever ad(D) generates the CC of ad(D 1 ), then D 1 generates the CC of D. However, as shown in the example below, many other parametrizations may exist. Reversing the arrows, we finally obtain: TORSION-FREE CRITERION 4.12: A differential module M having a finite free presentation F → F 0 → M → 0 is torsion-free, that is to say t(M ) = {m ∈ M | ∃0 = P ∈ D, P m = 0} = 0, if and only if there exists a free differential module E and an exact sequence F -→ E providing the parametrization M ⊆ E.

d 2 d 3 -d 4 d 1 d 4 d 2 d 2 d 3 -2 d 4 d 2 d 2 d 5 d 1 d 2 d 1 d 3 d 5 2 d 5 d 4 d 1 d 2 0 -d 4 d 1 0 0 -d 4 d 1 -d 1 d 5 -2 d 4 d 1 d 1 d 5 d 2 5 -2 d 5 d 4 d 5 5 d 5 d 4 d 4 d 2 0 0 d 4 d 2 d 2 d 5 6 d 4 d 2 -2 d 2 d 5 3 d 2 3 -3 d 2 5 -4 d 3 d 4 6 d 3 d 5 -d 2 1 -4 d 2 2 + 5 d 3 2 -2 d 2 4 + 2 d 2 5 -4 d 5 d 4 -6 d 4 d 2 2 d 2 d 5 -5 d 3 2 + 5 d 2 5 6 d 3 d 4 -10 d 3 d 5 4 d 1 2 + 4 d 2 2 -8 d 2 3 + 3 d 2 4 -3 d 5 2 6 d 5 d 4 -d 3 d 4 -d 3 d 5 -d 2 d 3 2 d 4 d 2 -d 2 d 5 -3 d 2 d 3 -d 2 d 5 -d 4 d 2 -d 5 d 4 d 3 d 5 d 3 d 4 0 d 2 2 -d 2 3 0 -d 2 d 3 0 d 5 d 4 d 2 2 -d 2 4 -d 3 d 5 d 3 d 4 0 -d 3 d 4 0 -d 2 d 5 0 0 d 2 d 3 -d 5 d 4 -d 2 3 + d 2 5 -d 2 d 5 0 d 2 d 3 d 2 d 5 -d 4 d 2 1 d 3 -d 3 d 5 d 3 d 5 0 d 2 d 3 -d 2 d 3 d 4 d 4 d 1 -d 1 d 5 0 d 1 d 3 -d 4 d 1 d 1 d 5 -d 1 d 2 5 d 4 2 d 3 d 5 d 5 d 4 -2 d 3 d 5 0 d 5 d 4 d 3 d 5 0 d 5 d 4 d 1 d 3 0 d 3 d 5 -d 4 d 1 d 4 d 1

 32335421555455523512245452542234244525352232443255233313524551 have ker D (U 5 ) = im D (V 5 ), where V 5 ∈ D 5×14 is the full row rank matrix defined by: D-module M = D 5 /(D 14 R 5 ) = D 5 /(D 15 W 5 ) admits the following finite free resolution:0 → D 5 V5 -→ D 14 U5 -→ D 35 T5 -→ D 35 S5 -→ D 14 R5 -→ D 5 → M 5 → 0

  The computation of S 3 takes 0.026 CPU seconds with Maple 18 on Mac OS 10.10.5 equipped with 2.8 GHz Intel Core i7 and 16 Go. For more details on the left kernel/syzygy computation, see Algorithm 1 on page 330 of ([1]