Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian geometry. We also discovered in the meantime that, in our first book of 1978, we had already used a new way for studying the various compatibility conditions (CC) of an operator that may not be necessarily formally integrable (FI) in order to construct canonical formally exact differential sequences on the jet level. The purpose of this paper is to prove that the combination of these two facts clearly shows the specific importance of the Spencer operator and the Spencer $\delta$-cohomology, totally absent from mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide for the first time the purely intrinsic interpretation of the respective numbers of successive first, second, third and higher order generating CC. However, if they of course agree with the linearized Killing operator over the Minkowski metric, they largely disagree with recent publications on the respective numbers of generating CC for the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular the only symbol existing in the literature which is 2-ayclic witout being of finite type, contrary to the conformal situation.


1) INTRODUCTION
The present study is mainly local and we only use standard notations of differential geometry. For simplicity, we shall also adopt the same notation for a vector bundle (E, F, . . . ) and its set of sections (ξ, η, ζ, . . . ). Now, if X is the ground manifold X with dimension n and local coordinates (x 1 , . . . , x n ) and E is a vector bundle over X with local coordinates (x, y), we shall denote by J q (E) the q-jet bundle of E with local coordinates (x, y q ) and sections ξ q transforming like the q-derivatives j q (ξ) of a section ξ = ξ 0 of E. If F with section η is another vector bundle over X and Φ : J q (E) → F is an epimorphism with kernel the linear system R q ⊂ J q (E), we shall associate the differential operator D = Φ • j q : E → F : ξ → η and set Θ = ker(D). All the operators considered will be locally defined over a differential field K whith n derivations (∂ 1 , . . . , ∂ n ) and we shall indicate the order of an operator under its arrow. It is well known and we shall provide many explicit examples, that, if we want to solve, at least locally the linear inhomogeneous system Dξ = η, one usually needs compatibility conditions (CC) of the form D 1 η = 0 defined by another differential operator D 1 : F = F 0 → F 1 : η → ζ that may be of high order in general but still locally defined over K. However, two types of " phenomena " can arise for exhibiting such CC but, though they can be quite critical in actual practice, we do not know any other reference on the possibility to solve them effectively because most people rely on the work of E. Cartan. 1) As shown in ( [11], Introduction) or ( [13]) with the Janet system (ξ 33 − x 2 ξ 11 = 0, ξ 22 = 0) over the differential field K = Q(x) and in ( [22]), it may be possible to find no CC of order one, no CC of order two, one CC of order three, then nothing new but one additional CC of order six and so on with no way to know when to stop. For the fun, when we started computer algebra around 1990, we had to ask a special permit to the head of our research department for running the computer a full night and were not even able after a day to go any further on. Hence, a first basic problem is to establish a preliminary list of generating CC and know their maximum order.
2) Once the previous problem is solved, we do know a generating D 1 of order q 1 and may start anew with it in order to obtain a generating D 2 of order q 2 and so on as a way to work out a differential sequence. Contrary to what can be found in the Poincaré sequence for the exterior derivative where all the successive operators are of order one, things may not be so simple in actual practice and " jumps " may apear, that is the orders may go up and down in a apparently surprising manner that only the use of " acyclicity " through the Spencer cohomology can explain. As we shall see with more details in the case of the conformal Killing operator of order 1, the successive orders are (1, 3, 1) when n = 3, (1, 2, 2, 1) when n = 4, (1, 2, 1, 2, 1) when n = 5 ( [27]).
A we have shown in our seven books, the only possibility to escape from these two types of problems is to start with an involutive operator D and construct in an intrinsic way two canonical differential sequences, namely the linear Janet sequence ( [8], p 185, 391 for a global definition): As in both cases, the central operator is the Spencer operator but not the exterior derivative, contrary to what is done in ( [1]) and the corresponding references, in particular ( [8]), we do not agree on the effectivity of their definition of " involutivity " (p 1608/1609). In fact, the most important property of theses two sequences is that they are formally exact on the jet level as follows. Introducing the (composite) r-prolongation by means of the formal derivatives d i : ρ r (Φ) : J q+r (E) → J r (J q (E)) → J r (F 0 ) : (x, y q+r ) → (x, z ν = d ν Φ, 0 ≤| ν |≤ r) with kernel R q+r = ρ r (R q ) = J r (R q ) ∩ J q+r (E) ⊂ J r (J q (E)), we have the long exact sequences: and so on till the similar ones stopping at J r (F n ), ∀r ≥ 0. As shown by the counterexample exhibited in ( [18], p 119-126), all these sequences may be absolutely useful till the last one. We shall also define the symbol g q = R q ∩ S q T * ⊗ E and its r-prolongations g q+r = ρ r (g q ) only depends on g q in a purely algebraic way, that is no differentiation is involved. On the contrary, we shall say that R q or D is formally integrable (FI) if R q+r is a vector bundle ∀r ≥ 0 and all the epimorphisms π q+r+1 q+r : J q+r+1 (E) → J q+r (E) : (x, y q+r+1 ) → (x, y q+r ) are inducing epimorphisms R q+r+1 → R q+r of constant rank ∀r ≥ 0, which is a true purely differential property.
Of course, for people familar with functional analysis, the definition of Θ could seem strange and uncomplete as it is not clear where to look for solutions. In our opinion (See [12] and review Zbl 1079.93001) it is mainly for this reason that differential modules or simply D-modules have been introduced but we shall explain why such a procedure leads in fact to a (rather) vicious circle as follows. Working locally for simplicity with dim(E) = m, dim(F ) = p, we may turn the definition backwards by introducing the non-commutative ring D = K[d 1 , . . . , d n ] = K[d] of differential polynomials (P, Q, . . . ) with coefficients in K. Then, instead of acting on the " left " of column vectors of sections by differentiations as in the previous differential setting, we shall use the same operator matrix still denoted by D but now acting on the " right " of row vectors by composition. Introducing the canonical projection onto the residual module M , we obtain the exact sequence D p D −→ q D m → M → 0 of differential modules also called " free resolution " of M because D m and D p are clearly free differential modules. However, as D is filtred by the order of operators, then I = im(D) ⊂ D m is filtred too and, as we shall clearly see on the motivating examples, the induced filtration of M = D m /I can only been obtained in any applications if and only if R q or D is FI. Accordingly, all the difficulty will be to use the following key theorem (For Spencer cohomology and acyclicity or involutivity, see [8]- [13], [18], [19]): THEOREM 1.1: There is a finite Prolongation/Projection (PP) algorithm providing two integers r, s ≥ 0 by successive increase of each of them such that the new system R (s) q+r = π q+r+s q+r (R q+r+s ) has the same solutions as R q but is FI with a 2-acyclic or involutive symbol and first order CC. The order of a generating D 1 is thus bounded by r + s + 1 as we used r + s prolongations. EXAMPLE 1.2: In the Janet example we have R 2 → R (1) 3 → R (2) 4 → R (2) 5 with 8 < 11 < 12 = 12 and dim K (M ) = 12 ⇒ rk D (M ) = 0. The final system is trivially involutive because it is FI with a zero symbol, a fact highly not evident a prori because it needs 5 prolongations and the maximum order of the CC is thus equal to 3 + 2 + 1 = 6. We obtain therefore a minimum resolution of the D → M → 0 (See the introduction of [11] or [13] for details).
When a system is FI, we have a projective limit R = R ∞ → · · · → R q → · · · → R 1 → R 0 . As we are dealing with a differential field K, there is a bijective correspondence: and we obtain the injective limit 0 ⊆ M 0 ⊆ M 1 ⊆ · · · ⊆ M q ⊆ . . . M ∞ = M providing the filtration of M . We have in particular d i M q ⊆ M q+1 and M = DM q for q ≫ 0. THEOREM 1.3: R = hom K (M, K) is a differential module for the Spencer operator.
Proof: As the ring D is generated by K and T = {a i d i | a i ∈ K}, we just need to define: (af )(m) = a(f (m)) = f (am), (d i f )(m) = ∂ i (f (m)) − f (d i m), ∀a ∈ K, ∀m ∈ M, ∀d i ∈ T, ∀f ∈ R and obtain d i a = ad i + ∂ i a in the operator sense. Choosing m ∈ M to be the residue of d µ y k = y k µ and setting f (y q ) = ξ q : f (y k µ ) = ξ k µ ∈ K, we obtain in actual practice exactly the Spencer operator: with a slight abuse of language. We notice that a "section" ξ q ∈ R q has in general, particularly for the non-commutative case (See [27] for examples), nothing to do with a "solution", a concept missing in ([1]- [4]). ✷ As we shall see in the motivating examples, once a differential module M or the dual system R = hom K (M, K) is given, there may be quite different differential sequences or quite different resolutions and the problem will be to choose the one that could be the best in the application considered. During the last world war, many mathematicians discovered that a few concepts, called extension modules, were not depending on the sequence used in order to compute them but only on M . A (very) delicate theorem of (differential) homological algebra even proves that no others can exist ( [28]). Let us explain in a way as simple as possible these new concepts.
As a preliminary crucial definition, if P = a µ d µ ∈ D, we shall define its (formal) adjoint by the formula ad(P ) = (−1) |µ| d µ a µ where we have set | µ |= µ 1 + · · · + µ n whenever µ = (µ 1 , . . . , µ n ) is a multi-index. Such a definition can be extended by linearity in order to define the formal adjoint ad(D) to be the transposed operator matrix obtained after taking the adjoint of each element. The main property is that ad(P Q) = ad(Q)ad(P ), ∀P, Q ∈ D ⇒ ad(D 1 • D) = ad(D) • ad(D 1 ).
In the operator framework, we have the differential sequences: 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: where the lower sequence is not exact at D 2 . The "extension modules " have been introduced in order to study this kind of " gaps ".
Therefore, we have to prove that the extension modules vanish, that is ad(D) generates the CC of ad(D 1 ) and, conversely, that D 1 generates the CC of D. We also remind the reader that it has not been easy to exhibit the CC of the Maxwell or Morera parametrizations when n = 3 and that a direct checking for n = 4 should be strictly impossible ( [17]). It has been proved by L. P. Eisenhart in 1926 (Compare to [8]) that the solution space Θ of the Killing system has n(n+1)/2 infinitesimal generators {θ τ } linearly independent over the constants if and only if ω had constant Riemannian curvature, namely zero in our case. As we have a Lie group of transformations preserving the metric, the three theorems of Sophus Lie assert than [θ ρ , θ σ ] = c τ ρσ θ τ where the structure constants c define a Lie algebra G. We have therefore ξ ∈ Θ ⇔ ξ = λ τ θ τ with λ τ = cst. Hence, we may replace the Killing system by the system ∂ i λ τ = 0, getting therefore the differential sequence: which is the tensor product of the Poincaré sequence for the exterior derivative by the Lie algebra G. Finally, as the extension modules do not depend on the resolution used and that most of them do vanish 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 ( [10]). However, we have proved in ( [14], [21], [24], [25]) why neither the Janet sequence nor the Poincaré sequence can be used in physics and must be replaced by another resolution of Θ called Spencer sequence ( [19]).
After this long introduction, the content of the paper will become clear: In section 2 we provide the mathematical tools from homological algebra and differential geometry needed for finding the generating CC of various orders. Then, section 3 will provide motivating examples in order to illustrate these new concepts. They are finally applied to the Killing systems for the S and K metrics in section 4 in such a way that the results obtained, though surprising they are, cannot be avoided because they will only depend on diagram chasing and elementary combinatorics.

A) HOMOLOGICAL ALGEBRA
We now need a few definitions and results from homological algebra ( [12], [13], [28]). In all that follows, A, B, C, ... are modules over a ring or vector spaces over a field and the linear maps are making the diagrams commutative. We introduce the notations rk = rank, nb = number, dim = dimension, ker = kernel, im = image, coker = cokernel. When Φ : A → B is a linear map (homomorphism), we may consider the so-called ker/coker exact sequence where where coker(Φ) = B/im(Φ): In the case of vector spaces over a field k, we successively have rk(Φ) = dim(im(Φ)), dim(ker(Φ)) = dim(A) − rk(Φ), dim(coker(Φ)) = dim(B) − rk(Φ) = nb of compatibility conditions, and obtain by substraction: In the case of modules, using localization, we may replace the dimension by the rank and obtain the same relations because of the additive property of the rank. The following result is essential: SNAKE LEMMA 2.A.1: When one has the following commutative diagram resulting from the the two central vertical short exact sequences by exhibiting the three corresponding horizontal ker/coker exact sequences: then there exists a connecting map M −→ Q both with a long exact sequence: Proof: We start constructing the connecting map by using the following succession of elements: Indeed, starting with m ∈ M , we may identify it with c ∈ C in the kernel of the next horizontal map. As Ψ is an epimorphism, we may find b ∈ B such that c = Ψ(b) and apply the next horizontal map to get b ′ ∈ B ′ in the kernel of Ψ ′ by the commutativity of the lower square. Accordingly, there is a unique a ′ ∈ A ′ such that b ′ = Φ ′ (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 Φ of a certain a ∈ A having zero image in Q by composition. The remaining of the proof is similar and left to the reader as an exercise. The above explicit procedure will not be repeated. ✷ We may now introduce cohomology theory through the following definition: 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 into the following two commutative and exact diagrams by taking into account the relations im(Ψ) = ker(Ω), im(Ψ ′ ) = ker(Ω ′ ): Using the snake theorem, we successively obtain: Comparing the sequences obtained in the previous examples, we may state: DEFINITION 2.B.1: A differential sequence is said to be formally exact if it is exact on the jet level composition of the prolongations involved. A formally exact sequence is said to be strictly exact if all the operators/systems involved are FI (See [5], [8], [11], [16] and [19] for more details). A strictly exact sequence is called canonical if all the operators/systems are involutive. Fourty years ago, we did provide the link existing between the only known canonical sequences, namely the Janet and Spencer sequences ( [8], See in particular the pages 185 and 391).
With canonical projection Φ 0 = Φ : J q (E) ⇒ J q (E)/R q = F 0 , the various prolongations are described by the following commutative and exact "introductory diagram " often used in the sequel: Chasing along the diagonal of this diagram while applying the standard "snake" lemma, we obtain the useful "long exact connecting sequence " also often used in the sequel: which is thus connecting in a tricky way FI (lower left) with CC (upper right).
We finally recall the "fundamental diagram I " that we have presented in many books and papers, relating the (upper) canonical Spencer sequence to the (lower) canonical Janet sequence, that only depends on the left commutative square D = Φ • j q with Φ = Φ 0 when one has an involutive system R q ⊆ J q (E) over E with dim(X) = n and j q : E → J q (E) is the derivative operator up to order q while the epimorphisms Φ 1 , ..., Φ n are successively induced by Φ: This result will be used in order to compare the M, S and K metrics when n = 4 but it is important to notice that this whole diagram does not depend any longer on the (a, m) parameters of S and K ( [20], [23]).
Proof: First we notice that necessarily we must have π q+1 q (R q+1 ) ⊂ R q because, as ρ 1 (R q ) may not project onto R q , it is nevertheless defined by (maybe) more equations of strict order q than R q .
The converse way is similar.
✷ The next key idea has been discovered in ( [8]) as a way to define the so-called Janet bundles and thus for a totally different reason.
However, a chase in this diagram proves that the kernel of this epimorphism is not im(σ r+1 (Φ) unless R q is FI (care). For this reason, we shall define it to be exactly g ′ r+1 .
is the number of new generating CC of order r + 1 .
Proof: First of all, we have the following commutative and exact diagram obtained by applying the Spencer operator to the top long exact sequence: Cutting" the diagram in the middle as before while using the last definition, we obtain the induced map R ′ r+1 d −→ T * ⊗ R ′ r and the first inclusion follows from the last proposition. Such a procedure cannot be applied to the top row of the introductory diagram through the use of δ instead of d because of the comment done on the symbol in the last definition. Now, using only the definition of the prolongation for the system and its symbol, we have the following commutative and exact diagram: and obtain the following commutative and exact diagram: The computation of y = dim(A) = dim(ρ 1 (R ′ r )) − dim(R ′ r+1 ) only depends on x = dim(Q ′ 1 ) and is rather tricky as follows (See the motivating examples): As we shall see with the motivating examples and with the S or K metrics, the computation is easier when the system is FI but can be much more difficult when the system is not FI. However, the number of linearly independent CC of order r + 1 coming from the CC of order r is dim(J 1 (Q r )) − x while the total number of CC of order r + 1 is: The number of new CC of strict order r + 1 is equal to y because dim(J r+1 (F 0 )) disappears by difference. For a later use in GR, we point out the fact that, if the given system R q ⊂ J q (E) depends on parameters that must be contained in the ground differential field K (only (m) for the S metric but (a, m) for the K metric), all the dimensions considered may highly depend on them even if the underlying procedure is of course the same.
As an alternative proof, we may say that the number of CC of strict order r + 1 obtained from the CC of order r is equal to dim(S r+1 T * ⊗ F 0 ) − dim(ρ 1 (g ′ r )) while the total number of CC of order r + 1 is equal to . The number of new CC of strict order r + 1 is thus also equal to y = dim((ρ 1 (g ′ r )) − dim(g ′ r+1 ) because dim(S r+1 T * ⊗ F 0 ) also disappears by difference. However, unless R q is FI, we have in general g ′ r = im(σ r (Φ)) and it thus better to use the systems rather than their symbols. ✷ COROLLARY 2.B.5: The system R ′ r ⊂ J r (F 0 ) becomes FI with a 2-acyclic or involutive symbol and R ′ r+1 = ρ 1 (R ′ r ) ⊂ J r+1 (F 0 ) when r is large enough.
Proof: According to the last diagram, we have g ′ r+1 ⊆ ρ 1 (g ′ r ) and g ′ r+1 is thus defined by more linear equations than ρ 1 (g ′ r ). We are facing a purely algebraic problem over commutative polynomial rings and well known noetherian arguments are showing that g ′ r+1 = ρ 1 (g ′ r ) or, equivalently, y = 0 when r is large enough. Chasing in the last diagram, we obtain therefore R ′ r+1 = ρ 1 (R ′ r ) for r large enough and R ′ r is a vector bundle because because R q+r is a vector bundle. If we denote by M ′ the differential module obtained from the system R ′ r ⊂ J r (F 0 ) exactly like we have denoted by M the differential module obtained from the system R q ⊂ J q (E), we have the short exact sequence 0 → M ′ → D m → M → 0. Accordingly, M ′ ≃ I ⊂ D m is a torsion-free differential module and there cannot exist any specialization as an epimorphism M ′ → M " → 0 with rk D (M ′ ) = rk D (M ") because the kernel should be a torsion differential module and thus should vanish. This comment is strengthening the fact that the knowledge of M and thus of I can only be done through Theorem 1.1. Therefore, if (r, s) are the ones produced by this theorem, then the order of the CC system must be r + s + 1. We obtain 3 + 2 + 1 = 6 for the Janet system with systems R ′ r of successive dimensions 2, 8, 20, 39, 66, 102, 147 and ask the reader to find dim(R ′ 7 ) = 202 (Hint: [11]). ✷ We are now ready for working out the generating CC D 1 : F 0 → F 1 and start afresh in a simpler way because this new operator is FI (Compare to [8], Proposition 2.9, p 173). However, contrary to what the reader could imagine, it is precisely at this point that troubles may start and the best example is the conformal Killing operator. Indeed, it is known that the order of the generating CC for a system of order q which is FI is equal to s + 1 if the symbol g q+s becomes 2-acyclic before becoming involutive. This fact will be illustrated in a forthcoming motivating example but we recall that the conformal Killing symbolĝ 1 ⊂ T * ⊗ T is such thatĝ 2 is 2-acyclic when n ≥ 4 whilê g 3 = 0, a fact explaining why the Weyl operator is of order 2 but the Bianchi-type operator is also of order 2, a result still neither known nor even acknowledged today ( [18], [27]).

3) MOTIVATING EXAMPLES.
We now provide three motivating examples in order to illustrate both the usefulness and the limit of the previous procedure. EXAMPLE 3.1: With m = 1, n = 3, K = Q, we revisit the nice example of Macaulay ([7]) presented in ( [22]), namely the homogeneous second order linear system R 2 ⊂ J 2 (E) defined by ξ 33 = 0, ξ 13 − ξ 2 = 0 which is far from being formally integrable. We let the reader prove the strict inclusions R The respective symbols are involutive but only the final system R (2) 2 is involutive. It follows that the generating CC of the operator defined by R 2 are at most of order 3 but there is indeed only one single generating second order CC ( [22]). Elementary combinatorics allows to prove the formulas dim(g r+2 ) = r + 4, dim(R r+2 ) = 4r + 8, ∀r ≥ 0. We have the short exact sequences: an the following commutative diagrams: )) = (r + 4)(r 2 + 17r + 54)/6, ∀r ≥ 0. Then, counting the dimensions, it is easy to check that the two prolongation sequences are exact on the jet level but that the upper symbol sequence is not exact at S 3 T * ⊗ F 0 with coboundary space of imension 21 − 7 = 14, cocycle space of dimension 20 − 3 = 17 and thus cohomology space 4 ) as we check that 7 − 20 + 16 − 3 = 0. The reader may use the snake theorem to find this result directly through a chase not evident at first sight. We have then dim(R r+2 ) = 2, ∀r ≥ 0. This result is of course coherent with the fact that the involutive system with the same solutions as R 2 is R (2) 2 which is defined by ξ 33 = 0, ξ 23 = 0, ξ 22 = 0, ξ 13 − ξ 2 = 0 .

EXAMPLE 3.2:
With m = 1, n = 3, q = 2, K = Q and the commutative ring of PD operators with coefficients in K, we revisit another example of Macaulay ([7]), namely the homogeneous second order formally integrable linear system R 2 ⊂ J 2 (E) defined in operator form by P ξ ≡ ξ 33 = 0, Qξ ≡ ξ 23 − ξ 11 = 0, Rξ ≡ ξ 22 = 0 and an epimorphism R 2 → J 1 (E) → 0. As for the systems, we have dim(R 2 ) = 7, dim(R r+3 ) = 8, ∀r ≥ 0. As for the symbols, we have dim(g 2 ) = 3, dim(g 3 ) = 1, g r+4 = 0, ∀r ≥ 0. This finite type system has the very particular feature that g 3 is 2-acyclic but not 3-acyclic (thus involutive) with the short exact δ-sequence: and we have the three linearly independent equations: Collecting these results, we get the two following commutative and exact diagrams: ) with a strict inclusion because 27 < 30 and we have at least 30 − 27 = 3 generating second order CC. However, from the second diagram, we obtain dim( , a result showing that there are no new generating CC of order 3. As dim(E) = 1, we have S q T * ⊗ E ≃ S q T * and the commutative diagram of δ-sequences: Using the fact that the upper sequence is known to be exact and dim(g ′ 1 ) = 9 < 10 = dim(S 3 T * ), an easy chase proves that the lower sequence cannot be exact and thus g ′ 2 cannot be 2-acyclic. The generating CC of D 1 is thus a second order operator D 2 : F 1 → F 2 where F 2 is defined by the long exact prolongation sequence: or by the long exact symbol sequence (by chance if one refers to the previous example !): [19]). We have thus obtained the following formally exact differential sequence which is nevertheless not a Janet sequence because R 2 is FI but not involutive as g 2 is finite type with g 4 = 0: Surprisingly, the situation is even quite worst if we start with R 3 ⊂ J 3 (E) which has nevertheless a 2-acyclic symbol g 3 which is not 3-acyclic (thus involutive because n = 3 ). Indeed, we know from the second section or by repeating the previous procedure for this new third order operator D that the generating CC are described by a first order operator D 1 . However, the symbol of this operator is only 1-acyclic but not 2-acyclic (exercise). Hence, one can prove that the new CC are described by a new second order operator D 2 which is involutive ... by chance, giving rise to a Janet sequence with first order operators as follows D 3 , D 4 , D 5 ([18], p 123,124): One could also finally use the involutive system R 4 ⊂ J 4 (E) in order to construct the canonical Janet sequence and consider the first order involutive system R 5 ⊂ J 1 (R 4 ) in order to obtain the canonical Spencer sequence with C r = ∧ r T * ⊗ R 4 and dimensions (8,24,24,8): To recapitulate, this example clearly proves that the differential sequences obtained largely depend on whether we use R 2 , R 3 or R 4 but also whether we look for a sequence of Janet or Spencer type. We invite the reader to treat similarly the example ξ 33 − ξ 11 = 0, ξ 23 = 0, ξ 22 − ξ 11 = 0. EXAMPLE 3.3: In our opinion, the best striking use of acyclicity is the construction of differential sequences for the Killing and conformal Killing operators which are both defined over the ground differential field K = Q for the Minkowski metric in dimension 4 or the Euclidean metric in dimension 5. We have indeed ( [18], [20]): with E = T, F 0 = S 2 T * and, successively, the Killing, Riemann and Bianchi operators acting on the left of column vectors. The differential module counterpart over D = K[d] is the resolution of the differential Killing module M : with the same operators as before but acting now on the right of row vectors by composition. The conformal situation for n = 4 is quite unexpected with a second order Bianchi-type operator:Ê The conformal situation for n = 5 is even quite different with the conformal differential sequence: Though these results and "jumps" highly depend on acyclicity, in particular the fact that the conformal symbolĝ 2 is 2-acyclic for n = 4 but 3-acyclic for n ≥ 5, and have been confirmed by computer algebra, they are still neither known nor acknowledged ( [18], [27]).

4) APPLICATIONS.
Considering the classical Killing operator D : is the Lie derivative with respect to ξ and ω ∈ S 2 T * is a nondegenerate metric with det(ω) = 0. Accordingly, it is a lie operator with Dξ = 0, Dη = 0 ⇒ D[ξ, η] = 0 and we denote simply by Θ ⊂ T the set of solutions with [Θ, Θ] ⊂ Θ. Now, as we have explained many times, the main problem is to describe the CC of Dξ = Ω ∈ F 0 in the form D 1 Ω = 0 by introducing the so-called Riemann operator D 1 : F 0 → F 1 . We advise the reader to follow closely the next lines and to imagine why it will not be possible to repeat them for studying the conformal Killing operator. Introducing the well known Levi-Civita isomorphism j 1 (ω) = (ω, ∂ x ω) ≃ (ω, γ) by defining the Christoffel symbols is the inverse matrix of (ω ij ) and the formal Lie derivative, we get the second order system R 2 ⊂ J 2 (T ): n(n − 1)/2 and is finite type because its first prolongation is g 2 = 0. It cannot be thus involutive and we need to use one additional prolongation. Indeed, using one of the main results to be found in ( [8], [10], [11], [18], [19]), we know that, when R 1 is FI, then the CC of D are of order s + 1 where s is the number of prolongations needed in order to get a 2-acyclic symbol, that is s = 1 in the present situation, a result that should lead to CC of order 2 if R 1 were FI. However, it is known that R 2 is FI, thus involutive, if and only if ω has constant Riemannian curvature, a result first found by L.P. Eisenhart in 1926 which is only a particular example of the Vessiot structure equations discovered b E. Vessiot in 1903 ( [29]), though in a quite different setting (See [8], [11], [18] and [19] for an explicit modern proof) and should be compared to ( [6]).

A) MINKOWSKI METRIC:
We have considered this situation in many books or papers and refer the reader to our arXiv page or to the recent references ( [20], [23]). All the operators are first order between the vector bundles  25]), depends on various chases in commutative diagrams that will be exhibited later on for comparing the respective dimensions. This is not a Janet sequence because R 1 is FI but g 1 is not involutive.
The main point is a tricky formula which is not evident at all. Indeed, using the well known properties of the Lie derivative, we have the following geometric objects (not necessarily tensors ) and their linearizations (generally tensors): Then, using r as a summation index, we shall see that we have in general: R kl,ij ≡ ρ rl,ij ξ r k + ρ kr,ij ξ r l + ρ kl,rj ξ r i + ρ kl,ir ξ r j + ξ r ∂ r ρ kl,ij = 0 ρ kl,ij = ω kr ρ r l,ij ⇒ R kl,ij = ω kr R r l,ij + ρ r l,ij Ω kr ⇒ ω rs R ri,sj = R ij + ω rs ρ t i,rj Ω st ρ ij = ρ r i,rj ⇒ R ij = R r i,rj = ω rs R ri,sj We prove these results using local coordinates and the formal Lie derivative obtained while replacing j 1 (ξ) by ξ 1 (See [8], [11], [18], [19] for details). First of all, from the tensorial property of the Riemann tensor and the Killing equations Ω us = ω ku ξ k s + ω ks ξ k u + ξ r ∂ r ω us , we have: We have for example, in this particular case: The only use of R 01,01 is allowing to get ξ 1 = 0 in the previous list, but we have also exactly: R 31,32 = − m 2r 3 Ω 12 = R 12 + ω 11 ρ 2 1,12 Ω 12 ⇒ R 12 = 0 The use of R 01,02 or R 13,23 is allowing to get ξ 1 2 = 0 in the previous list with: and thus also exactly: It follows that the 4 central second order CC of the list successively amounts to R 12 = 0, R 13 = 0, R 02 = 0, R 03 = 0, a result breaking the intrinsic/coordinate-free interpretation of the 10 Einstein equations and the situation is even worst for the other components of the Ricci tensor. Indeed, R 01 and R 23 only depend on the vanishing of R 02,12 , R 03,13 and R 02,03 , R 12,13 among the bottom CC of the list, while the diagonal terms R 00 , R 11 , R 22 , R 33 only depend, as we just saw, on the 6 non zero components of the Riemann tensor. We have thus obtained the totally unusual partition 10 = 4 + 4 + 2 along the successive blocks of the former list with: Finally, we notice that R 01,23 = 0, R 02,31 = 0 ⇒ R 03,12 = 0 from the identity in ∧ 3 T * ⊗ T * : R 01,23 + R 02,31 + R 03,12 = 0 and there is no way to have two identical indices in the first jets appearing through the (formal) Lie derivative just described. As for the third order CC, setting ξ 1 1 = A ′ 2A ξ 1 ∈ j 2 (Ω), we have at least the first prolongations of the previous second order CC to which we have to add the three new generating ones: provided by the Spencer operator, leading to the crossed terms d i ξ 1 j − d j ξ 1 i = 0 for i, j = 1, 2, 3 because the Spencer operator is not FI.
(Ω), we have to look for the CC of the system R Among the CC we must have d 2 V 3 − d 3 V 2 = 0 which is among the differential consequences of the Spencer operator as we saw but we must also have d 2 W 3 − d 3 W 2 = 0 and both seem to be new third order CC, together with the CC obtained by eliminating ξ 2 and ξ 3 from the three last equations after two prolongations as in ( [23]): However, things are not so simple, even if we have in mind that (V, W ) ∈ j 2 (Ω), because the central sign in the previous formula is opposite to the sign found after one prolongation in the formula: ξ 1 33 + sin(θ)cos(θ)ξ 1 2 − sin 2 (θ)ξ 1 22 = 0 and it is at this moment that we need introduce new differential geometric methods !.
With more details, we number the 20 linearly independent Bianchi identities as follows: We successively study a few situations without any, with one or with two vanishing linearized Riemann components, taking into account that the four Einstein equations are described by: 12 , 17 , 20 for the index 0, 4 , 9 , 19 for the index 1, 1 , 10 , 15 for the index 2, 2 , 6 , 11 for the index 3.

REMARK 4.B.1: Though a few conditions like
21 look like to be third order CC for Ω, we have thus proved that they come indeed from the first prolongations of the second order CC. The same comment is also valid for a few other striking CC. Using previous results, we have successively 6 other relations: 18 . Now, we notice that, among these 24 B, only 4 of them do contain three components R kl,ij that are not vanishing for the S-metric, namely 1 , 2 , 19 and 20 . They are providing the terms d r E rr for r = 0, 1, 2, 3 in the divergence type condition for the linearized Einstein equations implied by the linearized Bianchi identities over the Schwarzschild metric. Accordingly, it does not seem possible to obtain any other third order CC apart from these 4 divergence conditions. It remains to apply these results to the successive prolongations of the Killing equations, as we know from the intrinsic study achieved in ( [20], [23]) that we have the successive Lie algebroids: with respective dimensions 4 = 4 < 5 < 10 = 10 < 20 and R 1 does not depend any longer on the S-parameter m. The challenge will be to prove that ... the only knowledge of these numbers is sufficient !.
In an equivalent way as g 2 = 0 ⇒ g r+2 = 0, ∀r ≥ 0, we obtain successively: and shall use these results from now on. First of all, using the introductory diagram when q = 1, r =, we may apply the Spencer δ-map to the symbol top ro in order to obtain the left: Uing the Spencer δ-cohomology H r (g 1 ) = Z r (g 1 )/B r (g 1 ) at · · · → ∧ r T * ⊗ g 1 → . . . , we obtain: Proof: As there cannot be any CC of order one and thus Q 1 = 0, we have the long exact connecting sequence 0 → R 3 → R 2 → h 2 → Q 2 → 0 and counting the dimensions with F 0 = S 2 T * , we have: This result is confirmed by a circular chase proving that the left bottom δ-map is an epimorphism and a snake chase in the last diagram providing the short exact sequence: Indeed, as det(ω) = 0 we may use the metric for providing an isoorphism T ≃ T * : (ξ r ) → (ξ i = ω ri ξ r ) in such a way that g 1 ≃ ∧ 2 T * is defined by ξ i,j + ξ j,i = 0 for both the M, S and K metrics. However, introducing the conformal Killing system of infinitesimal Lie equations with symbolĝ 1 defined by the (n(n + 1)/2) − 1 linear equations ω rj ξ r i + ω ir ξ r j − 2 n ω ij ξ r r = 0 that do not depend on any conformal factor, we have theÒ fundamental diagram II ( [9], [11]): showing that we have the splitting sequence 0 → S 2 T * → H 2 (g 1 ) → H 2 (ĝ 1 ) → 0 providing a totally unusual interpretation of the successive Ricci, Riemann and Weyl tensors and the corresponding splitting. However, it must be noticed that the W eyl-type operator is of order 3 when n = 3 because n 2 (n 2 − 1)/12 − n(n + 1)/2 = n(n + 1)(n + 2)(n − 3)/12 but of order 2 for n ≥ 4 ( [18], [27]). Similar results could be obtained for the Bianchi-type operator as we shall see. ✷ Using now the same procedure for the introductory diagram with r = 2, we get the diagram: Using a snake chase and Theorem 3.2.3, we obtain the short exact sequence: A chase around the upper south-east arrow on the right is leading to the following corollary where g ′ 2 ⊂ S 2 T * ⊗ F 0 is the symbol of the system R ′ 2 ⊂ J 2 (F 0 ) which is the image of J 3 (T ) and Q ′ 1 is the cokernel of the central bottom map: allowing to use the Bianchi indentities as B ∈ F 2 ≃ H 3 (g 1 ) and we have dim(Q ′ 1 ) ≤ dim(H 3 (g 1 ).
Proof: Using the notations of the introductory diagram and the fact that Q 1 = 0, we have the following two commutative and exact diagrams obtained by choosing F 1 = Q 2 for the first, then F 1 = Q 3 for the second and so on, in a systematic manner as in the motivating examples: First, we have the short exact sequence 0 → R 2 → J 2 (T ) → J 1 (F 0 ) → 0 with 10 − 60 + 50 = 0 and get R ′ 0 = F 0 , R ′ 1 = J 1 (F 0 ) and dim(ρ 1 (R ′ 0 )) − dim(R ′ 1 ) = 0, that is no CC of order 1. Now, using the long exact sequence: and there are 15 second order CC. ✷ Then, with dim(Q ′ 1 ) = x, we obtain by counting the dimensions: and thus x = 4 ⇒ y = 3 if we only take into account the 4 divergence condition of the Einstein equations. The situation will be worst for the Kerr metric with y = 6. After one prolongation, we get: From this second diagram we obtain the commutative and exact diagram: Indeed, setting again dim(Q ′ 1 ) = x, we obtain now similarly: We find exactly dim(F 2 ) = 170 like in ( [20], p 1996) and the condition y = 0 just means that the CC of order 4 are generated by the CC of order 3. With one more prolongation, applying again the δ-map to the top symbol sequence, we get the following commutative diagram: where the right exact vertical column is 0 → 224 → 504 → 360 → 80 → 0. It just remains to replace in the two upper right epimorphisms h 5 by T * ⊗ Q 4 and h 4 by Q 4 along with the following commutative diagram where we have chosen F 1 = Q 4 : in order to obtain the long exaxt sequence 0 → S 6 T * ⊗ T → S 5 T * ⊗ F 0 → T * ⊗ Q 4 . Finally, chasing in the following commutative and exact introductory diagram: is involutive with dim(R ′ 5 ) = 840−4 = 836 and symbol g ′ 5 ≃ S 6 T * ⊗T . Unhappily, the reader will check at once that a similar procedure cannot be applied in order to prove that R ′ 4 = ρ 1 (R ′ 3 ). Indeed, if we still have a monomorphism 0 → h 4 → Q 4 we do not have a monomorphism h 3 → Q 3 because now this map has a kernel of dimension equal to dim(R 3 /R 3 ) = 5 − 4 = 1 according to the corresponding long exact connecting sequence.

IT IS THUS NOT POSSIBLE TO PROVE THAT THERE ARE ONLY SECOND AND THIRD ORDER GENERATING CC IN A SIMPLE INTRINSIC WAY.
However, like in the first motivating example in which we should be waiting for third order CC but a direct computation was proving that only second order ones could be used, we have: Proof: With F 0 = S 2 T * and F 1 = Q 3 while applying the Spencer operator, we obtain the following commutative diagram in which the two central vertical columns are locally exact ( [8], [11]): Chasing in this diagram by using the Snake lemma of the second section, we discover that the local exactness at F 0 of the top row is equivalent to the local exactness at T * ⊗ R 3 of the left column. Now, we have the commutative diagram: The top row is known to be locally exact as it is isomorphic to a part of the Poincaré sequence according to the commutative diagram with R 4 ≃ R 5 ≃ R 6 : The bottom row is purely algebraic as it is induced by the exact sequence obtained by applying the Spencer operator to the long exact connecting sequence and chasing along the south west diagonal: Changing the confusing notations used in ( [20]), we prove that the bottom Spencer operator is injective. Indeed, we have the following representative parametric jets for the various Lie equations: 2 ) = 5 ⇒ {ξ 1 , ξ 1 2 , ξ 1 3 , ξ 0 2 , ξ 0 3 } We also recall the definition of the Spencer operator d : T * ⊗ J q+1 (T ) → ∧ 2 T * ⊗ J q (T ): Accordingly, we may choose local coordinates (ξ 1 0,i ) for a representative and a representative of the image by d is for example (ξ 1 ,0i = ξ 1 0,i − ξ 1 i,0 ). Now, as dim(R 3 /R (1) the 6 × 5 = 30 local coordinates (ξ 1 ,ij , ξ 1 2,ij , ξ 1 3,ij , ξ 0 2,ij , ξ 0 3,ij ) in order to describe ∧ 2 T * ⊗ (R 2 /R (1) 2 ). In the kernel of d, we have in particular ξ 1 ,0i = ξ 1 0,i − ξ 1 i,0 = 0 ⇒ ξ 1 0,i = ξ 1 i,0 = 0, ∀i = 1, 2, 3 because ξ 1 1 = m 2Ar 2 ξ 1 in R 1 but also ξ 0 ,01 = ξ 0 0,1 − ξ 0 1,0 = 0 ⇒ ξ 0 1,0 = 0 ⇒ ξ 1 0,0 = 0 because {ξ 0 } is among the parametric jets of R 3 and thus ξ 1 0,i = 0, ∀i = 0, 1, 2, 3. The bottom Spencer operator is thus injective and the bottom sequence is thus exact. A circular chase ends the proof: If b ∈ T * ⊗ R 3 is killed by d, then its projection c ∈ T * ⊗ (R 3 /R (1) 3 ) is also killed by d and is such that c = 0. Accordingly, ∃a ∈ T * ⊗ R 4 with image b under the monomorphism T * ⊗ R 4 → T * ⊗ R 3 and such that da = 0. We may thus find e ∈ R 5 and f ∈ R 4 because R 5 ≃ R 4 with a = de ⇒ b = df . ✷ Like in the second motivating example, the sequence constructed in the previous theorem may have "jumps " in the order of the successive operators and we have therefore (Compare to [1]): Proof: Recapitulating the results so far obtained, we have successively R ′ r+1 ⊆ ρ 1 (R ′ r ) with: x ≥ 4 because of the divergence CC condition for Einstein equations implied by the Bianchi identities. It also follows that: ∀r ≥ 0 and we have the basic commutative and exact " defining diagram " of the system R 2 ⊂ J 2 (T ): allowing to obtain the central vertical short exact sequence 0 → g ′ 1 → R ′ 1 → F 9 → 0. Now, it is known that a symbol g q of finite type is involutive if an only if it is vanishing ( [8], [11], [12]). Using a similar proof, let us consider the commutative diagram of δ-sequences: Using the fact that the upper sequence is known to be exact as a δ-sequence and that we have dim(S 4 T * ⊗ T ) = 140 < 141 = dim(g ′ 3 ), an easy chase proves that the lower sequence cannot be exact and thus g ′ 3 cannot be involutive after counting the dimensions. The corollary follows from the fact that g ′ 4 = ρ 1 (g ′ 3 ) ≃ S 5 T * ⊗ T is indeed 3-acyclic one step ahead by chasing and even involutive. Finally, with vector bundles A, B such that dim(A) = 1, dim(B) = 5, we have the commutative diagram of δ-sequences in which we recall that g ′ 1 ≃ T * ⊗ F 0 : Taking into account that the top row is exact and proceeding as in the last theorem with similar local coordinates, we get: ξ 1 ,123 = ξ 1 1,23 + ξ 1 2,31 + ξ 1 3,12 = 0 + 0 + 0 = 0 always. ξ 1 ,012 = ξ 1 0,12 + ξ 1 1,20 + ξ 1 2,01 = ξ 1 0,12 + 0 + 0 = 0 ⇒ ξ 1 0,12 = 0 ⇒ ξ 1 0,ij = 0, ∀i, j = 1, 2, 3 ξ 0 ,10i = ξ 0 1,0i + ξ 0 0,i1 + ξ 0 i,01 = ξ 0 1,0i + 0 + 0 = 0 ⇒ ξ 0 1,0i = 0 ⇒ ξ 1 0,0i = 0, ∀i = 2, 3. We are thus only left with ξ 1 0,01 that may not vanish though ξ 1 0,01 + ξ 1 0,10 + ξ 1 1,00 = 0 in any case and the bottom map δ is not injective. Let us prove that g ′ 3 is not 2-acyclic because the central δ-sequence cannot be exact at ∧ 2 T * ⊗ g ′ 3 . Indeed, if it were, let c ∈ ∧ 2 T * ⊗ A be killed by δ. Then, we may lift c to b ∈ ∧ 2 T * ⊗ g ′ 3 such that δb = f ∈ ∧ 3 T * ⊗ S 3 T * ⊗ T and obtain by commutativity δf = 0 because the last vertial downarrow on the right is an isomorphism, thus a monomorphism. As the upper row is an exact sequence, we may thus find a ∈ ∧ 2 T * ⊗ S 4 T * ⊗ T such that f = δa. Chasing circularly, it follows from the exactness assumtion at ∧ 2 T * ⊗ g ′ 3 that we can find e ∈ T * ⊗ g ′ 4 ≃ T * ⊗ S 5 T * ⊗ T such that b = a + δe = a ′ ∈ ∧ 2 T * ⊗ S 4 T * ⊗ T . It should follow that c = 0 and a contradiction, that is g ′ 3 cannot be 2-acyclic.
As we know from ( [8], [11], [12]) that the order of the generating CC for D 1 is equal to s + 1 if one needs s prolongations in such a way that ρ s (g ′ 3 ) = g ′ 3+s becomes 2-acyclic. As we already know that g ′ 4 = ρ 1 (g ′ 3 ) ≃ T * ⊗ S 5 T * ⊗ T is involutive, we get s = 1 and the generating CC D 2 of D 1 are of order 2. We have just a " jump " in the order and, for the details, refer the reader to the quite delicate Example 3.14 of ( [18], p 119-125) in which it is already difficult to discover how many new second order CC should be introduced though the initial system is trivially FI with coefficients in Q. Such a result could not even be imagined while using the methods of ([1]- [4]). ✷ There are "natural" reason for which we do not believe that these results could be useful in physics. Indeed, considering like in the previous reference the long exact sequence of jet bundles allowing to define F 2 when F 1 = Q 3 , namely:

C) KERR METRIC:
We now write the Kerr metric in Boyer-Lindquist coordinates: where we have set ∆ = r 2 − mr + a 2 , ρ 2 = r 2 + a 2 cos 2 (θ) as usual and we check that we recover the Schwarschild metric when a = 0. We notice that t or φ do not appear in the coefficients of the metric. We shall change the coordinate system in order to confirm theses results by using computer algebra and the idea is to use the so-called " rational polynomial " coefficients as follows: We obtain over the differential field K = Q(a, m)(t, r, c, φ) = Q(a, m)(x): with now ∆ = (x 1 ) 2 − mx 1 + a 2 = r 2 − mr + a 2 and ρ 2 = (x 1 ) 2 + a 2 (x 2 ) 2 = r 2 + a 2 c 2 . For a later use, it is also possible to set ω 33 = −(1 − c 2 )((r 2 + a 2 ) 2 − a 2 ((1 − c 2 )(a 2 − mr + r 2 ))/(r 2 + a 2 c 2 ) and we have det(ω) = −(r 2 + a 2 c 2 ) 2 . Framing the leading derivatives, we obtain: 0 + ω 00 ξ 0 1 + ω 03 ξ 3 1 = 0 Ω 00 ≡ 2(ω 00 ξ 0 0 + ω 03 ξ 3 0 ) + ξ∂ω 00 = 0 Now, we know that if R q ⊂ J q (T ) is a system of infinitesimal Lie equations, then we have the algebroid bracket and its link with the prolongation/projection (PP) procedure ( [8], [11]- [13]): 1 , R 1 ) = 20 − 16 = 4 because we have obtained a total of 6 new different first order equations. Using the first general diagram of the Introduction, we discover that the operator defining R 1 has 10 + 4 = 14 CC of order 2, a result obtained totally independently of any specific GR technical object like the Teukolski scalars or the Killing-Yano tensors introduced in ([1]- [4], [6]). Like in the case of the S metric, two prolongations allow to obtain 6 additional equations (instead of 5) that we set on the left side in the following list obtained mod( 2 (Ω): We have on sections (care) the 16 (linear) equations mod(j 2 (Ω)) of R 1 as follows ( [23]): The coefficients of the linear equations lin involved depend on the Riemann tensor as in ( [23]). Accordingly, we may choose only the 2 parametric jets (ξ 1 0 , ξ 2 0 ) among (ξ 1 0 , ξ 1 3 , ξ 2 0 , ξ 2 3 ) to which we must add (ξ 0 , ξ 3 ) in any case as they are not appearing in the Killing equations. The system is not involutive because its symbol is finite type but non-zero.
Using one more prolongation, all the sections (care again) vanish but ξ 0 and ξ 3 , a result leading to dim(R (3) 1 ) = 2 in a coherent way with the only nonzero Killing vectors {∂ t , ∂ φ }. We have indeed: Taking therefore into account that the metric only depends on (x 1 = r, x 2 = cos(θ)) we obtain after three prolongations the first order system: Surprisingly and contrary to the situation found for the S metric, we have now an involutive first order system with only solutions (ξ 0 = cst, ξ 1 = 0, ξ 2 = 0, ξ 3 = cst) and notice that R 1 does not depend any longer on the parameters (m, a) ∈ K. The difficulty is to know what second members must be used along the procedure met for all the motivating examples. In particular, we have again identities to zero like d 0 ξ 1 − ξ 1 0 = 0, d 0 ξ 2 − ξ 2 0 = 0 and thus at least 6 third order CC coming from the 6 following components of the Spencer operator, namely: = 0 a result that cannot be even imagined from ([1]- [4]). Of course, proceeding like in the motivating examples, we must substitute in the right members the values obtained from j 2 (Ω) and set for example ξ 1 1 = − 1 2ω11 ξ∂ω 11 while replacing ξ 1 and ξ 2 by the corresponding linear combinations of the Riemann tensor already obtained for the right members of the two zero order equations.
We have the fundamental diagram I no longer depending on (m, a) with fiber dimensions: providing the Euler-Poincaré characteristic 4 − 18 + 32 − 28 + 12 − 2 = 0. However, the only intrinsic concepts associated with a differential sequence are the " extension modules " that only depend on the Kerr differential module but not on the differential sequence and it follows that ( [16]):

THE ONLY IMPORTANT CONCEPT IS THE GROUP INVOLVED, NOT THE SEQUENCE.
In an equivalent way as g 2 = 0 ⇒ g r+2 = 0, ∀r ≥ 0, we obtain successively: and shall use these results from now on. According to a cut of the preliminary diagram with now m = n = 4, q = 1, K = Q(m, a), we obtain the following commutative and exact diagrams: Denoting as before by y the number of additional CC of strict order 3 and by x the number dim(h ′ 1 ) = dim(Q ′ 1 ), we discover from the above diagram that the sum of the number of second order CC (that is 14) and the number of differentially independent third order CC obtained by one prolongation of these second order CC is equal to 70 − x. As now dim(Q 3 ) = 72, we obtain therefore 72 − y = 70 − x and thus y = x + 2. However, as x ≥ 4 because of the 4 divergence conditions implied on the Einstein tensor by the 20 Bianchi identities, we must have y ≥ 6. As we have already found effectively only 6 CC of order 3, we must have indeed x = 4 effectively and, in any case, we cannot have y = 4 as claimed in ( [1], [3]).
From the short exact sequence: we obtain the commutative and exact diagrams: As a byproduct, we have the commutative and exact diagrams: and to the formula y = x + 2. We obtain therefore the following most useful diagram with symbolic notations: , a result leading to the long exact connecting sequence of vector bundles: in agreement with the main theorem of section 2. We have the following dimensions: Prolonging once while taking into account that R 5 ≃ R 4 with common dimension 2, namely the dimension of the Kerr algebra generated by {∂ t , ∂ φ }, we obtain the following commutative and exact diagram in which Q 2 and Q 3 are replaced by Q 3 and Q 4 : showing that g ′ 4 ≃ S 5 T * ⊗ T with dim(R ′ 4 ) = 504 − 2 = 502 and dim(Q 4 ) = 700 − 502 = 198. It follows that R ′ 4 = ρ 1 (R ′ 3 ) is an involutive fourth order system allowing to construct a formally exact Janet sequence following the Killing operator as in ( [20]), namely (exercise !!): Of course, such a sequence is quite far from being minimum. However, as the Killing operator for the Kerr metric is not formally integrable as we saw, the corresponding free resolution of the Kerr differential module, namely: is not strictly exact though we have indeed: rk D (M ) = 4 − 10 + 198 − 568 + 652 − 348 + 72 = 0 As the maximum size of the matrices involved is dim(J 4 (198))×dim(J 3 (568)), that is 13860×19880, we hope to have convinced the reader that there is no hope for using computer algebra. As R ′ 3 ⊂ ρ 1 (R ′ 2 ) with a strict inclusion, the only posibility to escape from te above difficulty is to use only R ′ 3 and third order CC. However, as we have the strict inclusion S 4 T * ⊗ T ⊂ g ′ 3 with a strict inclusion because 140 < 142. As for the S metric, we have the crucial theorem: Proof: First of all, as R ′ 3 is strictly contained into ρ 1 (R ′ 2 ), we have at least one third order generating CC but we already know that we have the six (d i ξ 1 − ξ 1 i = 0, d i ξ 2 − ξ 2 i = 0) for i = 1, 2, 3.

5) CONCLUSION
To end this paper with a rather personal story, let me come back 60 years ago when I was preparing the competition for the french "Grandes Ecoles" at the State College Louis le Grand in Paris which is famous for one of his former student Evariste Galois. To give a few statistics, let us say that, for the one I had in mind, 30 000 students were trying, 3000 were selected after the written exam and 300 were only elected after oral exam !. This college was known to have the maximum number of success in France and the teachers were carefully selected for that purpose, in particular in the best class room where I was. Once, this teacher was writing on the board the text of the problem we had to solve for the next day about what is now called "Desargues theorem". Roughly, if you consider in a plane two triangles (ABC), (A ′ B ′ C ′ ) that are not flat and such that the 3 straight lines AA ′ , BB ′ , CC ′ have a common origin O (Center of perspective), then the intersection P of BC and B ′ C ′ , Q of AC and A ′ C ′ , R of AB and A ′ B ′ are on a straight line (Axis of perspective). Though I knew nothing about this result at that time, I suddenly "saw" the figure as a volume in space and shouted "P, Q, R are on a straight line", even before the teacher had been asking the question in front of the asthonished students. Surprisingly, and I will never forget, the teacher said "Pommaret, this is true but how did you find it". When I said "Well, Sir, I have seen in space that the common line is the intersection of the two planes containing the triangles" (the reader may draw the picture for fun), his only comment has been "Better don't do that on the day of the competition". I replied "Sir, a result is important but the way you find it may even be more important". As a byproduct he never asked me any question during the full academic year and became a "private ennemy" in my scholar life during 10 years till he retired.
In a similar way, we point out the fact that during a visit for lecturing at the Albert Einstein Institute (AEI, Berlin/Postdam) in october 23-27, 2017 ([21], arXiv:1802.02430), we discovered that the members of the inviting research team were not interested about the new tools we developed in the many books or papers already quoted, in particular the link existing between the Spencer operator and the bracket of Lie algebroids. We also claim that the few references they quote for defining involutive systems are not the best ones as it happens that we have been regularly lecturing in Aachen during more than fifteen years and we know that the authors involved are only using Janet, Gröbner or Pommaret bases for explicit computations but are unable to deal with acyclicity in general. The situation we met previously in the case of the Lie pseudogroup of conformal transformations is a good example. As a byproduct, it became a personal challenge to clarify the CC for the Killing operators over the Schwarzschild and Kerr metrics without using any of their tedious computations. The surprise is that, if we found again the 15 second order CC for the S metric and the 14 second order CC for the K metric, we also found explicitly 3 third order CC for the S metric and 6 third order CC for the K metric. All the formulas can be written within less than one line provided we use these new methods from differential homological algebra that have never been introduced in GR up to now, mainly because they prove that Einstein equations cannot be parametrized by a potential like Maxwell equations ... but this is surely another story !.