The Diffiety Institute Preprint Series

Preprint DIPS–2/2007 March 26, 2007

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Iterated Differential Forms VI: Differential Equations

by

A. M. Vinogradov and L. Vitagliano

DIPS-1/2007

                                                 arXiv:math.DG/0703761

Iterated Differential Forms VI: Differential Equations

A. M. VINOGRADOV AND L. VITAGLIANO

Abstract. We describe the first term of the Λ_{k−1}C–spectral sequence [1] of the diffiety (E, C), E being the infinite prolongation of an ℓ–normal system of partial differential equations, and C the Cartan distribution on it.

According to [1] secondary k–times iterated differential forms and, in particular, secondary k–th degree covariant tensors on a generic diffiety (O, C) are elements of the first term of the associated with it Λ_{k−1}C–spectral sequence. In this note we report results of computation of this term for diffieties that are infinite prolongations of normal systems of partial differential equations. To simplify the exposition the equations we deal with are assumed to be imposed on sections of a vector bundle π : E ⟶ M, i.e., the diffieties in consideration are of the form (E, C) with E ⊂ J^{∞}(π) being the infinite prolongation of a normal system of partial differential equations Y ⊂ J^{s}(π) and C being the Cartan distribution on E.

1. Preliminaries

Here we collect all necessary preliminaries concerning geometry of infinitely prolonged PDEs, iterated differential forms (shortly, IDFs) and secondary IDFs by following [2],

[3] and [4], respectively. Accordingly, we use the notation of these works. As in [4] we sometimes shorten the notation by using, for instance, J^{k} instead of J^{k}(π) for the k–th jet manifold of a fiber bundle π, etc. As usually x^{1}, …, x^{n} denote a local chart of M. Let ξ : V ⟶ M be a vector bundle, r = dim ξ. Denote by v^{1}, …, v^{r} a system of local linear fiber coordinates extending x^{1}, …, x^{n}. Any s–th order non–linear differential operator which sends sections of π to sections of ξ may be interpreted as a morphism of bundles F : J^{s} ⟶ V. Locally F is represented in the form

v^{a} = F^{a}(…, x^{μ}, …, u_{σ}^{j}, …), F^{a} ∈ C^{∞}(J^{k}), a = 1, …, r, |σ| ≤ s

and defines the differential equation E_{F} = {θ ∈ J^{s} | F (θ) = 0} ⊂ J^{s}. Further on E_{F} is assumed to be regular (see [2]). Denote by E = E_{F}^{∞} the infinite prolongation of E_{F} and by i_{E} : E ↪ J^{∞} the corresponding natural embedding. Put P = F ⊗_{C^{∞}(M)} Γ(ξ) and recall that there is a unique morphism of diffieties F̂ : J^{∞} ⟶ J^{∞}(ξ) such that ξ_{∞,0} ∘ F̂ = F ∘ π_{∞,k}. Let Λ_{k−1}(ξ) be the algebra of (k − 1)–times IDFs on J^{∞}(ξ) and C_{∘}Λ_{k−1}(ξ) ⊂ Λ_{k−1}(ξ) be the C^{∞}(V)–subalgebra generated by elements in the form d_{K}^{v}g, g ∈ C^{∞}(V) ⊂ C^{∞}(J^{∞}(ξ)), K ⊂ {1, …, k − 1}. The pull–back homomorphism F̂^{∗} : Λ_{k−1}(ξ) ⟶ Λ_{k−1} is locally given by

2000 Mathematics Subject Classification. 58A10, 58A20, 58J10.

Key words and phrases. Differential Forms, Jet Spaces, C–Spectral Sequence, Secondary Calculus.

F̂^{∗}(d_{L}x^{μ}) = d_{L}x^{μ}, F̂^{∗}(d_{K}^{v}v_{σ}^{a}) = (d_{K}^{v} ∘ D_{σ})(F^{a}),

with L, K ⊂ {1, …, k − 1}, μ = 1, …, n = dim M, a = 1, …, r and σ being a multi– index. Obviously, F̂^{∗} sends C_{∘}Λ_{k−1}(ξ) to C_{⋆}Λ_{k−1} (see [4]) and this way C_{⋆}Λ_{k−1} acquires a structure of a graded C_{∘}Λ_{k−1}(ξ)–module. By this reason the notion of a C_{⋆}Λ_{k−1}– valued derivation of the algebra C_{∘}Λ_{k−1}(ξ) is well–defined. By noticing that C^{∞}(M) ⊂ C_{∘}Λ_{k−1}(ξ) we call a derivation Y of C_{∘}Λ_{k−1}(ξ) vertical if Y (C^{∞}(M)) = 0. Denote by Λ_{k−1}P the C_{⋆}Λ_{k−1}–module of vertical C_{⋆}Λ_{k−1}–valued derivations of C_{∘}Λ_{k−1}(ξ) and note that Λ_{0}P ≃ P. Let W_{a}^{K} ∈ D(Λ_{k−1}(ξ), Λ_{k−1}(ξ)), a = 1, …, r, K ⊂ {1, …, k − 1}, be the derivations uniquely defined by relations

W_{a}^{K}(d_{L}^{v}v_{σ}^{b}) = \{ \begin{matrix}1 & if a = b, σ = ∅ and K = L \\ 0 & otherwise \end{matrix} ,
W_{a}^{K}(d_{L}x^{μ}) = 0,

with b = 1, …, r, μ = 1, …, n, L ⊂ {1, …, k − 1}. Then W_{a}^{K}(C_{∘}Λ_{k−1}(ξ)) ⊂ C_{∘}Λ_{k−1}(ξ), a = 1, …, r, K ⊂ {1, …, k − 1}. Observe that the module Λ_{k−1}P is locally generated by elements (W_{a}^{K})_{F} = F̂^{∗} ∘W_{a}^{K} and is horizontal (see [4]). Indeed, the C^{∞}(M)–module W_{k−1}(ξ) ⊂ Λ_{k−1}P composed of derivations that are locally of the form h_{K}^{a}(W_{a}^{K})_{F}, h_{K}^{a} ∈ C^{∞}(M), a = 1, …, r, K ⊂ {1, …, k − 1} is well defined and

Λ_{k−1}P ≃ C_{⋆}Λ_{k−1} ⊗_{C^{∞}(M)} W_{k−1}(ξ).

Let ℒ ∈ D(V) be the standard Liouville vector field on V, i.e., ℒ = v^{a} \frac{∂}{∂v^{a}}. It naturally extends to a vertical derivation ℒ ^{v} of C_{∘}Λ_{k−1}(ξ), which locally looks as

ℒ ^{v} =^{def} d_{K}^{v}v^{a} W_{a}^{K}.

If Λ_{k−1}F =^{def} F̂^{∗} ∘ ℒ ^{v} ∈ Λ_{k−1}P, then Λ_{k−1}F has the following local expression

Λ_{k−1}F = d_{K}^{v}F^{a} (W_{a}^{K})_{F} .

Put Λ_{k−1}μ_{E} =^{def} ker i_{E}^{∗} ⊂ Λ_{k−1}, i_{E}^{∗} : Λ_{k−1} ⟶ Λ_{k−1}(E), C_{⋆}Λ_{k−1}μ_{E} =^{def} Λ_{k−1}μ_{E} ∩ C_{⋆}Λ_{k−1}

and C_{⋆}Λ_{k−1}(E) =^{def} i_{E}^{∗}(C_{⋆}Λ_{k−1}) ≃ C_{⋆}Λ_{k−1}/C_{⋆}Λ_{k−1}μ_{E}. The restriction of a C_{⋆}Λ_{k−1}–module Q to E is defined to be Q|_{E} =^{def} Q/C_{⋆}Λ_{k−1}μ_{E} · Q and R_{E}^{Q} : Q ∋ q ⟼ [q] ∈ Q|_{E} is called the restriction homomorphism.

ITERATED DIFFERENTIAL FORMS V: C–SPECTRAL SEQUENCE ON J^{∞}(π) 3

2. Λ_{k−1}C–modules

Total derivatives D_{μ}, μ = 1, …, n, restricted to E extend canonically to derivations of the algebra C_{⋆}Λ_{k−1}(E). These extensions will be still denoted by D_{μ}’s. In what follows every C_{⋆}Λ_{k−1}(E)–module is assumed to be graded, locally free and of finite rank. Let Q_{1}, Q_{2} be such ones.

Definition 1. A linear differential operator Δ : Q_{1} ⟶ Q_{2} is called C–differential if for any local basis {e_{1}, …, e_{t}} of Q_{1}, Δ is locally of the form Δ(p) = (−1)^{|p|·|α|}Δ_{α}^{σ}D_{σ}(p^{α}) with Δ_{α}^{σ} ∈ Q_{2}, p = p^{α}e_{α} ∈ Q_{1}, |α| = |e_{α}| and p^{α} ∈ C_{⋆}Λ_{k−1}(E), α = 1, …, t.

The left multiplication transforms the totality of all C–differential operators Δ : Q_{1} ⟶ Q_{2} into a C_{⋆}Λ_{k−1}(E)–module denoted by CDiff(Q_{1}, Q_{2}). Similarly, the C_{⋆}Λ_{k−1}(E)–module of multiplicity l, Q_{2}–valued, multi–C–differential operators on Q_{1}, denoted by CDiff_{(l)}(Q_{1}; Q_{2}), is defined. For l = 0 we put CDiff_{(0)}(Q_{1}; Q_{2}) = Q_{2}. The sub–module of CDiff_{(l)}(Q_{1},; Q_{2}), composed of skew–symmetric operators, is denoted by CDiff_{(l)}^{alt}(Q_{1}; Q_{2}) and

alt_{l} : CDiff_{(l)}(Q_{1}, Q_{2}) ⟶ CDiff_{(l)}^{alt}(Q_{1}, Q_{2}).

stands for the alternation operator. Let P_{1}, P_{2} be C_{⋆}Λ_{k−1}–modules and □ : P_{1} ⟶ P_{2} a C–differential operator.

Proposition 2. □ restricts to E, i.e., there exists a unique □|_{E} ∈ CDiff(P_{1}|_{E}, P_{2}|_{E}) such that R_{E}^{P_{2}} ∘ □ = □|_{E} ∘ R_{E}^{P_{1}}.

The module CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)) becomes a unitary ring with respect to the composition operation. So, the ring C_{⋆}Λ_{k−1}(E) may be viewed as a subring of CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)) containing the identity operator. Moreover, if X ∈ CD(Λ_{k−1}(E)), then X(C_{⋆}Λ_{k−1}(E)) ⊂ C_{⋆}Λ_{k−1}(E). So, the restriction X|_{C_{⋆}Λ_{k−1}(E)} ∈ CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)) is well-defined. Put

CD(C_{⋆}Λ_{k−1}(E)) =^{def} {X|_{C_{⋆}Λ_{k−1}(E)} | X ∈ CD(Λ_{k−1}(E))} ⊂ CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)).

As a ring CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)) is generated by its submodules C_{⋆}Λ_{k−1}(E) and CD(C_{⋆}Λ_{k−1}(E)).

Definition 3. A couple composed of a left graded C_{⋆}Λ_{k−1}(E)–module Q and a homomorphism

CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)) ∋ Δ ⟼ Δ^{Q} ∈ CDiff(Q, Q)

of unitary rings is called a Λ_{k−1}C–module (see, e.g., [5, 6]).

Example 4. C_{⋆}Λ_{k}^{p}(E) = C_{⋆}Λ_{k}(E) ∩ Λ_{k}^{p}(E) ⊂ C^{p}Λ_{k}^{p}(E) is a Λ_{k−1}C–module for any p. Indeed, for Y = X|_{C_{⋆}Λ_{k−1}(E)} ∈ CD(C_{⋆}Λ_{k−1}(E)), X ∈ CD(Λ_{k−1}(E)), define the operator Y^{Q} by putting

Y^{Q}(Ω) = L_{X}^{{k}}(Ω) ∈ C_{⋆}Λ_{k}^{p}(E), Ω ∈ C^{p}Λ_{k}^{p}(E).

Then, in particular, (ωY)^{Q} = ωY^{Q}, ω ∈ C_{⋆}Λ_{k−1}(E), Y ∈ CD(C_{⋆}Λ_{k−1}(E)).

Note that the tensor product of Λ_{k−1}C–modules is a Λ_{k−1}C–module in a natural way. Let Q be a Λ_{k−1}C–module and P_{1}, P_{2} be C_{⋆}Λ_{k−1}(E)–modules. For Δ ∈ CDiff(P_{1}, P_{2}), p ∈ P_{1} and ϕ ∈ P_{2}^{∗} = Hom_{C_{⋆}Λ_{k−1}(E)}(P_{2}, C_{⋆}Λ_{k−1}(E)) define Δ(p, ϕ) ∈ CDiff(C_{⋆}Λ_{k−1}(E), C_{⋆}Λ_{k−1}(E)) by putting

Δ(p, ϕ)(ω) =^{def} (−1)^{|ϕ|(|p|+|Δ|)+|ω||p|}(ϕ ∘ Δ)(ωp), ω ∈ C_{⋆}Λ_{k−1}(E).

Tensor products we need now on are over the algebra C_{⋆}Λ_{k−1}(E) and we shall simplify the notation by using ⊗ instead of ⊗_{C_{⋆}Λ_{k−1}(E)}.

Proposition 5. There exists a unique C_{⋆}Λ_{k−1}(E)–module homomorphism

CDiff(P_{1}, P_{2}) ∋ Δ ⟶ Δ^{Q} ∈ CDiff(P_{1} ⊗ Q, P_{2} ⊗ Q)

such that

Δ^{Q}(p ⊗ q, ϕ ⊗ ψ)(ω) = (−1)^{|ψ|(|q|+|p|+|Δ|+|ϕ|)+|q|(|ϕ|+|ω|)}ψ(Δ(p, ϕ)^{Q}(ωq)),

p ∈ P_{1}, q ∈ Q, ϕ ∈ P_{2}^{∗}, ψ ∈ Q^{∗}, ω ∈ C_{⋆}Λ_{k−1}(E).

Now, let P_{3} be another C_{⋆}Λ_{k−1}(E)–module. Then

(Δ_{2} ∘ Δ_{1})^{Q} = Δ_{2}^{Q} ∘ Δ_{1}^{Q}

for any Δ_{1} ∈ CDiff(P_{1}, P_{2}), Δ_{2} ∈ CDiff(P_{2}, P_{3}). This simple fact allows to associate with a complex of C_{⋆}Λ_{k−1}(E)–modules

⋯ ⟶ P_{i−1} ⟶^{Δ_{i−1}} P_{i} ⟶^{Δ_{i}} P_{i+1} ⟶^{Δ_{i+1}} ⋯

connected by C–differential operators Δ_{i}’s a new one, namely,

⋯ ⟶ P_{i−1} ⊗ Q ⟶^{Δ_{i−1}^{Q}} P_{i} ⊗ Q ⟶^{Δ_{i}^{Q}} P_{i+1} ⊗ Q ⟶^{Δ_{i+1}^{Q}} ⋯

3. (k − 1)–IDF–symmetries and Λ_{k−1}C–Spectral Sequence of a System of PDEs

Let E be as above. The universal linearization ℓ_{Λ_{k−1}F}^{{k}} : Λ_{k−1}κ ⟶ Λ_{k−1}P of Λ_{k−1}F is a C–differential operator and as such (proposition 2) restricts to E.

Theorem 6. There is a Lie–algebra isomorphism Λ_{k−1}Sym(E) ≃ ker(ℓ_{Λ_{k−1}F}^{{k}} |_{E}) ⊂ Λ_{k−1}κ|_{E}.

An exact description of this isomorphism is as follows. Let \overline{χ} ∈ ker(ℓ_{Λ_{k−1}F}^{\{k\}} |_{E}) ⊂ Λ_{k−1}κ|_{E}. Then \overline{χ} = R_{E}^{Λ_{k−1}κ}(χ) for some χ ∈ Λ_{k−1}κ. The evolutionary derivation �_{χ} : Λ_{k−1} ⟶ Λ_{k−1} can be restricted to E, i.e., there exists a unique derivation �_{χ}|_{E} ∈ D(Λ_{k−1}(E), Λ_{k−1}(E)) such that i_{E}^{∗} ∘ �_{χ} = �_{χ}|_{E} ∘ i_{E}^{∗}. Moreover, �_{χ}|_{E} ∈ D_{C}(Λ_{k−1}(E)). Then the derivation that corresponds to \overline{χ} via the isomorphism of theorem 6 is [�_{χ}|_{E}]_{CD(Λ_{k−1}(E))} ∈ Λ_{k−1}Sym(E).

Put

Λ_{k−1}Q =^{def} {∇|_{E} ∘ ℓ_{Λ_{k−1}F}^{{k}} |_{E} | ∇ ∈ CDiff(Λ_{k−1}P, Λ_{k−1})} ⊂ CDiff(Λ_{k−1}κ|_{E}, Λ_{k−1}(E))

and note that CDiff_{(p−1)}(Λ_{k−1}κ|_{E}; Λ_{k−1}Q) is naturally embedded into CDiff_{(p)}(Λ_{k−1}κ|_{E}; Λ_{k−1}(E)). In view of this embedding the following definition

Λ_{k−1}Q_{p} =^{def} alt_{p}(CDiff_{(p−1)}(Λ_{k−1}κ|_{E}; Λ_{k−1}Q)) ⊂ CDiff_{(p)}^{alt}(Λ_{k−1}κ|_{E}; Λ_{k−1}(E))

makes sense. In particular, Λ_{k−1}Q_{1} = Λ_{k−1}Q.

Proposition 7. There is a Λ_{k−1}(E)–module isomorphism

C^{p}Λ_{k}^{p}(E) ≃ CDiff_{(p)}^{alt}(Λ_{k−1}κ|_{E}; Λ_{k−1}(E))/Λ_{k−1}Q_{p}.

According to this proposition the homomorphism

λ_{Λ_{k−1}F} : CDiff(Λ_{k−1}P |_{E}; Λ_{k−1}(E)) ⟶ CDiff(Λ_{k−1}κ|_{E}; Λ_{k−1}(E)),

sending ∇ ∈ CDiff(Λ_{k−1}P |_{E}; Λ_{k−1}(E)) to ∇ ∘ ℓ_{Λ_{k−1}F}^{{k}}, makes the following sequence of Λ_{k−1}(E)–homomorphisms

CDiff(Λ_{k−1}P |_{E}; Λ_{k−1}(E)) ⟶^{λ_{Λ_{k−1}F}} CDiff(Λ_{k−1}κ|_{E}; Λ_{k−1}(E)) ⟶ CΛ_{k}^{1}(E) ⟶ 0,

exact. Recall (see [2, 5]) that the equation E_{F} is said ℓ–normal if the sequence of C^{∞}(E)– homomorphisms

0 ⟶ CDiff(P |_{E}; C^{∞}(E)) ⟶^{λ_{F}} CDiff(κ|_{E}; C^{∞}(E)) ⟶ CΛ^{1}(E) ⟶ 0

with λ_{F} (∇) =^{def} ∇ ∘ ℓ_{F}, ∇ ∈ CDiff(P |_{E}; C^{∞}(E)), is exact. The “iterated”analogue of this notion is as follows.

Definition 8. An equation E_{F} is said ℓ^{{k}}–normal if the sequence of Λ_{k−1}(E)– homomorphisms

0 ⟶ CDiff(Λ_{k−1}P |_{E}; Λ_{k−1}(E)) ⟶^{λ_{Λ_{k−1}F}} CDiff(Λ_{k−1}κ|_{E}; Λ_{k−1}(E)) ⟶ CΛ_{k}^{1}(E) ⟶ 0

is exact, i.e., ker λ_{Λ_{k}F} = 0.

Proposition 9. If E is ℓ–normal, then it is ℓ^{({k}}–normal as well for any k > 1.

The zeroth column differential d_{0}^{0,•} : HΛ_{k}(E) ⟶ HΛ_{k}(E) in the zeroth term of the Λ_{k−1}C–spectral sequence of E is a C–differential operator. Consider the operator

[d_{0}^{0,•}]_{p} =^{def} (d_{0}^{0,•})^{C_{⋆}Λ_{k}^{1}(E)⊗C_{⋆}Λ_{k}^{p−1}(E)},

[d_{0}^{0,•}]_{p} : C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E) ⟶ C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E)

and denote by

alt_{p} : C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E) ⟶ C_{⋆}Λ_{k}^{p}(E) ⊗ HΛ_{k}(E)  (1)

the alternation map. Then a natural isomorphism C_{⋆}Λ_{k}^{p}(E)⊗HΛ_{k}(E) ≃ Λ_{k−1}CE_{0}^{p,•} takes place. Moreover, the inclusion C_{⋆}Λ_{k}^{p}(E) ⊗ HΛ_{k}(E) ⊂ C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E) is a right inverse of alt_{p} so that Λ_{k−1}CE_{0}^{p,•} is a direct summand in C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E). More precisely, the complex (Λ_{k−1}CE_{0}^{p,•}, d_{0}^{p,•}) is a direct summand in the complex (C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E), [d_{0}^{0,•}]_{p}). Now consider the restriction to E of the adjoint operator of ℓ_{Λ_{k−1}F}^{{k}}:

ℓ̂_{Λ_{k−1}F}^{\{k\}} |_{E} : { \atop Λ_{k−1}P} |_{E} ⟶ { \atop Λ_{k−1}κ}|_{E}

and its multiple extension

[ℓ̂_{Λ_{k−1}F}^{\{k\}} |_{E}]_{p} : { \atop Λ_{k−1}P} |_{E} ⊗ C_{⋆}Λ_{k}^{p}(E) ⟶ { \atop Λ_{k−1}κ}|_{E} ⊗ C_{⋆}Λ_{k}^{p}(E)

defined as [ℓ̂_{Λ_{k−1}F}^{{k}} |_{E}]_{p} =^{def} (ℓ̂_{Λ_{k−1}F}^{{k}} |_{E})^{C_{⋆}Λ_{k}^{p}(E)}.

Proposition 10.

• H^{q}(C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E), [d_{0}^{0,•}]_{p}) ≃ 0 for q < n − 1.

• H^{n−1}(C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E), [d_{0}^{0,•}]_{p}) ≃ ker [ℓ̂_{Λ_{k−1}F}^{{k}} |_{E}]_{p−1}.

• H^{n}(C_{⋆}Λ_{k}^{1}(E) ⊗ C_{⋆}Λ_{k}^{p−1}(E) ⊗ HΛ_{k}(E), [d_{0}^{0,•}]_{p}) ≃ coker [ℓ̂_{Λ_{k−1}F}^{{k}} |_{E}]_{p−1}.

In the following theorem that collects the above results the map alt_{p}, abusing the notation, stands for the induced by projection (1) map in cohomology.

Theorem 11 (Two Lines Theorem). Let E = E_{F}^{∞} be the infinite prolongation of an ℓ–normal equation E_{F} = {F = 0}. Then E is ℓ^{{k}}–normal and

• Λ_{k−1}CE_{1}^{0,•}(E) ≃ Λ_{k−2}CE_{1}^{•,•}(E).

• Λ_{k−1}CE_{1}^{p,q}(E) = 0 for any p > 0 and q < n − 1.
• Λ_{k−1}CE_{1}^{p,n−1}(E) ≃ alt_{p}(ker [ℓ̂_{Λ_{k−1}F}^{{k}} |_{E}]_{p−1}) for any p > 0.
• Λ_{k−1}CE_{1}^{p,n}(E) ≃ alt_{p}(coker [ℓ̂_{Λ_{k−1}F}^{{k}} |_{E}]_{p−1}) for any p > 0.

4. Short Conclusion

Limits of a short note do not allow us neither to be completely conceptual in the exposition, nor to present all obtained results in full generality. Both require introducing numerous new notions and constructions that can be done only in frames of the forthcoming detailed and systematical exposition. The same concerns various applications, even quite direct ones as, for instance, secondary versions of notes [3, 7, 8]. By concluding we state that the “iterated”analogues of the l–lines theorem and all results contained in the last chapter of book [5] take place.

References

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[6] I. S. Krasil’shchik, A. M. Verbovetsky, Homological Methods in Equation of Mathematical Physics Open Education (Opava) 1998. See also The Diffiety Inst. Preprint Series, DIPS 7/98, http:

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Alexandre Vinogradov, DMI, Facoltà di Scienze MM. FF. NN. Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy

E-mail address: vinograd@unisa.it

Luca Vitagliano, DMI, Facoltà di Scienze MM. FF. NN. Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy

E-mail address: luca vitagliano@fastwebnet.it