The Diffiety Institute Preprint Series

Preprint DIPS–1/96 October 1, 1996

Available via INTERNET: http://ecfor.rssi.ru/˜ diffiety/ http://www.botik.ru/˜ diffiety/ anonymous FTP: ftp://ecfor.rssi.ru/general/pub/diffiety ftp://www.botik.ru/˜ diffiety/preprints

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Calculus over commutative algebras: a concise user guide

by

I.S.Krasil’shchik

Calculus over commutative algebras: a concise user guide

I.S.Krasil’shchik Moscow Institute for Municipal Economy & the Diffiety Institute^{∗}

Abstract

A short introduction to the theory of linear differential operators over commutative algebras is exposed.

Key words and phrases: linear differential operators, jets, forms, Spencer complexes, commutative algebras.

1991 AMS Subject Classification: 13N10, 16S32, 58A20, 58G05.

When dealing with coordinate-free treatment of nonlinear differential equations, one often starts with various aspects of the Calculus over commutative algebras. To help the reader, we expose here basic facts and definitions mutual for majority of the papers below. The exposition is based on [1] and originates from [2]. The contents of this introductory text is as follows:

1. Linear differential operators over commutative algebras.

2. Diff-prolongations and compositions of differential operators.

3. Derivations.

4. The Spencer Diff-complex.

5. Jets.

6. Jet-prolongations and co-compositions.

7. Differential forms and the de Rham complex.

8. The Spencer Jet-complex.

9. Generalizations and specializations of the basic constructions.

∗For correspondence: 1st Tverskoy-Yamskoy per. 14, Apt.45, 125047 Moscow, Russia. Email: josephk@glas.apc.org

LINEAR DIFFERENTIAL OPERATORS. We consider a commutative ring K with a unit and a commutative associative unitary K-algebra A. The theory below is the theory of linear differential operator in the category M(A) of all A-modules.

Let P, Q be A-modules. Then two A-module structures exist in the K-module hom_{K}(P, Q):

(af)p = a(fp), (a^{+}f)p = f(ap), a ∈ A, f ∈ hom_{K}(P, Q), p ∈ P. (1)

We set δ_{a}(f) = af − a^{+}f. Obviously, for any a, b ∈ A one has

δ_{ab}f = aδ_{b}f + bδ_{a}f (2)

and if R is another A-module and g ∈ hom_{K}(Q, R), then

δ_{a}(g ∘ f) = δ_{a}g ∘ f + g ∘ δ_{a}f. (3)

Definition 1 An K-homomorphism Δ ∈ hom_{K}(P, Q) is called a linear differential operator of order ≤ k acting from P to Q, if

(δ_{a_{0}} ∘ δ_{a_{1}} ∘ … ∘ δ_{a_{k}})Δ = 0

for any a_{0}, a_{1}, …, a_{k} ∈ A.

The set of all such operators inherits both A-module structures (1) and thus forms an A-bimodule. This bimodule is denoted by Diff_{k}^{(+)}(P, Q). When considering only one of these module structures, we write Diff_{k}(P, Q) or Diff_{k}^{+}(P, Q). From the definition it follows that Diff_{0}^{(+)}(P, Q) = hom_{A}(P, Q).

The correspondence (P, Q) ⇒ Diff_{k}^{(+)}(P, Q) is a bifunctor from M(A) to the category of A-bimodules. We also use the notation Diff_{k}^{(+)}(P) for the bimodule Diff_{k}^{(+)}(A, P).

Consider the maps

i_{+} : Diff_{k}(P, Q) → Diff_{k}^{+}(P, Q), i^{+} : Diff_{k}^{+}(P, Q) → Diff_{k}(P, Q)

induced by the identity id : hom_{K}(P, Q) → hom_{K}(P, Q).

Proposition 1 The maps i_{+} and i^{+} are differential operators of order ≤ k.

Let Δ : P → Q, ∇ : Q → R be differential operators. Then from (3) one obtains

Proposition 2 For any two differential operators Δ ∈ Diff_{k}^{(+)}(P, Q) and ∇ ∈ Diff_{l}^{(+)}(Q, R) their composition ∇ ∘ Δ : P → R is an operator order ≤ k + l.

For any k ≤ l one has obvious embeddings Diff_{k}^{(+)}(P, Q) ⊂ Diff_{l}^{(+)}(P, Q) and consequently the module Diff_{∗}^{(+)}(P, Q) = ⋃_{k≥0} Diff_{k}^{(+)}(P, Q) is defined. Using Proposition 2, it can be seen that Diff_{∗}^{(+)}(P, P) is a filtered associative A-algebra.

Diff -PROLONGATIONS AND COMPOSITIONS. Let us fix an A-mo-dule Q and consider the functor Diff_{k}^{+}( · , Q) : M(A) ⇒ M(A). This functor is representable. Namely, one has the following Proposition 3 For any differential operator Δ : P → Q of order ≤ k, there exists a uniquely defined homomorphism f_{Δ} : P → Diff_{k}^{+}(Q) such that the diagram

[figure: \begin{aligned}P & Diff_{k}^{+}(Q) \\ & Q \end{aligned}; Δ; f_{Δ}; D_{k}]

where D(∇) = ∇(1), ∇ ∈ Diff_{k}^{+}(Q), is commutative. The correspondence Δ ↦ f_{Δ} establishes an isomorphism between the functors Diff_{k}^{+}( · , Q) and hom_{A}( · , Diff_{k}^{+}(Q). We say the homomorphism f_{Δ} to be Diff - associated with the operator Δ. The homomorphism f_{Δ} is defined by (f_{Δ}(p))a = Δ(ap).

Note that the above defined D_{k} : Diff_{k}^{+}(Q) → Q is a differential operator of order ≤ k. Hence, for any two integers k, l ≥ 0, using Propositions 2 and 3 one obtains the commutative diagram

Diff_{k}^{+}Diff_{l}^{+}Q  Diff_{l}^{+}Q

Diff_{k+l}^{+}Q  Q

c_{k,l}

[figure]

D_{k} [figure]

D_{l}

[figure] D_{k+l} [figure]

where c_{k,l} = c_{k,l}(Q) =^{def} f_{D_{l}∘D_{k}}. The correspondence Q ⇒ c_{k,l}(Q) is a natural transformation of functors Diff_{k}^{+}Diff_{l}^{+}( ·) and Diff_{k+l}^{+}( ·); the homomorphism c_{k,l}(Q) is called the gluing, or the composition homomorphism. Let ∇ ∈ Diff_{k}^{+}Diff_{l}^{+}Q and a ∈ A. Then from the definitions it follows that

c_{k,l}(∇)(a) = f_{D_{l}∘D_{k}}(∇)(a) = D_{l}D_{k}(a^{+}∇) = ∇(a)(1). (4)

In the sequel we use the notation Diff_{i_{1},...,i_{n}}^{(+)}( ·) for the iterated functor Diff_{i_{1}}^{(+)}(… (Diff_{i_{n}}^{(+)}( ·)) …).

Consider now an operator Δ : P → Q of order ≤ k. Then the operator Δ_{(l)} = Δ ∘ D_{l} : Diff_{l}^{+}(P) → Q is of order ≤ k + l and is called the lth Diff -prolongation of Δ. Since for any l ≤ l′ one has D_{l′} |_{Diff_{l}^{+}(P)} = D_{l}, from where it follows that

Δ_{(l′)} |_{Diff_{l}^{+}(P)} = Δ_{(l)}. Hence, one obtains the infinite Diff -prolongation Δ_{(∗)} : Diff_{∗}^{+}(P) → Q of the operator Δ. From the definition it also follows that there exists the commutative diagram

[figure: \begin{aligned}Diff_{l}^{+}P & Diff_{k+l}^{+}P \\ & Q \end{aligned}; Δ_{(l)}; f_{Δ_{(l)}}; D_{k+l}]

and f_{Δ_{(l′)}} |_{Diff_{l}^{+}P} = f_{Δ_{(l)}} for any l′ ≥ l. Thus we obtain the filtered A-module homomorphism f_{Δ_{(∗)}} : Diff_{∗}^{+}(P) → Diff_{∗}^{+}(Q) of filtration k. It is called the Diff -prolongation homomorphism. In particular, the composition homomorphism c_{∗,l} : Diff_{∗,l}^{+}Q → Diff_{∗}^{+}Q coincides with the prolongation homomorphism for the operator D_{l}. Note also that the commutative diagram

Diff_{l,l′}^{+}P  Diff_{l+l′}^{+}P

Diff_{l}^{+}P  Q

D_{l′}

[figure]

c_{l′,l} [figure]

(Δ_{(l)})_{(l′)}

[figure]

Δ_{(l+l′)}

[figure] Δ_{(l)} [figure]

establishes the relations between iterated prolongations and compositions.

DERIVATIONS.

Definition 2 Let P be an A-module. A K-linear map X : A → P satisfying the Leibnitz rule

X(ab) = aX(b) + bX(a), a, b ∈ A,

is called a P -valued derivation of A, or P -derivation.

The set of all such derivation is denoted by D(P). Obviously, any X ∈ D(P) is a differential operator of order ≤ 1 and D(P) forms a submodule in Diff_{1}P.

Example. Fix an operator Δ ∈ Diff_{k}^{(+)}(P, Q). Then, due to (2) the correspondence a ↦ δ_{a}(Δ) is a Diff_{k−1}^{(+)}(P, Q)-derivation.

Obviously, an operator Δ ∈ Diff_{1}P lies in D(P) if and only if Δ(1) = 0. Hence, we have the exact sequence

0 → D(P) →^{S_{1}^{1}} Diff_{1}^{+}P →^{D_{1}} P → 0, (5)

where S_{1}^{1} is the composition of the natural embedding D(P) ↪ Diff_{1}P with the operator i_{+} : Diff_{1}P → Diff_{1}^{+}P. This the 1st order Spencer Diff -complex. In the next subsection we construct its nth order generalizations. To do this, we shall need the notion of polyderivations.

Let P be an A-module and N ⊂ P be a subset. Define

D(N ⊂ P) =^{def} {X ∈ D(P) | X(A) ⊂ N}.

Now we inductively define modules D_{i}(P), i ≥ 0, together with monomorphic differential operators D_{i}(P) ↪ (Diff_{1}^{+})^{i}P, where (Diff_{1}^{+})^{i} =^{def} Diff + ; ^{1,...,1}_{i times}. Namely,

we set D_{0}(P) = P, D_{1}(P) = D(P) and assuming that the embedding D_{i−1}(P) ↪ (Diff_{1}^{+})^{i−1}P has been defined, set

D_{i}(P) =^{def} D(D_{i−1}(P) ⊂ (Diff_{1}^{+})^{i}P).

Corresponding embedding is defined as

D_{i}(P) = D(D_{i−1}(P) ⊂ (Diff_{1}^{+})^{i}P) ⊂ D((Diff_{1}^{+})^{i−1}P ⊂ (Diff_{1}^{+})^{i}P).

From the definition one obtains

Proposition 4 For any i ≥ 0 the set D_{i}(P) is an A-module while the correspondence P ⇒ D_{i}(P) is a functor in M(A).

Note that for any X ∈ D_{i}(P) and a ∈ A one has X(a) ∈ D_{i−1}(P) and by induction X(a_{1}, …, a_{j}) ∈ D_{i−j}(P) for any a_{1}, …, a_{j} ∈ A and i ≥ j. For j > i we formally set X(a_{1}, …, a_{j}) = 0.

Proposition 5 Let i ≥ 2 and X ∈ hom_{K}(A, D_{i−1}(P)). Then X ∈ D_{i}(P) if and only if

X(ab) = aX(b) + bX(a)

and

X(a, b) + X(b, a) = 0

for any a, b ∈ A.

This result motivates the following

Definition 3 Elements of D_{i}(P) are called P -valued i-derivations of A.

SPENCER Diff -COMPLEX. The construction of the Spencer Diff -complex in general situation is based on the following

Lemma 1 Let P^{(+)} be an A-bimodule and P_{1}, P_{2} ⊂ P be submodules with respect to the first A-module structure. Consider a homomorphism φ : P^{+} → P^{+} such that φ |_{P_{1}} : P_{1} → P_{2} is a differential operator of order ≤ 1. Then the map Dφ : D(P_{1} ⊂ P^{+}) → D(P_{2} ⊂ P^{+}) defined by (Dφ)Δ =^{def} φ ∘ Δ is an operator of order ≤ 1 as well. Moreover, ker φ = 0 implies ker Dφ = 0.

Now we inductively define the sequence of injective operators δ^{i} : D_{i}(P) → D_{i−1}Diff_{1}^{+}P of order ≤ 1, starting with δ^{1} =^{def} S_{1}^{1} (see (5)). Assume that δ^{1}, …, δ^{i−1}, i ≥ 2, have been defined. By definition, one has

D_{i}(P) = D(D_{i−1}(P) ⊂ (Diff_{1}^{+})^{i−1}P)

and

D_{i−1}Diff_{1}^{+}P = D(D_{i−2}Diff_{1}^{+}P ⊂ (Diff_{1}^{+})^{i−1}P).

Hence, using Lemma 1, we may set δ^{i} =^{def} Dδ^{i−1}.

Let operators S_{i+k}^{i} : D_{i}Diff_{k}^{+}P → D_{i−1}Diff_{k+1}^{+}P be defined by commutativity of the diagram

D_{i}Diff_{k}^{+}P  D_{i−1}Diff_{k+1}^{+}P

D_{i−1}Diff_{1,k}^{+}P

S_{i+k}^{i} [figure]

δ^{i}

[figure]
D_{i}(c_{1,k})

[figure]

Proposition 6 For any i, k ≥ 0 one has S_{i+k}^{i} ∘ S_{i+k}^{i+1} = 0.

Definition 4 The sequence

0 → D_{i+k}(P) →^{S_{i+k}^{i+k}} D_{i+k−1}Diff_{1}^{+}P → … → D_{i}Diff_{k}^{+}P →^{S_{i+k}^{i}}

D_{i−1}Diff_{k+1}^{+}P → … → Diff_{i+k}^{+}P ⟶^{S_{i+k}^{0}=D_{i+k}} P → 0  (6)

is called the Spencer Diff -complex of order k + i for the module P.

Denote this complex by S_{i+k}P. Consider now the natural embeddings

D_{i}Diff_{k}^{+}P ↪ D_{i}Diff_{k+1}^{+}P

induced by the embeddings Diff_{k}^{+}P ↪ Diff_{k+1}^{+}P. Then the diagram

D_{i}Diff_{k}^{+}P  D_{i−1}Diff_{k+1}^{+}P

D_{i}Diff_{k+1}^{+}P  D_{i−1}Diff_{k+2}^{+}P

[figure]

[figure]

S_{i+k}^{i} [figure]

[figure]

[figure] S_{i+k+1}^{i} [figure]

is commutative. Consequently, morphisms S_{i+k}P → S_{i+k+1} of complexes exist and one can define the complex S_{∗}P

… → D_{i}Diff_{∗}^{+}P →^{S_{∗}^{i}} D_{i−1}Diff_{∗}^{+}P → … → D_{1}Diff_{∗}^{+}P →^{S_{∗}^{0}} Diff_{∗}^{+}P → 0

as the inductive limit of S_{i+k} with respect to these morphisms. Both S_{i+k}( ·) and S_{∗}( ·) are functors from M(A) to the category of differential complexes over A.

Moreover, let P, Q be A-modules and Δ : P → Q be an sth order differential operator. Then the map f_{Δ_{(∗)}} : Diff_{∗}^{+}P → Diff_{∗}^{+}P (see above) induces corresponding cochain map of corresponding Spencer complexes:

D_{i}Diff_{k}^{+}P  D_{i−1}Diff_{k+1}^{+}P

D_{i}Diff_{k+s}^{+}Q  D_{i−1}Diff_{k+s+1}^{+}Q

D_{i}(f_{Δ(k)})

[figure]

S_{i+k}^{i} [figure]

D_{i−1}(f_{Δ(k+1)})

[figure] S_{i+k+s}^{i} [figure]

Denote by C_{i,k}(Δ) the cokernel of the map D_{i}(f_{Δ_{(k)}}). Then one obtains the following complex

… → C_{i,k}(Δ) →^{S_{i+k}^{i}} C_{i−1,k+1}(Δ) → … → C_{0,i+k} → coker Δ → 0

which is called the (i + k)th order Spencer Diff -complex of the operator Δ.

JETS. Let P be an A-module. Consider the tensor product A ⊗_{K} P endowed with an A-module structure by

a′(a ⊗ p) =^{def} (a′a) ⊗ p, a, a′ ∈ A, p ∈ P.

For any a′ ∈ A set

δ^{a′}(a ⊗ p) =^{def} a ⊗ a′p − a′a ⊗ p

and denote by μ_{k} the submodule in A ⊗_{K} P spanned by

(δ^{a_{0}} ∘ … ∘ δ^{a_{k}})(a ⊗ p), a, a_{0}, …, a_{k} ∈ A, p ∈ P.

Set

J^{k}(P) =^{def} (A ⊗_{K} P)/μ_{k}, j_{k}(p) =^{def} (1 ⊗ p) mod μ_{k}.

The map j_{k} : P → J^{k}(P) is a differential operator of order ≤ k.

Proposition 7 Fix a module P. Then the functor Diff_{k}(P, ·) is representable. Namely, for any differential operator Δ : P → Q there exists a uniquely defined homomorphism f^{Δ} : P → J^{k}(P) such that the diagram

[figure: \begin{aligned}P & J^{k}(P) \\ & Q \end{aligned}; Δ; j_{k}; f^{Δ}]

is commutative. The correspondence Δ ↦ f^{Δ} establishes an isomorphism between the functors Diff_{k}(P, ·) and hom_{A}(J^{k}(P), ·). We say the homomorphism f_{Δ} to be Jet-associated with the operator Δ.

Definition 5 J^{k}(P) is called the module of k-jets of the module P.

Obviously, J^{k}(P) as an A-module is generated by the elements of the form j_{k}(p), p ∈ P.

Introduce a second A-module structure into J^{k}(P) by setting

a^{+}(a′j_{k}(p)) =^{def} a′j_{k}(ap)

and denote corresponding bimodule by J_{(+)}^{k}(P). Let Q be another A module. For

any φ ∈ hom_{A}(P, Q) set J^{k}(φ)(aj_{k}(p)) =^{def} aj_{k}(φ(p)). Then J_{(+)}^{k}( ·) is a functor from M(A) to the category of A-bimodules.

Denote by J_{(+)}^{k} the bimodule J_{(+)}^{k}(A). For any θ = aj_{k}(b) ∈ J_{(+)}^{k} and ρ =

a′j_{k}(p) ∈ J^{k}(P) set θ · ρ =^{def} aa′j_{k}(bp). Then J_{(+)}^{k} acquires the structure of a commutative unitary A-algebra while J_{(+)}^{k}(P) becomes a J_{(+)}^{k}-module. Define the maps τ_{k} : J_{(+)}^{k} → J_{(+)}^{k} and i_{k} : A → J_{(+)}^{k} by setting

τ_{k}(aj_{k}(b)) =^{def} bj_{k}a), i_{k}(a) =^{def} aj_{k}(1), a, b ∈ A.

Then τ_{k} is a homomorphism of the algebra J_{(+)}^{k},

τ_{k} : J^{k} → J_{+}^{k}, τ_{k} : J_{+}^{k} → J^{k}

are A-module homomorphisms and the following identities hold

τ_{k}^{2} = id, τ_{k} ∘ j_{k} = i_{k}, τ_{k} ∘ i_{k} = j_{k}.

Consider an A-bimodule P^{(+)} and a module Q. Then two module structures exist in the tensor product P^{+} ⊗ Q:

a^{+}(p ⊗ q) = a^{+}p ⊗ q, a(p ⊗ q) = ap ⊗ q, a ∈ A, p ∈ P, q ∈ Q.

The module endowed with the second structure will be denoted by P^{+} ⊗_{in} Q.

Proposition 8 Let P be an A-module. Then the following isomorphisms take place

J^{k}(P) = J_{+}^{k} ⊗_{in} P, J_{+}^{k}(P) = J_{+}^{k} ⊗ P.

Note also that the modules J_{+}^{k} and J^{k} are representative objects for the functors Diff_{k}( ·) and Diff_{k}^{+}( ·) respectively.

Jet-PROLONGATIONS AND CO-COMPOSITIONS. Take two integers k, l ≥ 0. Then from Propositions 2 and 7 one obtains the following commutative diagram

[figure: \begin{matrix}P & J^{l}(P) \\ J^{k+l}(P) & J^{k}(J^{l}(P)) \end{matrix}; j_{k+l}; j_{l}; j_{k}; c^{k,l}]

where c^{k,l} = c^{k,l}(P) =^{def} f^{j_{k}∘j_{l}}. The correspondence P ⇒ c^{k,l}(P) is a natural transformation of functors J^{k+l}( ·) and J^{k}(J^{l}( ·)); the homomorphism c^{k,l}(P) is called the co-composition homomorphism.

Let now Δ : P → Q be an operator of order k. Then the operator Δ^{(l)} =^{def} j_{l} ∘ Δ : P → J^{l}(Q) is called the lth Jet-prolongation of the operator Δ. For such a prolongation one obtains the commutative diagram

[figure: \begin{matrix}P & Q \\ J^{k+l}(P) & J^{l}(Q) \end{matrix}; j_{k+l}; Δ; Δ^{(l)}; j_{l}; f^{Δ^{(l)}}]

Let k ≤ l. Then, since any kth order operator can be considered as an operator of order l, we have the commutative diagram

[figure: \begin{aligned}P & J^{l}(P) \\ & J^{k}(P) \end{aligned}; j_{k}; j_{l}; ν_{l,k}] (7)

where ν_{l,k} =^{def} f^{j_{k}}. The maps ν_{l,k} are epimorphisms dual to the embeddings Diff_{k}(P, Q) ↪ Diff_{l}(P, Q). For any k ≤ l ≤ s one has ν_{s,k} = ν_{l,k} ∘ ν_{s,l} and the projective limit of the system {J^{k}(P), ν_{l,k}}_{l≥k} is denoted by J^{∗}(P) and is called the module of infinite jets of the module P. For any k ≥ 0 a projection ν_{∗,k} : J^{∗}(P) → J^{k}(P) is defined such that ν_{∗,k} = ν_{l,k} ∘ ν_{∗,l}, l ≥ k.

All natural constructions presented here for the modules J^{k}(P) are carried over to J^{∗}(P). Namely, due to diagram (7) there exists an operator j_{∗} : P → J^{∗}(P) such that j_{k} = ν_{∗,k} ∘j_{∗}. Further, let k′ ≥ k, l′ ≥ l be non-negative integers. Then the diagrams

[figure: \begin{matrix}J^{k′+l}(P) & J^{k′}(J^{l}(P)) \\ J^{k+l}(P) & J^{k}(J^{l}(P)) \end{matrix}; ν_{k′+l,k+l}; c^{k′,l}; ν_{k′,k}; c^{k,l}; \begin{matrix}J^{k+l′}(P) & J^{k}(J^{l′}(P)) \\ J^{k+l}(P) & J^{k}(J^{l}(P)) \end{matrix}; ν_{k+l′,k+l}; c^{k,l′}; J^{k}(ν_{l′,l}); c^{k,l}]

are commutative. Consequently, one obtains the map

c^{∗,∗} : J^{∗}(P) → J^{∗}(J^{∗}(P)).

Finally, for any operator Δ : P → Q of order ≤ k from the definitions we have commutative diagrams

[figure: \begin{aligned}P & J^{l}(Q) \\ & J^{l}(Q) \end{aligned}; Δ^{(l)}; Δ^{(l′)}; ν_{l′,l}]

and

[figure: \begin{matrix}J^{k+l′}(P) & J^{l′}(Q) \\ J^{k+l}(P) & J^{l}(Q) \end{matrix}; ν_{k+l′,k+l}; f^{Δ^{(l′)}}; ν_{l′,l}; f^{Δ^{(l)}}]

Hence, the infinite prolongation Δ^{(∗)} : P → J^{∗}(Q) is defined, as well as the corresponding homomorphism f^{Δ^{(∗)}} : J^{∗}(P) → J^{∗}(P). Note also that for any l, l′ ≥ 0 the diagram

[figure: \begin{matrix}P & J^{l′}(Q) \\ J^{l+l′}(Q) & J^{l}(J^{l′}(Q)) \end{matrix}; Δ^{(l′+l)}; Δ^{(l′)}; (Δ^{(l′)})^{(l)}; j_{l}; c^{l,l′}]

is commutative and thus c^{∗,∗} ∘ Δ^{(∗)} = (Δ^{(∗)})^{(∗)}.

FORMS AND DE RHAM COMPLEX. Set Λ^{1}(A) =^{def} J^{1}/im i_{1} and define d : A → Λ^{1}(A) as the composition of j_{1} : A → J^{1} with the natural projection J^{1} → Λ^{1}(A). Set also Λ^{i}(A) =^{def} \underbrace{Λ^{1}(A) ∧ … ∧ Λ^{1}(A)}_{i times}.

Definition 6 The elements of Λ^{i}(A) are called differential i-forms over the algebra A.

Proposition 9 The modules Λ^{i}(A) are representative objects for the functors D_{i}( ·). In the case i = 1 an isomorphism between the modules D_{1}(P) and hom_{A}(Λ^{1}(A), P) is established in the following way: for any derivation X ∈ D_{1}(P) there exists a uniquely defined homomorphism h_{X} : Λ^{1}(A) → P such that the diagram

[figure: \begin{aligned}A & Λ^{1}(A) \\ & P \end{aligned}; X; d; h_{X}]

is commutative.

In the sequel we use the notation Λ^{i} =^{def} Λ^{i}(A).

Note that Λ^{1} as an A-module is generated by elements d(a) with the relations d(ab) = ad(b) + bd(a). Note also that the wedge product ∧ : Λ^{i} ⊗_{A} Λ^{j} → Λ^{i+j} is the representing morphism for the natural transformation of functors D_{i+j}( ·) ⇒ D_{j}(D_{i}( ·)).

Now we define by induction on i the 1st order operators d : Λ^{i} → Λ^{i+1}. For i = 0 we take for d the operator d : A → Λ^{1} introduced above. If i = 1, we set d(ad(b)) =^{def} da ∧ db and for i > 1 and any decomposable element ω ∧ θ ∈ Λ^{i} set

d(ω ∧ θ) =^{def} dω ∧ θ + (−1)^{ω}ω ∧ dθ.

Proposition 10 d ∘ d = 0.

Definition 7 The sequence

0 → A →^{d} Λ^{1} → … → Λ^{i} →^{d} Λ^{i+1} → …

is called the de Rham complex of the algebra A.

SPENCER Jet-COMPLEX. Consider the composition Λ^{i−1} →^{d} Λ^{i} →^{j_{k}} J^{k}(Λ^{i}). Since it is a (k +1)st order operator, from Proposition 7 one obtains the following commutative diagram

[figure: \begin{matrix}Λ_{i−1} & Λ_{i} \\ J^{k+1}(Λ^{i−1}) & J^{k}(Λ^{i}) \end{matrix}; j_{k+1}; d; j_{k}; S̃_{i}^{i+k}]

where S̃_{i}^{i+k} =^{def} f^{j_{k}∘d}. Thus we get the sequence of homomorphisms

0 → J^{k+i} → … → J^{k+1}(Λ^{i−1}) ⟶^{S̃_{i}^{i+k}} J^{k}(Λ^{i}) → … → Λ^{i+k} → 0 (8)

which is a complex and is called the Spencer Jet-sequence. Denote it by SJ^{i+k}.

Let P^{(+)} be an A-bimodule and P be an A-module. Consider the module hom_{A}(P, Q) and introduce the second module structure by setting

(a^{+}φ)(p) =^{def} φ(a^{+}p), a ∈ A, p ∈ P, φ ∈ hom_{A}(P, Q).

Denote such a module by hom_{A}^{+}(P, Q).

Proposition 11 The functors S_{i+k}( ·) and hom_{A}^{+}(SJ^{i+k}, ·) are isomorphic, i.e. the Spencer Jet-sequence is the representative object for the functor S_{i+k}( ·).

Let now P, Q be A-modules. Define the map τ_{k}^{P,Q} : J^{k}(P) ⊗_{A} Q by setting τ_{k}^{P,Q}(j_{k}(p) ⊗ q) =^{def} j_{k}(q) ⊗ p. Obviously,

τ_{k}^{P,Q} ∈ Diff_{k}^{(+)}(J^{k}(P) ⊗_{A} Q, J^{k}(Q) ⊗_{A} P), τ_{k}^{P,Q} ∘ τ_{k}^{Q,P} = id.

Introduce the operators S_{i}^{i+k} : J^{k+1}(P) ⊗_{A} Λ^{i−1} → J^{k}(P) ⊗_{A} Λ^{i} defined by commutativity of the diagram

[figure: \begin{matrix}J^{k+1}(Λ^{i−1}) ⊗_{A} P & J^{k}(Λ^{i}) ⊗_{A} P \\ J^{k+1}(P) ⊗_{A} Λ^{i−1} & J^{k}(P) ⊗_{A} Λ^{i} \end{matrix}; τ_{k+1}^{Λ^{i−1},P}; S̃_{i}^{i+k}⊗id; τ_{k}^{Λ^{i},P}; S_{i}^{i+k}] (9)

Proposition 12 The maps S_{i}^{i+k} : J^{k+1}(P) ⊗_{A} Λ^{i−1} → J^{k}(P) ⊗_{A} Λ^{i} defined by (9) are operators of order ≤ 1 and S_{i+1}^{i+k} ∘ S_{i}^{i+k} = 0.

Definition 8 The sequence

0 → P →^{j_{k+i}} J^{k+i}(P) → … → S_{i}^{i+k}J^{k+1}(P) ⊗_{A} Λ^{i−1} ⟶^{S_{i}^{i+k}}

J^{k}(P) ⊗_{A} Λ^{i} → … → P ⊗_{A} Λ^{i+k} → 0  (10)

is called the Spencer Jet-complex of the module P.

The action of the Spencer Jet-operators is expressed by the following formula

S_{i}^{i+k}(j_{k+1}(p) ⊗ ω) = j_{k}(p) ⊗ dω, p ∈ P, ω ∈ Λ^{i−1}. (11)

Consider an lth order operator Δ : P → Q. Then the diagram

[figure: \begin{matrix}J^{k+l+1}(P) ⊗_{A} Λ^{i−1} & J^{k+l}(P) ⊗_{A} Λ^{i} \\ J^{k+1}(Q) ⊗_{A} Λ^{i−1} & J^{k}(Q) ⊗_{A} Λ^{i} \end{matrix}; f^{Δ^{(k+1)}}⊗id; S_{i}^{k+l+i}; f^{Δ^{(k)}}⊗id; S_{i}^{k+i}]

is commutative. Hence, one can define the Spencer Jet-complex of the operator Δ as the corresponding kernel complex and introduce the Spencer Jet-cohomology groups of Δ.

GENERALIZATIONZ AND SPECIALIZATIONS. There are several directions to generalize or specialize the above described theory. Probably, the most important one, giving rize to various interesting specializations, is associated with the following concept.

Definition 9 An abelian subcategory D of M(A) is said to be differentially closed, if it is closed under tensor product over A and

(i) it is closed under the action of the functors Diff_{k}^{(+)}( · , ·) and D_{i}( ·), (ii) the functors Diff_{k}^{(+)}(P, ·), Diff_{k}^{(+)}( · , Q) and D_{i}( ·) are representable in

D, where P, Q are objects of D.

When it is necessary to indicate the category explicitely, we use the notations Diff_{k}^{(+)}(P, Q; D), D_{i}(P; D), etc. In the case D = M(A) we write Diff_{k}^{(+)}(P, Q; A), D_{i}(P; A), etc.

As an example consider the following situation. Let M be a smooth (i.e. C^{∞}-class) finite-dimensional manifold and set A =^{def} C^{∞}(M). Let π : E → M, ξ : F → M be two smooth locally trivial finite-dimensional vector bundles over M and P = Γ(π), Q = Γ(ξ) be corresponding A-modules of smooth sections.

Proposition 13 The module Diff_{k}^{(+)}(P, Q) coincides with the module of kth order differential operators acting from the bundle π to ξ. The module D(A) coincides with the module of vector fields on the manifold M.

However if one constructs representative objects for the functors Diff_{k}(P, ·) and D_{i}( ·) in the category M(A), the modules J^{k}(P) and Λ^{i} will not coincide with ”geometrical” jets and differential forms.

Definition 10 A module P over C^{∞}(M) is called geometrical, if

⋂_{x∈M} μ_{x}P = 0,

where μ_{x} is the ideal in C^{∞}(M) consisting of functions vanishing at the point x ∈ M.

Denote by G(M) the full subcategory of M(A) whose objects are geometrical C^{∞}(M)-modules. Let P be an A-module and set

G(P) =^{def} P/ ⋂_{x∈M} μ_{x}P.

Evidently, G(P) is a geometrical module while the correspondence P ⇒ G(P) is a functor form M(C^{∞}(M)) to G(M), if for any homomorphism φ ∈ hom_{A}(P, Q) one sets G(φ) = φ.

Proposition 14 Let M be a smooth finite-dimensional manifold and A = C^{∞}(M). Then

(i) The category G(A) of geometrical A-modules is differentially closed.

(ii) The representative objects for the functors Diff_{k}(P, ·) and D_{i}( ·) in G(A) coincide with G(J^{k}(P)) and G(Λ^{i}) respectively.

(iii) The module G(Λ^{i}) coincide with the module of differential i-forms on M. (iv) If P = Γ(π) for a smooth locally trivial finite-dimensional vector bundle

π : E → M, then the module G(J^{k}(P)) coincide with Γ(π_{k}), where π_{k} : J^{k}(π) → M is the bundle of k-jets for the bundle π.

Another example of a differentially closed category is the category of filtered geometrical modules over a filtered algebra. This category is essential to construt differential calculus over manifolds of infinite jets and infinitely prolonged differential equations (see details in [1]).

Note in conclusion that the logical structure of the above described theory is generalized onto a noncommutative case as it is shown in the paper by A.M. Verbovetsky published in this issue of Acta Applicandae Mathematicae.

References

[1] Krasil’shchik, I.S., Lychagin, V.V. and Vinogradov, A.M.: Geometry of Jet Spaces and Nonlinear Differential Equations, Gordon and Breach, New York–London, 1986.

[2] Vinogradov, A.M.: The logic algebra of linear differential operators, Soviet Math. Dokl. 13(4) (1972), 1058–1062.