The Diffiety Institute Preprint Series

Preprint DIPS-7/96 October 1, 1996

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The Diffiety Institute

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Residues without duality

by

Michael M. Vinogradov

RESIDUES WITHOUT DUALITY

Michael M. Vinogradov

Diffiety Institute (Pereslavl-Zalessky, Russia), Institute of Economic Forecasting (Moscow, Russia)

Abstract. The aim of this paper is to suggest an approach of constructing a residue theory. In particular, this approach may be used in case of infinite dimension of an algebra of differential forms. The results of the paper was announced without proofs in [4]. The Russian version of the paper was published in 1985 in [12].

Introduction

An algebraic approach to the residue theory was suggested by A. Grothendieck in [1]. It was realized by R. Hartshorne [2] in the case of zero characteristic and P. Berthelot [3] in the case of positive characteristic. This approach is based on the principle of duality and is suitable only for differential form sheaves of finite dimension. Another approach to constructing a residue theory was announced in [4]. This approach may be used in the case of infinite dimensional algebra of differential forms. In this paper we expose a detailed presentation of the results sketched in [4].

The paper is organized as follows. Section 1 contains the division lemma and its corollaries. In Sections 2 and 3 we consider the cases of zero and positive characteristic respectively.

As a rule, we omit a pure technical proofs. Begin of a proof is signed by ◀, end or absence is signed by ▶.

Necessary notions and notations. Let K be a commutative ring with a unit, A be a commutative associative unitary K-algebra and M, N be an A-modules. We deal with the functors of differential calculus arising in the category M(A) of modules over A. The definition of differential operators over a commutative algebra and other necessary notions and notations up to the definition of differentially closed subcategories of M(A) can be found in the paper [13] of this issue. See also [6,7].

Let K be a differentially closed subcategory of M(A), then modules Λ_{K}^{i} of differential i-forms and the De Rham complex

[figure: 0  A  Λ_{K}^{1}  …  Λ_{K}^{i−1}  Λ_{K}^{i}  …; d_{1}; d_{2}; d_{i−1}; d_{i}; d_{i+1}]

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16S32, 16S80, 17B37, 16W30, 18D10, 35A99, 58A99, 58G05, 58G99, 81R50.

Key words and phrases. Differential operator, jets, differential forms, residues, division lemma, spectral sequence.

can be defined. The homology groups of this complex are called the De Rham cohomology of algebra A in the category K and denoted by H^{i}(A; K). For simplicity of notation, set H^{i}(A) = H^{i}(A; M(A)).

Below we assume K to be a differentially closed subcategory of M(A).

1. The division lemma

Localization. Let S ⊂ A be a multiplicative subset, and Δ : M → N be a DO. Denote by S^{−1}A the ring of quotients of A with respect to S. The modules of quotients of A-modules M, N with respect to S is denoted by S^{−1}M, S^{−1}N respectively.

Proposition 1. There exists a unique DO S^{−1}Δ : S^{−1}M → S^{−1}N such that the diagram

[figure: \begin{matrix}M & N \\ S_{−1}M & S_{−1}N \end{matrix}; S^{−1}; Δ; S^{−1}; S^{−1}Δ]

is commutative.

◀ The proof is a direct calculation based on the definition of a DO. See also [9] ▶

To any category K assign the category S^{−1}K; it is called the localization of K by the multiplicative subset S. Objects of the category S^{−1}K are localizations of objects K, and morphisms in the category S^{−1} are localizations of morphisms in K.

Proposition 2. Let a subcategory K of M(A) be differentially closed; then S^{−1}K is a differentially closed subcategory of M(S^{−1}A), and Λ_{S^{−1}K}^{i} = S^{−1}Λ_{K}^{i}.

◀ The proof is a trival implication of the definitions. ▶

The division lemma for differential forms. Suppose s ∈ A be an irreversible nonnilpotent element and S = {s^{n}}_{n≥0}. Let Λ^{∗} =^{def} Im S^{−1} ⊂ S^{−1}Λ_{K}^{∗}, and Λ_{s}^{∗} =^{def} Λ^{∗}/(sΛ^{∗}+ds∧Λ^{∗}). The image of a differential form ω ∈ Λ_{s}^{∗} under the factorization map Λ^{∗} → Λ_{s}^{∗} is called the restricton of ω on S and is denoted by ω|_{s}.

Below we suppose that the following conditions hold:

(1) either the field of rational number ℚ or the field 𝔽_{p} of residue classes modulo p is a subfield of K;

(2) the division lemma is true for the form ds ∈ Λ_{s}^{∗}, i.e. the following complex

[figure: 0  A  Λ_{1}  …  Λ_{i−1}  Λ_{i}  …; ∧ds; ∧ds; ∧ds; ∧ds; ∧ds]

is acyclic.

For simplicity set p = ∞ if char A = 0.

Proposition 3. Let ω ∈ Λ^{∗} and φ = \frac{ω}{s^{n}} , n < p. Suppose there exists a form α ∈ Λ^{∗} such that dφ = \frac{α}{s^{n}}; then there exist forms ψ, θ ∈ Λ^{∗} such that φ = \frac{ds}{s^{n}} ∧ ψ + \frac{θ}{s^{n−1}} .

◀ Since dφ = \frac{α}{s^{n}}, it follows that

nds ∧ ω = sdω − sα  (1)

Therefore, we have ds∧dω = ds∧α. By the division lemma, there exists a form θ ∈ Λ^{∗} such that dω = ds ∧ nθ + α. Substituting the last relation in (1), we obtain nds ∧ ω = nds ∧ sω. Again using the division lemma, we find a form ψ ∈ Λ^{∗} such that ω = sθ + dsψ. ▶

Corollary. If conditions of Proposition 1 hold and n > 1, then there exist forms β, γ ∈ Λ^{∗} such that φ = d(\frac{β}{s^{n−1}} ) + \frac{γ}{s^{n−1}} . ▶

Proposition 4. Let ω ∈ Λ^{∗} and the form φ = \frac{ω}{s} is closed, then there exist forms ψ, θ ∈ Λ^{∗} such that φ = \frac{ds}{s} ∧ ψ + θ. The restrictions ψ|_{s}, θ|_{s} of the forms ψ, θ on S are uniquely determined, and ψ|_{s} is closed. ψ|_{s} are called the residue form of φ on S and is denoted by res[φ]_{s}.

◀ Existence of forms φ, θ follows from Proposition 3. The proof of the remaining statements are similar to one in the analytical case and may be found, for example, in chapter III of [10]. ▶

Obviously, if φ ∈ Λ^{∗} then res[φ]_{s} = 0.

Proposition 5. Let φ = \frac{ω}{s} , ω ∈ Λ^{∗}; ω be an exact form, and char K = 0, then res[φ]_{s} is also an exact form.

◀ By definition, \frac{ω}{s} = d(\frac{α′}{s^{n}}), α′ ∈ Λ^{∗}. If n > 1, then, by the corollary to Proposition 3, there exist forms β, γ ∈ Λ^{∗} such that \frac{α′}{s^{n}} = d(\frac{β}{s^{n−1}} ) + \frac{γ}{s^{n−1}} . Therefore, \frac{ω}{s} = d(\frac{γ}{s^{n−1}} ). Hence, by induction on n, we can find a form α ∈ Λ^{∗} such that \frac{ω}{s} = d(\frac{α}{s}). By Proposition 3, there exist forms ψ, θ ∈ Λ^{∗} such that \frac{α}{s} = \frac{ds}{s} ∧ ψ + θ. Thus \frac{ω}{s} = d(\frac{α}{s}) = −\frac{ds}{s} ∧ dψ + dθ, and res[\frac{ω}{s}]_{s} = −dψ|_{s}. ▶

Proposition 6. Let φ = \frac{ω}{s^{n}} , ω ∈ Λ^{∗}, 0 ≤ n < p, be a closed form. Then there exists a form α ∈ Λ^{∗} such that φ is cohomologous with \frac{α}{s}.

◀ If n = 0, then α = sω. The proof of this proposition for the case n > 1 is similar to the proof of the Proposition 5. ▶

2. Residues for the case of characteristic zero

General scheme. Let char A = 0. If φ ∈ Λ_{S^{−1}K}^{∗} is a closed form, then, by Proposition 6, there exists a form α ∈ Λ^{∗} such that φ is cohomologous with \frac{α}{s}. The cohomology class of the form res[\frac{α}{s}]_{s} is uniquely determined by Proposition 4. It is called the residue class of φ, and is denoted by Res[φ]_{s}.

Consider a cohomology class Θ ∈ H^{l}(S^{−1}A; S^{−1}K). Let φ, ψ be arbitrary elements of Θ. Then by Proposition 5 Res[φ]_{s} = Res[ψ]_{s}. Thus K-homomorphism

Res_{s} : H^{l}(S^{−1}A; S^{−1}K) → H^{l−1}(Λ_{s}^{∗}), Res_{s}(Θ) = Res[φ]_{s}, φ ∈ Θ,

is well defined.

Theorem 1. The sequence

0 ⟶ H^{l}(Λ^{∗}) ⟶^{i} H^{l}(S^{−1}A; S^{−1}K) ⟶^{Res_{s}} H^{l−1}(Λ_{s}^{∗})

is exact.

◀ Obviously, Im i ⊂ Ker Res_{s}. Let φ be an arbitrary representative of a cohomology class Θ ∈ H^{l}(S^{−1}A; S^{−1}K). By definition, the kernel of the factorization homomorphism Λ^{∗} → Λ_{s}^{∗} is generated by s and ds. This implies that there exist forms α, β, γ, δ ∈ Λ^{∗} such that φ = \frac{ds}{s} ∧ (dα + sβ + ds ∧ γ) + δ. This means that the form φ is cohomologous with ds ∧ β + δ. Therefore Θ ∈ Im i. ▶

Remark. Let R ⊂ A be a multiplicative subset, and s ∉ R. Suppose R^{−1} : Λ_{S^{−1}K}^{∗} → R^{−1}Λ_{S^{−1}K}^{∗} = Λ_{R^{−1}S^{−1}K}^{∗} is a monomorphism. Then the division lemma is true for the element d_{R^{−1}S^{−1}K} ∈ R_{−1}Λ^{∗}, and the residue homomorphism

Res_{R^{−1}s} : H^{l}(R^{−1}S^{−1}A; R^{−1}S^{−1}K) → H^{l−1}(Λ_{R^{−1}s}^{∗})

can be constructed. The diagram

H^{l}(S^{−1}A; S^{−1}K)  ⟶^{Res_{s}} H^{l−1}(Λ_{s}^{∗})
↓  ↓
H^{l}(R^{−1}S^{−1}A; R^{−1}S^{−1}K) ⟶^{Res_{R^{−1}s}} H^{l−1}(Λ_{R^{−1}s}^{∗})

is commutative. This means that it is not difficult to generalize all constructions described above to the case of schemes and sheaves.

Examples. Let K be a field of zero characteristic, A be a discrete valuation ring, ν ⊂ A be corresponding maximal ideal, and Ā be its field of fractions. Suppose an element s is a generator of the ideal ν. Then S^{−1}A = Ā, and S^{−1}M(A) = M(Ā). For simplicity of notation, set Λ_{S^{−1}M(A)}^{∗} = Λ^{∗}(Ā).

Definition. Let Ω = {ω_{j}} be a finite subset of Λ^{∗} . A finite subset G = {g_{i}} is called a ν-system for Ω if the following conditions hold:

(1) The elements dg_{i} are linearly independent;

(2) If dG is a subalgebra of L generated by dg_{i}, then Ω ⊂ dG.

It is not difficult to to prove the following two statements.

Proposition 7. The forms dg_{i} are linearly independent over Ā iff the elements g_{i} are algebraicly independent over K. ▶

Proposition 8. For any finite subset of Ω = {ω_{j}} ⊂ Λ^{∗} there exists a ν-system. ▶ Proposition 9. \frac{ds}{s} ∉ Λ^{∗}.

◀ Assume the converse. Then there exist elements f_{i}, g_{i} ∈ A such that

\frac{ds}{s} = ∑_{i=1}^{k} f_{i}dg_{i}  (2)

By Propositions 7, 8 the elements g_{i} are algebraically independent over K. On the other hand, it follows from the last condition that the elements {s, g_{1},…, g_{k}} are algebraically dependent over K.

Let F (x, y_{1},…, y_{k}) ∈ K[x, y_{1},…, y_{k}] be an irreducible polinomial over K such that s, g_{1},…, g_{k} = 0. It is easily shown that (2) is true iff the polinomial F (x, y_{1},…, y_{k}) can be presented in the form

F (x, y_{1},…, y_{k}) = xF′(x, y_{1},…, y_{k}) + C,

F′(x, y_{1},…, y_{k}) ∈ K[x, y_{1},…, y_{k}], 0 ≠ C ∈ K. Then sF′(s, g_{1},…, d_{k}) + C = 0, and the element s is invertible in the algebra A: \frac{1}{s} = −\frac{F′(s,g_{1},...,d_{k})}{C}. This contradiction proves the proposition. ▶

Proposition 10. For any finite subset of Ω = {ω_{j}} ⊂ Λ^{∗} there exists a ν-system containing s.

◀ Let {g_{i}} be any ν-system for set Ω. If the set {s, g_{i}} is algebraicly independent over K, then it is just the required ν-system. Otherwise, there exists a nontrivial constraint α_{0}ds + ∑_{i=1}^{k} α_{i}dg_{i} = 0, α_{0} ≠ 0, α_{i} ∈ A. Each of nonzero coefficients α_{i} may be presented in the form α_{i} = β_{i}s^{r_{i}}, β_{i} ∈ A. Let r = min_{i>0}{r_{i}} and l be an integer such that r_{l} = r. The assumption r_{0} < r_{l} contradicts to Proposition 9, and thus the set G\{g_{l}} is the required ν-system. ▶

Proposition 11. The division lemma is true for the complex (Λ^{∗}, d).

◀ Let ω ∈ Λ^{∗}. By the previous proposition, for any ω there exists a ν-system {s, g_{i}}. Obviously, if ds ∧ ω = 0, then ω = ∑_{(i_{1},...,i_{k})} h_{(i_{1},...,i_{k})}ds ∧ dg^{i_{1}} ∧ ⋯ ∧ dg^{i_{k}}. ▶

Thus Theorem 1 is true for discrete valuation rings. In this case the complexes Λ^{∗} and Λ_{s}^{∗} may be identified with the complexes Λ_{M(A)}^{∗} and Λ_{M(A/ν)}^{∗} respectively. The latter identification is obvious. To prove the former one it is sufficient to prove that Proposition 9 still holds if one replaces Λ^{∗} by Λ_{M(A)}^{∗} . Therefore any form ω ∈ Λ_{M(A)}^{∗} can be represented as ω = ∑_{(i_{1},...,i_{k})} h_{(i_{1},...,i_{k})}dg^{i_{1}} ∧ ⋯ ∧ dg^{i_{k}}, where the set {g_{i}} is algebraically independent of K and homomorphism S^{−1} : Λ_{M(A)}^{∗} → Λ_{M(Ā)}^{∗} has no kernel. Finally we note that the homomorphism Res_{s} is independent of a choice of a generator s of the ideal ν. Indeed, let a ∈ A be an invertible element, and φ = \frac{d(as)}{as} ∧ ψ + θ. Then φ = \frac{ds}{s} ∧ ψ + \frac{da}{a} ∧ ψ + θ. Thus res[φ]_{as} = ψ|_{s} = res[φ]_{s}. It follows that the exact sequence of Theorem 1 has the form

0 ⟶ H^{l}(A; M(A)) ⟶^{i} H^{l}(Ā; M(Ā)) ⟶^{Res_{s}} H^{l−1}(A/ν; M(A/ν))

The following example is interesting from two points of a view. First, we consider a non-Noetherian algebra. Second, this example shows that validity of the division lemma is dependent on a choice of a “coefficient” category K. Let A be a non-Noetherian local algebra such that its maximal ideal ν is principal, s be a generator of ν, A_{s} = S^{−1}A, and Ā be a field of fractions of the algebra A. Denote by K, K_{s} complete subcategories of M(A), M(A_{s}) respectively such that objects of K and K_{s} has no torsion. It is easy to prove that categories K and K_{s} are differentially closed. Natural embeddings A ⊂ A_{s} ⊂ Ā induce homomorphisms Λ_{K}^{∗} →^{i} Λ_{K_{s}}^{∗} →^{j} Λ_{M(Ā)}^{∗}. Since Λ_{K}^{∗} and Λ_{K_{s}}^{∗} are torsion-free, homomorphisms i and j have no kernels.

Proposition 12. Under the above described conditions the division lemma holds for the complex (Λ^{∗}, d).

◀ Consider any differential form ω = ∑_{(i_{1},...,i_{k})} h_{(i_{1},...,i_{k})}dg^{i_{1}} ∧ ⋯ ∧ dg^{i_{k}}. Let R̄ =^{def}

K(s, h_{1},…, g^{1} …) and R =^{def} R̄ ⋂ A. Therefore R is a discrete valuation ring, whence the division lemma is true for the complex Λ_{M(R)}^{∗} . As we already know, homomorphism α : Λ_{M(R)}^{∗} → Λ_{M(R̄)}^{∗} has no kernel. The extension R̄ ⊂ Ā is separable, therefore the homomorphism α : Λ_{M(R̄)}^{∗} → Λ_{M(Ā)}^{∗} has no kernel also. Let ω′ = ∑_{(i_{1},...,i_{k})} h_{(i_{1},...,i_{k})}dg_{M(R)}^{i_{1}} ∧

⋯ ∧ dg_{M(R)}^{i_{k}}. If ds ∧ ω = 0 then d_{M(R)}s ∧ ω′ = 0. By the division lemma for the complex Λ_{M(R)}^{∗} , there exists a form λ′ ∈ Λ_{M(R)}^{∗} such that ω′ = d_{M(R)}s ∧ λ′. Set λ = β ∘ α(λ′). Then ω = β ∘ α(ω′) = ds ∧ λ. ▶

3. Residues in the case of positive characteristic

Let char K = p > 0. If n ≡ 0 mod p, then Propositions 3, 6 are not true. For this reason the scheme of constructing the residue homomorphism described above is not valid in the case char K > 0. However one can construct a spectral sequence replacing a residue homomorphism.

Let k ≥ 0. By definition, put

F^{−2k,q} = \{\frac{α}{s^{kp}} | α ∈ Λ^{q−2k−2}\}
F^{−(2k+1),q} = \{\frac{α}{s^{kp+p+1}} + \frac{ds}{s^{kp+p}} ∧ β | α ∈ Λ^{q−2k−1}, β ∈ Λ^{q−2k}\}
F^{k,q} = 0

It is readily seen that dF^{l,q} ⊂ F^{l,q+1}, the filtration is descending, and ⋃_{l} F^{l} = Λ_{S^{−1}K}^{∗}. Using this filtration we can construct a spectral sequence.

Proposition 13. (1) E_{1}^{0,q} = H^{q}(Λ^{∗}).

(2) E_{1}^{−2k,q} = H^{q−2k}(Λ_{s}^{∗}), k > 0. Suppose ω|_{s}, ω ∈ Λ^{q−2k}, is a closed form; then the cohomology class [ω|_{s}] ∈ H^{q−2k}(Λ_{s}^{∗}) corresponds to [\frac{ω}{s^{kp}}] ∈ E_{1}^{−2k,q}.

(3) E_{1}^{−(2k−1),q} = H^{q−2k}(Λ_{s}^{∗}), k > 0. Suppose ω|_{s}, ω ∈ Λ^{q−2k}, is a closed form; then the class [ω|_{s}] ∈ H^{q−2k}(Λ_{s}^{∗}) corresponds to [\frac{ds}{s^{kp−p+1}} ∧ ω] ∈ E_{1}^{−(2k−1),q}.

◀ Statements 1 and 2 are trivial. To prove statement 3 we note that multiplication by s^{p(k−1)} establishes an isomorphism of complexes E_{0}^{−(2k−1),∗} and E_{0}^{−1,∗}. Therefore we may examine only the case k = 1, i.e. we may consider a cohomology group for forms “having singularity on S of order < p” modulo complex Λ^{∗} . The restriction on singularity order allows us transfer word for word the proof of Theorem 1 to this case. ▶

Corollary. d_{1}^{−(2k+1),q} = 0 for k > 0. ▶

Discrete valuation ring. Let, as above, K be a field, A be a discrete valuation ring, ν ⊂ A be corresponding maximal ideal, s be a generator of ν, and Ā be the field of fractions of A. If char K = p > 0 then the division lemma may not be valid. If the algebra A is a finitely generated over K, then Λ^{1} is a free A-module of a finite dimension. Let Ω = {ω_{1}, …, ω_{l}} be a free system of its generators, and T = {t_{1}, …, t_{k}} be a ν-system for Ω. Each of elements t_{j}, 1 ≤ j ≤ k, may be reduced to the form s^{r_{j}}f_{j}, where f_{j} ∉ ν.

Proposition 14. The division lemma is valid iff min{r_{j}} = 0. ▶

Now assume that A is finitely generated over K and the division lemma is valid. Under this conditions one can compute the term E_{∞} of the spectral sequence.

We begin with some notation. Denote a differential form dt_{i_{1}} ∧… ∧ dt_{i_{n}}, t_{i_{m}} ∈ T, by dt_{α}, α = (α_{1},…, α_{k}) ∈ ℤ_{+}^{k}, where

α_{i} = \{ \begin{matrix}1, & i ∈ \{i_{1},…, i_{n}\} \\ 0, & i ∉ \{i_{1},…, i_{n}\}. \end{matrix}

A polinomial t_{1}^{β_{1}} · . . . · t_{k}^{β_{k}} will be denoted by t^{β}, β = (β_{1},…, β_{k}) ∈ ℤ_{+}^{k}. Fix a vector σ ∈ ℤ_{+}^{k}. A differential form ω_{σ} = ∑_{α+β=σ} h_{α}t^{β}dt_{α}, h_{α} ∈ K(Ā^{p}), β_{i} < p ∀i = 1,…, k is called a homogeneous form of the weight σ. Since the extension K(t_{1},…, t_{k}) ⊂ Ā is separable algebraic, it follows that any form ω ∈ Λ_{M(Ā)}^{∗} can be represented as ω = ∑_{σ} ω_{σ}, where ω_{σ} are homogeneous forms of the weight σ. Moreover, if ω ∈ Λ^{∗}, then ∀σ ω_{σ} ∈ Λ^{∗}.

Proposition 15. If ω is a homogeneous form of the weight σ, then dω is a homogeneous form of the weight σ too. ▶

Corollary. Any closed (exact) form is a sum of closed (exact) homogeneous forms. ▶

Proposition 16. If the weight σ of a closed form ω_{σ} does not belong to pℤ_{+}^{k}, then the form ω_{σ} is exact. Closed forms h_{α}t^{pσ−α}dt_{α} are not exact. ▶

Corollary. (1) For B = A, Ā, H^{0}(B; M(B)) = K(B^{p}).

(2) H^{n}(B, M(B)) is a free K(B^{p})-module. Its generators has a form t^{pσ−α}dt_{α}. ▶

Now we can prove a theorem describing the spectral sequence.

Theorem 2. Under the above described conditions, the equality E_{1} = E_{∞} holds.

◀ It is sufficient to prove that each class Θ ∈ E_{1}^{r,q} contains a closed form. For the case r = 0 there is nothing to prove.

Since the division lemma is valid, it follows that there exists ν-system T containing s. Let t_{1} = s. According to Proposition 13 we must prove that for each class Θ̃ ∈ H^{∗}(Λ_{s}^{∗}) there exists a closed form ω ∈ Λ^{∗} such that [ω|_{s}] = Θ. Let ω′ ∈ Θ̃. By the corollary from Proposition 15 we may assume ω′ to be homogeneous. If σ is a weight of the form ω′, then σ_{1} = 0. Otherwise, ω′|_{s} = 0 and Θ̃ = 0. Let us consider two cases.

Case 1: σ ∈ ℤ_{+}^{n}. Then dω′ = 0 and ω′ is the desired form. Case 2: σ ∉ ℤ_{+}^{n}. Since dω′|_{s} = 0, it follows that there exists a form φ′ ∈ Λ^{∗}, φ′ ∉ sΛ^{∗}

such that dω′ = s^{l}φ′, l > 0. A coefficient of the form dω′ at dt_{α} is equal to a_{α}t^{σ−α}, where a_{α} ∈ A^{p} and t^{σ−α} ∉ ν. Therefore l ≡ 0 mod p. Set φ = s^{l−p}φ′, then dω′ = s^{p}φ. Now we note that the form φ is closed and its weight is equal σ. Since σ ∉ ℤ_{+}^{n}, from Proposition

16 it follows that φ is exact. Let ψ ∈ Λ^{∗} be a form such that dψ = φ. Set ω =^{def} ω′ − s^{p}ψ. Then ω|_{s} = ω′|_{s} and dω = dω′ − s^{p}dψ = 0. ▶

Finally we note that a theory of multiple residues can be constructed in just the same way. In this case the de Rham lemma [11] plays a central role.

Acknowledgments. The author is grateful to A. M. Vinogradov for constant attention to this work and to A. V. Samokhin and I. S. Krasil’shchik for their help in translating this paper in English.

References

1. Grothendieck A., The theory of abstract algebraic varieties, Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge Univ. Press, New York, 1960, pp. 103 – 118.

2. Hartshorne R., Residues and duality, Lect. Notes Math. No 20, Springer Verlag, 1966.

3. Berthelot P., Cohomologie cristalline des schemas de caracteristique p > 0, Lect. Notes Math. No 407, Springer Verlag, 1974.

4. Vinogradov M.M., Residue theory on commutative algebras, Uspehi mat. nauk, 35, No 2 (212) (1980), 203 – 204. (Russian)

5. Vinogradov M.M., On De Rham and Spencer algebraic cohomology of smooth manifolds, Doklady Akad. Nauk SSSR, 242, No 5 (1978), 989 – 992 (Russian); English translation in Soviet Math. Dokl. 19, No 5, 1220 – 1224.

6. Vinogradov A.M., The logic algebra for the theory of linear differential operators, Soviet Math. Dokl. 13 (1972), 1058–1062.

7. Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics, 1, Gordon and Breach, 1986.

8. Vinogradov A.M., Some homological systems associated with differential calculus over commutative algebras, Russian Math. Surveys 34 No 6 (1979), 250–255..

9. Suzuki S., Differentials of commutative rings Queen’s papers in pure and appl. math. No 29, Kingston, Queen’s university, 1971,.

10. Pham F., Introduction a l’étude topologique des singularités de Landau, Memorial des sciences mathematiques, fascicule CLXIV, Gauthier-Villars editeur, 1967.

11. De Rham G., Sur la division de formes et de courants par une forme lineare, Comment. Math. Helv. 28 (1954), 346 – 352.

12. Vinogradov M.M., Residues without duality, ”Works of seminar ’Algebra and geometry differential equations”, VINITI, No 858/86B, Moscow, 1986, pp. 183 – 201, (in Russian).

13. Krasil’shchik I. S., Calculus over commutative algebras: a concise user guide, Preprint DIPS 1/96.

Institute of Economic Forecasting, 32, Krasikova str., Moscow 117418, Russia E-mail address: vin@ecfor.msk.su