The Diffiety Institute Preprint Series

Preprint DIPS–3/96 October 1, 1996

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Poincaré δ-lemma for smooth algebras

by

I.S.Krasil’shchik

Poincaré δ-lemma for smooth algebras

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

Abstract

The Poincaré δ-lemma (stable triviality of Spencer cohomology groups) for smooth algebras is proved.

Key words and phrases: Spencer cohomology, smooth algebras, algebraic models.

1991 AMS Subject Classification: 58G05, 58H10.

We prove here a well-known fact: Spencer δ-complex for a linear differential operator is acyclic in stable dimensions [2, 5]. A very elegant proof belonging to J.-P. Serre was published in [3] and is purely algebraic. We show here that the result itself is of a purely algebraic nature, provided the initial setting be chosen in appropriate way.

Basic constructions related to linear differential operators over commutative algebras [4, 6] are assumed to be known below.

SYMBOLS. We start with defining this setting. Let K be a commutative ring with a unit and A be a unitary K-algebra.

Definition 1 Algebra A is called smooth, if the module Λ^{1}(A) of one-forms is projective and of finite type.

We fix a smooth algebra A and consider the category M(A) of all A-modules. The first consequence of smoothness is

Proposition 1 For any A-module P there exists an isomorphism

D_{i}(P) ≃ P ⊗_{A} D_{i}(A),

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

while

D_{i}(A) ≃ ⋀^{i} D_{1}(A) = \underbrace{D_{1}(A) ∧ … ∧ D_{1}(A)}_{i times} .

where D_{i}(P) denotes the module of P -valued i-derivations of the algebra A.

Proof. Let Λ^{i} denote the module of i-forms over A. Since Λ^{i} = \underbrace{Λ^{1} ∧ … ∧ Λ^{1}}_{i times},

this module is projective and of final type too. But for any A-module P one has D_{i}(P) ≃ hom_{A}(Λ^{i}, P) and consequently

D_{i}(P) ≃ hom_{A}(Λ^{i}, P) ≃ P ⊗_{A} hom_{A}(Λ^{i}, A) ≃ P ⊗_{A} D_{i}(A).

The second statement is proved in a similar manner.

Consider now two A-modules P, Q and bimodules Diff_{k}^{(+)}(P, Q), k ≥ 0, of kth order linear differential operators acting from P to Q. Due to embeddings Diff_{k−1}^{(+)}(P, Q) ⊂ Diff_{k}^{(+)}(P, Q) quotient modules

S_{k}(P, Q) =^{def} Diff_{k}^{(+)}(P, Q)/Diff_{k−1}^{(+)}(P, Q)

are defined and are called symbol modules of operators acting from P to Q. The image |Δ| = |Δ|_{k} of an operator Δ ∈ Diff_{k}^{(+)}(P, Q) under the natural projection Diff_{k}^{(+)}(P, Q) → S_{k}(P, Q) is called the symbol of this operator. We use the notations S_{k}(P) =^{def} S_{k}(A, P) and S_{k} =^{def} S_{k}(A) as well as S_{∗}(P) =^{def} ∑_{k≥0} S_{k}(P), S_{∗} =^{def} ∑_{k≥0} S_{k}.

If R is the third A-module, then the composition of differential operators induces an A-module homomorphism S_{l}(Q, R) ⊗_{A} S_{k}(P, Q) → S_{k+l}(P, R) defined by

|Δ|_{l} · |∇|_{k} =^{def} |Δ ∘ ∇|_{k+l}, Δ ∈ Diff_{l}^{(+)}(Q, R), ∇ ∈ Diff_{k}^{(+)}(P, Q). (1)

In particular, S_{∗}(P, P) is an associative A-algebra. The main result of this subsection is

Proposition 2 Let A be a smooth algebra and P be an A-module. Then:

(i) The algebra S_{∗} is commutative and equality (1) determines an S_{∗}-module structure in S_{∗}(P).

(ii) If A is an algebra over the field of rational numbers, then S_{∗}(P) ≃ P ⊗_{A}S_{∗}, while

S_{∗} ≃ S^{⊗}(S_{1}) = S^{⊗}(D_{1}(A)),

where S^{⊗}(·) denotes the symmetric algebra of corresponding module.

Proof. The first part is trivial (see [4, 6, 7]).

To prove the second part note that for smooth algebras

S^{⊗k}D_{1}(A) ≃ S^{⊗k}((Λ^{1})^{∗}) ≃ (S^{⊗k}(Λ^{1}))^{∗},

where (·)^{∗} =^{def} hom_{A}(·, A) and S^{⊗k} is the kth symmetric power. Let s = |Δ| ∈ S_{k}(P), where Δ ∈ Diff_{k}^{(+)}(A, P). Define an A-homomorphism μ_{P} : S_{k}(P) → hom_{A}(S^{⊗k}(Λ^{1}), P) by setting

μ_{P} (s)(da_{1} · . . . · da_{k}) =^{def} (δ_{a_{1}} ∘ … ∘ δ_{a_{k}})(Δ), (2)

where δ_{a}(Δ) = aΔ − Δ · a, a ∈ A, and dots in the definition of μ_{P} denote multiplication in S^{⊗}(Λ^{1}).

We start with proving that μ is well-defined. First note that by definition of differential operators, (δ_{a_{1}}∘…∘δ_{a_{k}})(Δ) ∈ Diff_{0}(A, P) = hom_{A}(A, P) = P. Hence, the right-hand side of (2) lies in P. Show now that if a′_{1}, …, a′_{k} is another set of elements such that da_{i} = da′_{i}, then the right-hand side of (2) remains the same. In fact, for any Δ ∈ Diff_{k}^{(+)}(A, P) the correspondence a ↦ δ_{a}(Δ) is a derivation lying in D_{1}(Diff_{k−1}^{(+)}(A, P)). But due to the universal property of the de Rham differential d : A → Λ^{1}, any X ∈ D_{1}(Q), Q being an A-module, is representable in the form X = f_{X} ∘ d for some f_{X} ∈ hom_{A}(Λ^{1}, Q). Consequently, da = 0 implies X(a) = 0. Finally, μ_{P} (s) is independent on the choice of Δ, |Δ| = s: if Δ′ is another representative of s, then Δ − Δ′ ∈ Diff_{k−1}^{(+)}(A, P) and therefore (δ_{a_{1}} ∘ … ∘ δ_{a_{k}})(Δ − Δ′) = 0 for any set a_{1}, …, a_{k}.

The map μ_{P} is injective. Let μ_{P} (s) = 0. Then from (2) it follows that (δ_{a_{1}} ∘…∘δ_{a_{k}})(Δ) = 0 for any set a_{1}, …, a_{k}. Hence, Δ is an operator of the order k − 1 which means that s = 0.

Let now σ = p⊗X_{1}·. . .·X_{k} ∈ P ⊗_{A}S^{⊗k}(D_{1}(A)), where p ∈ P and X_{1}, …, X_{k} ∈ D_{1}(A). Set

μ′_{P} (σ) =^{def} \frac{1}{k!} |1_{p} ∘ X_{1} ∘ … ∘ X_{k}| ,

where 1_{p} ∈ hom_{A}(A, P) acts as 1_{p}(a) = ap. Obviously, μ_{P} ∘ μ′_{P} = id and consequently μ_{P} is an isomorphism. It is also obvious that μ_{P} preserves composition.

δ-COMPLEX. Let P be an A-module. Then the Spencer Diff-complex of the order i + k is defined [4]:

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

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

Due to embeddings D_{i}(Diff_{k−1}^{+}(A, P)) ↪ D_{i}(Diff_{k−1}^{+}(A, P)) one obtains the quotient complex

… → \frac{D_{i}(Diff_{k}^{+}(A, P))}{D_{i}(Diff_{k−1}^{+}(A, P))} ⟶^{δ_{i,k}} \frac{D_{i−1}(Diff_{k+1}^{+}(A, P))}{D_{i−1}(Diff_{k}^{+}(A, P))} → …

and since the functors D_{i}(·) are exact from the right when A is a smooth algebra, this complex transforms to

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

… → D_{1}(S_{i+k−1}(P)) → S_{i+k}(P) → 0.  (3)

Definition 2 Let A be a smooth algebra and P be an A-module. Complex (3) is called the (i + k)-order Spencer δ-complex of the module P.

We shall also consider the total complex corresponding to (3)

… → D_{i}S_{∗}(P) →^{δ_{i}} D_{i−1}S_{∗}(P) → … → D_{1}S_{∗}(P) →^{δ_{1}} S_{∗}(P) → 0.

Denote by H_{δ}^{i,k}(P) the cohomology of (3) at the term D_{i}S_{k}(P) and set H_{δ}(P) = ∑_{i,k} H_{δ}^{i,k}(P).

¿From the definition of Spencer Diff-operators [4] one immediately obtains

Proposition 3 Let A be a smooth algebra and P be an A-module. Then: (i) The operator δ_{1} : D_{1}S_{∗}(P) ≃ S_{∗}(P) ⊗ S_{1} → S_{∗}(P) coincides with the composition map; (ii) For i > 1 and any Δ ∈ D_{i}S_{∗}(P) one has (δ_{i}(Δ))a = δ_{i}(Δ(a)), a ∈ A.

An easy consequence of Proposition 3 is an explicit description of Spencer δoperators:

Corollary 4 Let A be a smooth algebra over the field of rational numbers and P be an A-module. In accordance with Propositions 1 and 3 represent D_{i}S_{k}(P) in the form P ⊗_{A} S^{⊗k}(D_{1}(A)) ⊗ ⋀^{i} D_{1}(A) and consider Δ = p ⊗ X_{1} · . . . · X_{k} ⊗ Y_{1} ∧ … ∧ Y_{i} ∈ D_{i}S_{k}(P), X_{1}, …, X_{k}, Y_{1}, …, Y_{i} ∈ D_{1}(A). Then

δ_{i,k}(Δ) = ∑_{α=1}^{i}(−1)^{α−1}p⊗X_{1} ·. . .·X_{k} ·Y_{α} ⊗Y_{1} ∧…∧Y_{α−1} ∧Y_{α+1} ∧…∧Y_{i}.(4)

Consider now a linear differential operator ∇ : P → Q of the order l. Then it generates an S_{∗}(A)-module homomorphism smbl_{∗}(∇) : S_{∗}(P) → S_{∗}(Q) by smbl_{∗}(∇)(|Δ|_{k}) =^{def} |∇∘Δ|_{k+l} for any Δ ∈ Diff_{k}^{(+)}(A, P). Since D_{i}(·) are functors, we also get S_{∗}-module homomorphisms σ_{i}(∇) =^{def} D_{i}(smbl_{∗}(∇)) : D_{i}S_{∗}(P) → D_{i}S_{∗}(Q) and from Proposition 3 it follows that the diagram

[figure: \begin{matrix}D_{i}S_{k}(P) & D_{i−1}S_{k+1}(P) \\ D_{i}S_{k+l}(Q) & D_{i−1}S_{k+l+1}(Q) \end{matrix}; σ_{i}(∇); δ_{i,k}; σ_{i−1}(∇); δ_{i,k+s}]

is commutative. Consequently, the quotient complex

… → C_{i}(∇) →^{δ_{i}} C_{i−1}(∇) → … → C_{1}(∇) → C_{0}(∇) → 0 (5)

is defined, where C_{i}(∇) =^{def} D_{i}S_{∗}(Q)/σ_{i}(∇)(D_{i}S_{∗}(P)).

Definition 3 The group H_{δ}(∇) = ∑_{i,k} H_{δ}^{i,k}, where H_{δ}^{i,k} is the cohomology group of complex (5) at the term D_{i}S_{k+l}(Q)/σ_{i}(∇)(D_{i}S_{k}(P)), is called Spencer δ-coho-mology of the operator ∇.

ACTIONS. Define now an action

ν_{P} : D_{i}S_{k}(P) ⊗_{A} D_{j}S_{l}(A) → D_{i+j}S_{k+l}(P)

of modules D_{∗}S_{∗}(A) on D_{∗}S_{∗}(P). We do it using induction with respect to i + j. Namely, for i + j = 0 we take for ν_{P} the above defined module structure of S_{∗}(P) over S_{∗}. We use the notation ν_{P} (Δ ⊗ Δ′) =^{def} Δ · Δ′. If i + j > 0 and a ∈ A, we set

(Δ · Δ′)(a) = Δ · Δ′(a) + (−1)^{j}Δ(a) · Δ′, (6)

where Δ ∈ D_{i}S_{∗}(P), Δ′ ∈ D_{j}S_{∗}(A).

Proposition 5 The action ν_{P} is well-defined by (6). Moreover, ν_{A} determines a graded commutative S_{∗}-algebra structure in D_{∗}S_{∗}(A), i.e.

Δ · Δ′ = (−1)^{ij}Δ′ · Δ, Δ ∈ D_{i}S_{∗}(A), Δ′ ∈ D_{j}S_{∗}(A),

while ν_{P} is the structure of D_{∗}S_{∗}(A)-graded module in D_{∗}S_{∗}(P).

The proof reduces to straightforward check of definitions.

We shall also need the following properties of ν_{P}.

Proposition 6 Let A be a smooth algebra. Then:

(i) For any operator ∇ : P → Q the map σ_{∗}(∇) : D_{∗}S_{∗}(P) → D_{∗}S_{∗}(Q) is a homomorphism of D_{∗}S_{∗}(A)-modules.

(ii) For any Δ ∈ D_{i}S_{∗}(P) and Δ′ ∈ D_{j}S_{∗}(A) one has

δ_{i+j}(Δ · Δ′) = δ_{i}(Δ) · Δ′ + (−1)^{i}Δ · δ_{j}(Δ′). (7)

Corollary 7 The groups H_{δ}(∇) inherit the action ν_{P}. To be precise, H_{δ}(A) is a graded commutative ∑_{k} H_{δ}^{0,k}-algebra, while H_{δ}(∇) is a graded H_{δ}(A)-module.

δ-LEMMA. We prove now the main result known as the Poincaré δ-lemma. The proof is based on the following two lemmas.

Lemma 1 Let A be a smooth Noetherian algebra over the field of rational numbers. Then S_{∗} is a Noetherian algebra and D_{∗}S_{∗}(P) is a Noetherian module for any finitely generated A module P.

Proof. We use standard fact concerning Noetherian rings and modules, [1].

Since the algebra A is smooth, the module D_{1}(A) is finitely generated. Let x_{1}, …, x_{n} be its generators. Due to Proposition 2, S_{∗} is a quotient algebra of the polynomial ring A[x_{,} …, x_{n}] and consequently is Noetherian.

Now, in accordance to Propositions 1 and 2 one has D_{∗}S_{∗}(P) ≃ P ⊗_{A}D_{∗}(A)⊗_{A} S_{∗}. But the module P ⊗_{A} D_{∗}(A) is finitely generated and hence D_{∗}S_{∗}(P) is Noetherian.

Note now that elements of S_{∗} are cocycles of complex (3). Therefore, S_{∗} acts on cohomology group H_{δ}(∇) (Proposition 6).

Corollary 8 Under the conditions of Lemma 1 H_{δ}(∇) is a Noetherian S_{∗}-module.

Lemma 2 Under the conditions of Lemma 1 the ideal of S_{∗} generated by S_{1} = D_{1}(A) acts trivially on H_{δ}(∇).

Proof. It follows from Proposition 3 (i) and equality (4).

Theorem 9 (Poincaré δ-lemma) Let A be a smooth Noetherian algebra over the field of rational numbers and ∇ : P → Q be a linear differential operator, where P, Q are A-modules, Q being finitely generated. Then there exists an integer k_{0} such that H_{δ}^{i,k}(∇) = 0 for all i ≥ 0 and k ≥ k_{0}.

Proof. ¿From Lemma 2 it follows that H_{δ}^{k}(∇) = ∑_{i} ∑_{α≤k} H_{δ}^{i,k}(∇) are S_{∗}- submodules. Hence, due to Lemma 1, there exists a k_{0} such that H_{δ}^{k_{0}}(∇) = H_{δ}^{k_{0}+1}(∇) = . . ..

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