Journal of Algebra

COHOMOLOGY OPERATORS DEFINED BY A DEFORMATION

LUCHEZAR L. AVRAMOV AND LI–CHUAN SUN

Introduction

A lot of (co)homological information on modules over a commutative ring R is encoded in terms of composition products of various Ext and Tor modules. Two main difficulties in using this information are that the resulting algebra and module structures are seldom finite, and the products are almost never commutative. One significant exception occurs when R = Q/(x) for a Koszul-regular set x = {x_{1}, …, x_{c}} in a commutative ring Q; we think of Q as a deformation of R over a regular base.

Indeed, Gulliksen [8] then constructs a set of commuting operators {X_{1}, …, X_{c}} acting on Ext_{R}^{∗}(M, N) by increasing degrees by 2 and on Tor_{∗}^{R}(M, N) by decreasing degrees by 2, making Ext and Tor graded modules over a polynomial ring S = R[X_{1}, …, X_{c}] with variables of cohomological degree 2. He proves that if Ext_{Q}^{n}(M, N) is noetherian over Q for each n and vanishes for n ≫ 0, then the graded S–module Ext_{R}^{∗}(M, N) is noetherian. This partly overcomes the first obstacle referred to above.

Under more restrictive conditions on Q and x, Mehta [9] interprets Gulliksen’s operators as composition products: this makes the second difficulty manageable, as the actions of Ext_{R}^{∗}(M, M) and Ext_{R}^{∗}(N, N) on Ext_{R}^{∗}(M, N) are “essentially central.”

These results have provided the basis for extensive studies of the (co)homological properties of modules over local complete intersections by Eisenbud [7], and of more general classes of modules of infinite projective dimension by Avramov [1].

Each of the papers quoted above also uses alternative construction(s) of cohomology operators. It has been repeatedly stated in the literature that they yield the same result, but a close inspection of the arguments supporting such claims has revealed serious defects. On the other hand, no single approach seems to give all the essential properties of the cohomology operators which have been extensively used in [4].

In an attempt to provide complete proofs for the coincidence of the various operators, we were led to two new constructions. Both of them come from viewing R–modules as DG (=differential graded) modules over a Koszul complex K resolving R over Q, and our arguments make a full-fledged use of techniques of DG homological algebra, summarized in Section 1. As a bonus, we construct the operators and establish their main properties directly for complexes of R–modules.

We first introduce operators à la Gulliksen, as connecting homomorphisms. Unlike previous approaches, both module arguments are involved from the start. This is important in proving that “left” and “right” versions of earlier operators agree— at least up to sign—which suffices for the applications in [7], [1].

Our second approach produces cohomology operators from chain-level maps, as did Eisenbud. However, our chain endomorphisms arise not from lifting the differential of a complex of R–modules, but from descending the differential of a DG K–module.

1991 Mathematics Subject Classification. 13D99, 13c40, 18G15.

L.L.A. was partially supported by a grant from the National Science Foundation.

These constructions are presented in Section 2. They are used in Section 3 to provide direct proofs of many formal properties of the cohomology operators, in particular, of their centrality. Section 4 compares our approach to earlier ones, filling in gaps and correcting misconceptions concerning the relations between the various operators. Trying to avoid further inaccuracies, we include sufficient detail for most of the essential computations.

In Section 5 we note that some of the existing proofs of the noetherian nature of Ext_{R}^{∗}(M, N) over the ring of cohomology operators carry over to complexes, and establish a new property of these operators—their primitivity. When k is the residue field of a local ring R, we prove that for certain finite R–modules M the graded module Ext_{R}^{∗}(M, k) is finite over the subalgebra of Ext_{R}^{∗}(k, k) generated by the central and primitive elements of degree 2, even though R itself may not have a deformation. This plays a key role in the investigation of finite CI-dimension in the recent paper [4].

1. Background

This section contains a synopsis of DG homological algebra, taken from [3]. All modules in sight are defined over a fixed commutative ring 𝕜, which is usually suppressed from the notation. In particular, ⊗ stands for ⊗_{𝕜} and Hom for Hom_{𝕜}.

1.1. Complexes and DG modules. Differentials have degree −1 and are ubiquitously denoted ∂. The notation m ∈ M means that m is a homogeneous element of M; we denote its degree by |m|. If m is a cycle, then [m] is its homology class. The functor forgetful of differentials is denoted (−)^{♮}.

Let M and N be complexes. The complex Hom(M, N) has d’th component ∏_{i∈ℤ} Hom(M_{i}, N_{i+d}), and differential ∂(γ) = ∂∘γ − (−1)^{|γ|}γ∘∂. A chain map γ is a cycle in Hom(M, N), so that ∂∘γ = (−1)^{|γ|}γ∘∂. A chain map of degree 0 is called a morphism, and a quasi-isomorphism if it induces an isomorphism in homology; the latter is indicated by the appearance of the symbol of ≃ next to its arrow. The complex M ⊗ N has d’th component ∐_{i}(M_{i} ⊗ N_{d−i}) and differential ∂(m ⊗ n) = ∂(m) ⊗ n + (−1)^{|m|}m ⊗ ∂(n).

A commutative DG algebra A is a complex together with a morphism A⊗A ⟶ A denoted by juxtaposition which has the usual associativity and unit properties, and satisfies ab = (−1)^{|a||b|}ba for all a, b ∈ A and a^{2} = 0 when |a| is odd. An A–module is a complex M, together with a morphism A ⊗ M ⟶ M, satisfying the usual conditions. If A_{i} = 0 for i ≠ 0, then a DG A–module is simply a complex of modules over the ring A_{0}, and we call it accordingly.

For A–modules M and N, the subcomplex Hom_{A}(M, N) of Hom(M, N) consists of those γ for which γ(am) = (−1)^{|γ||a|}aγ(m) for all a ∈ A and m ∈ M; it is an A– module with (aγ)(m) = a(γ(m)) = (−1)^{|γ||a|}γ(am). Dually, the complex M ⊗_{A} N is the quotient of M ⊗N by its subcomplex generated by {(am)⊗n−(−1)^{|a||m|}m⊗ (an) | a ∈ A, m ∈ M, n ∈ N}; it is an A–module, with a(m ⊗ n) = (am) ⊗ n = (−1)^{|a||m|}m ⊗ (an).

1.2. Resolutions. An A–module P is DG-projective if Hom_{A}(P, −) transforms surjective quasi-isomorphisms into surjective quasi-isomorphisms. This is equivalent to the property that Hom_{A}(P, −) preserves all quasi-isomorphisms and the graded A^{♮}–module P^{♮} is projective. In that case (P ⊗_{A} −) preserves quasi-iso-morphisms.

An A–module I is DG-injective if Hom_{A}(−, I) transforms injective quasi-iso-morphisms into surjective quasi-isomorphisms. This is equivalent to the properties that Hom_{A}(−, I) preserves quasi-isomorphisms and the graded A^{♮}–module I^{♮} is injective.

A bounded below (respectively, above) A–module M such that M^{♮} is projective (respectively, injective) over A^{♮} is DG-projective (respectively, DG-injective).

Let X be an arbitrary A–module. There exists a quasi-isomorphism P ⟶^{≃} X from a DG-projective module P; any such morphism is called a DG-projective resolution of X. Dually, X has a DG-injective resolution, that is, a quasi-iso-morphism X ⟶^{≃} I with I DG-injective. For each X we fix a DG-projective resolution ε_{X}^{A}: P_{X}^{A} ⟶^{≃} X and a DG-injective resolution η_{A}^{X}: X ⟶^{≃} I_{A}^{X}.

Let φ: A′ ⟶ A be a homomorphism of DG algebras, let M and N be A– modules, and let M′ and N′ be A′–modules. A φ-contravariant homomorphism ν: N ⟶ N′ is a 𝕜–linear such that ν(φ(a′)n) = (−1)^{|ν||a′|}a′ν(n) for all a′ ∈ A′ and all n ∈ N. A φ-covariant homomorphism μ: M′ ⟶ M is a 𝕜–linear map such that μ(a′m′) = (−1)^{|a′||μ|}φ(a′)μ(m′) for all a′ ∈ A′ and all m′ ∈ M′. In other words, ν and μ are homomorphisms of A′–modules for the structures induced on N and M through φ.

For φ, μ, ν, as above, and a φ–covariant chain map λ: L′ ⟶ L, by 1.2 there exist unique up to homotopy φ–equivariant chain maps λ^{•}, μ^{•}, ν_{•}, such that the squares

\begin{matrix}P_{L′}^{A′} & ⟶^{λ^{•}} & P_{L}^{A} \\ ^{≃}↓ & & ↓^{≃} \\ L′ & ⟶^{λ} & L \end{matrix}  \begin{matrix}P_{M′}^{A′} & ⟶^{μ^{•}} & P_{M}^{A} \\ ^{≃}↓ & & ↓^{≃} \\ M′ & ⟶^{μ} & M \end{matrix}  \begin{matrix}N & ⟶^{ν} & N′ \\ ^{≃}↓ & & ↓^{≃} \\ I_{A}^{N} & ⟶^{ν_{•}} & I_{A′}^{N′} \end{matrix}

commute up to homotopy. They induce a φ-contravariant chain map

Hom_{φ}(μ^{•}, ν_{•}): Hom_{A}(P_{M}^{A}, I_{A}^{N}) ⟶ Hom_{A′}(P_{M′}^{A′}, I_{A′}^{N′}) , γ ↦ (−1)^{|μ|(|γ|+|ν|)}ν_{•}γμ^{•}

and a φ–covariant chain map

λ^{•} ⊗_{φ} μ^{•}: P_{L′}^{A′} ⊗_{A′} P_{M′}^{A′} ⟶ P_{L}^{A} ⊗_{A} P_{M}^{A} , p′ ⊗ q′ ↦ (−1)^{|μ||p′|}λ^{•}(p′) ⊗ μ^{•}(q′)

both of which are unique up to homotopy.

1.3. Derived functors. With the resolutions chosen in 1.2, set Ext_{A}^{∗}(M, N) = H Hom_{A}(P_{M}^{A}, I_{A}^{N}) and Tor_{∗}^{A}(L, M) = H(P_{L}^{A} ⊗_{A} P_{M}^{A}). The morphisms from 1.2 uniquely define homomorphisms of graded 𝕜–modules

Ext_{φ}^{∗}(μ, ν): Ext_{A}^{∗}(M, N) ⟶ Ext_{A′}^{∗}(M′, N′)

and

Tor_{∗}^{φ}(λ, μ): Tor_{∗}^{A′}(L′, M′) ⟶ Tor_{∗}^{A}(L, M)

which make Ext^{∗} and Tor_{∗} functors of three arguments. If φ, λ, μ, ν, are quasiisomorphisms, then Ext_{φ}^{∗}(μ, ν) and Tor_{∗}^{φ}(λ, μ) are isomorphisms.

When A = A_{0} is a ring, the construction yields the usual hyper(co)homology of complexes. By further specialization to modules M = M_{0} and N = N_{0}, it produces the classical derived functors of Hom_{A} and ⊗_{A}.

Let γ ∈ Ext_{A}^{∗}(M, N) and δ ∈ Ext_{A}^{∗}(L, M) be represented by chain maps γ̃: P_{M}^{A} ⟶ I_{A}^{N} and δ̃: P_{L}^{A} ⟶ I_{A}^{M}. By 1.2, choose unique up to homotopy chain maps γ^{•}: P_{M}^{A} ⟶ P_{N}^{A} and δ^{•}: P_{L}^{A} ⟶ P_{M}^{A} such that γ̃ = η_{A}^{N}∘ε_{N}^{A}∘γ^{•} and δ̃ = η_{A}^{M}∘ε_{M}^{A}∘δ^{•}.

The composition product is the degree 0 homomorphism of graded 𝕜–modules

Ext_{A}^{∗}(M, N) ⊗ Ext_{A}^{∗}(L, M) ⟶ Ext_{A}^{∗}(L, N) ,
γ ⊗ δ ↦ γ · δ = [γ̃∘δ^{•}] .

If t is the homology class of a cycle _{∑k}e_{k} ⊗_{A} f_{k} ∈ P_{L}^{A} ⊗_{A} P_{M}^{A}, then

Ext_{A}^{∗}(M, N) ⊗ Tor_{∗}^{A}(L, M) ⟶ Tor_{∗}^{A}(L, N) ,

γ ⊗ t ↦ γ · t = [∑_{k}(−1)^{|γ||e_{k}|}e_{k} ⊗_{A} γ^{•}(f_{k})]

defines a degree 0 homomorphism of graded 𝕜–modules.

Similarly, if u is the homology class of _{∑k}e_{k} ⊗_{A} g_{k} ∈ P_{L}^{A} ⊗_{A} P_{N}^{A}, then

Ext_{A}^{∗}(L, M) ⊗ Tor_{∗}^{A}(L, N) ⟶ Tor_{∗}^{A}(M, N) ,

δ ⊗ u ↦ δ · u = [∑_{k}δ^{•}(e_{k}) ⊗_{A} g_{k}]

is a degree 0 homomorphism of graded 𝕜–modules.

These pairings are associative, and so define a structure of graded 𝕜–algebra on Ext_{A}^{∗}(M, M) with unit given by the class of the map η_{A}^{M}∘ε_{M}^{A}: P_{M}^{A} ⟶ I_{A}^{M}; they make Ext_{A}^{∗}(M, N) into a graded left Ext_{A}^{∗}(N, N)–, right Ext_{A}^{∗}(M, M)–bimodule, while Tor_{∗}^{A}(L, M) becomes a graded left Ext_{A}^{∗}(L, L)–, left Ext_{A}^{∗}(M, M)–bimodule.

2. Construction

We start by describing the setup for much of the discussion in this paper.

2.1. Koszul regularity. Let x = {x_{1}, …, x_{c}} be a set of elements in Q, 𝔞 = (x) the ideal it generates, and R = Q/𝔞 the quotient ring. We denote K the Koszul complex K(x, Q); thus, K^{♮} is the exterior algebra on a free Q–module with basis ξ_{1}, …, ξ_{c} with |ξ_{j}| = 1, and ∂(ξ_{j}) = x_{j} for 1 ≤ j ≤ c. We denote 𝔄 the kernel of the canonical augmentation κ: K ⟶ H_{0}(K) = R.

We assume that x is Koszul-regular that is, that H_{i}(K) = 0 for i ≠ 0; equivalently, κ is a quasi-isomorphism. The cotangent module C = 𝔞/𝔞^{2} is then free over R on \overline{x}, where \overline{x}_{j} = x_{j} + 𝔞^{2}, and for each R–module M and each i ∈ ℤ there are isomorphisms

Ext_{Q}^{i}(R, M) ≅ H_{−i} Hom_{Q}(K, M) = Hom_{Q}(K_{i}, M) ≅ ⋀^{i} Hom_{R}(C, R) ⊗_{R} M

and

Tor_{i}^{Q}(R, M) ≅ H_{i}(K ⊗_{Q} M) = K_{i} ⊗_{Q} M ≅ ⋀^{i} C ⊗_{R} M .

The canonical isomorphisms κ^{∗} = Ext_{κ}^{∗}(M, N) and κ_{∗} = Tor_{∗}^{κ}(L, M) are computed as follows. For each R–module X factor its DG resolutions over K as

P_{X}^{K} ⟶^{(id_{X})^{•}} P_{X}^{K}/𝔄P_{X}^{K} ⟶^{(ε_{X}^{K})^{•}} X ⟶^{(η_{K}^{X})_{•}} (0 : 𝔄)_{I_{K}^{X}} ⟶^{(id_{X})_{•}} I_{K}^{X} .

Note that the intermediate complexes of R–modules are (respectively) DG-projec-tive and DG-injective and the induced maps (ε_{X}^{K})^{•} and (η_{K}^{X})_{•} are quasi-isomor-phisms, then choose ε_{X}^{R} = (ε_{X}^{K})^{•} and η_{R}^{X} = (η_{K}^{X})_{•} as the DG resolutions of X over R.

2.2. Fundamental sequences. We consider for each j the DG subalgebra K^{[j]} ⊂ K generated over K_{0} = Q by ξ_{1}, …, ξ̂_{j}, …, ξ_{c}, and the multiplication map

β_{j}: K ⊗_{K^{[j]}} K ⟶ K  with  β_{j}(b ⊗ c) = bc .

Proposition. Let L, M, N, be K–modules.

(a) The canonical inclusion β_{j}|_{N}^{M} and the degree one map α_{j}|_{N}^{M} defined by

γ ⟼ (m ↦ (−1)^{|γ|}γ(ξ_{j}m) − ξ_{j}γ(m))

yield an exact sequence of chain maps

0 ⟶ Hom_{K}(M, N) ⟶^{β_{j}|_{N}^{M}} Hom_{K^{[j]}}(M, N) ⟶^{α_{j}|_{N}^{M}} Hom_{K}(M, N)

in which α_{j}|_{N}^{M} is surjective if M^{♮} is projective or N^{♮} is injective over K^{♮}. (b) The canonical projection β_{j}|_{LM} and the degree one map α_{j}|_{LM} defined by

l ⊗_{K} m ↦ (−1)^{|l|}l ⊗_{K^{[j]}} (ξ_{j}m) − (ξ_{j}l) ⊗_{K^{[j]}} m

yield an exact sequence of chain maps

L ⊗_{K} M ⟶^{α_{j}|_{LM}} L ⊗_{K^{[j]}} M ⟶^{β_{j}|_{LM}} L ⊗_{K} M ⟶ 0

in which α_{j}|_{LM} is injective if L^{♮} or M^{♮} is flat over K^{♮}.

Proof. To simplify notation, we set A = K^{[j]} and ξ = ξ_{j}.

A direct computation yields equalities

Ker β_{j} = (K ⊗_{A} K)(1 ⊗_{A} ξ − ξ ⊗_{A} 1) = (0 :_{K⊗_{A}K} (1 ⊗_{A} ξ − ξ ⊗_{A} 1)) .

Thus, left multiplication by 1 ⊗_{A} ξ − ξ ⊗_{A} 1 on K ⊗_{A} K defines a degree 1 homomorphism of graded K ⊗_{A} K–modules

α_{j}: K ⟶ K ⊗_{A} K  given by  a ↦ 1 ⊗_{A} (ξa) − ξ ⊗_{A} a

which fits into a fundamental exact sequence

0 ⟶ K ⟶^{α_{j}} K ⊗_{A} K ⟶^{β_{j}} K ⟶ 0 .

More verifications show that the action ((b ⊗_{A} c)γ)(m) = (−1)^{|c||γ|}bγ(cm) gives Hom_{A}(M, N) a structure of K⊗_{A}K–module, and that the action (b⊗_{A}c)(l⊗_{A}m) = (−1)^{|c||l|}bl ⊗_{A} cm produces such a structure for L ⊗_{A} M. Furthermore, if X is a K ⊗_{A} K–module and K acts on X ⊗_{K} M by a(x ⊗ m) = (ax) ⊗ m, then there is a canonical isomorphism

Hom_{(K⊗_{A}K)}(X, Hom_{A}(M, N)) ≅ Hom_{K}(X ⊗_{K} M, N)

given by γ ⟼ ((x ⊗_{K} m) ↦ γ(x)(m)), and a canonical isomorphism

(L ⊗_{A} M) ⊗_{(K⊗_{A}K)} X ≅ L ⊗_{K} (X ⊗_{K} M)

given by (l ⊗ m) ⊗ x ↦ (−1)^{|m||x|}l ⊗ (x ⊗ m). To obtain the exact sequence in (a), apply the functor Hom_{(K⊗_{A}K)}(−, Hom_{A}(M, N)) to the fundamental exact sequence, and set α_{j}|_{N}^{M} = Hom(α_{j}, Hom(M, N)). For the sequence in (b), apply (L⊗_{A} M)⊗_{(K⊗_{A}K)} − to the fundamental sequence, and set α_{j}|_{LM} = (L⊗M)⊗α_{j}. Finally, use the isomorphisms above with X = K to identify the outer terms.

The first isomorphism shows that the sequence in (a) can be obtained in two stages. The first is an application of − ⊗_{K} M to the fundamental exact sequence, with the K–module structure on K ⊗_{A} K given by a(b ⊗ c) = (−1)^{|a||b|}b ⊗ (ac). The map β_{j}^{♮} is then split by a ↦ 1 ⊗ a, so we get an exact sequence of K–modules

0 ⟶ M ⟶^{α_{j}|_{KM}} K ⊗_{A} M ⟶^{β_{j}|_{KM}} M ⟶ 0 .

Thus, α_{j}|_{N}^{M} is surjective if N^{♮} is injective over K^{♮} or M^{♮} is projective over K^{♮}. A similar argument proves the injectivity of α_{j}|_{LM} if L^{♮} or M^{♮} is flat over K^{♮}. □

2.3. Cohomology operators. Let x = {x_{1}, …, x_{c}} ⊂ Q be a Koszul-regular set. Let L, M, N, be complexes of R–modules. We view them as K–modules via κ and introduce shorthand notation for their resolutions over K

U = P_{L}^{K}  V = P_{M}^{K}  Y = I_{K}^{N} .

Then we choose as in 2.1 resolutions over R

E = U/𝔄U = P_{L}^{R}  F = V/𝔄V = P_{M}^{R}  J = (0 : 𝔄)_{Y} = I_{R}^{N} ,

and let ε_{U}: U ⟶ E, ε_{V} : V ⟶ F, and η^{Y} : J ⟶ Y be the canonical maps. By 2.2 there are exact sequences of K–modules

0 ⟶ Hom_{K}(V, Y) ⟶^{β_{j}|_{Y}^{V}} Hom_{K^{[j]}}(V, Y) ⟶^{α_{j}|_{Y}^{V}} Hom_{K}(V, Y) ⟶ 0

and

0 ⟶ U ⊗_{K} V ⟶^{α_{j}|_{UV}} U ⊗_{K^{[j]}} V ⟶^{β_{j}|_{UV}} U ⊗_{K} V ⟶ 0 .

The connecting homomorphisms ð_{j} of the associated long (co)homology exact sequences define operators χ_{j} by commutativity of the squares

Ext_{R}^{i}(M, N) ⟶^{≅} Ext_{K}^{i}(M, N)
χ_{j} ↓  ↓^{ð_{j}}
Ext_{R}^{i+2}(M, N) ⟶^{≅} Ext_{K}^{i+2}(M, N)

Tor_{i}^{K}(L, M) ⟶^{≅} Tor_{i}^{R}(L, M)
ð_{j} ↓  ↓^{χ_{j}}
Tor_{i−2}^{K}(L, M) ⟶^{≅} Tor_{i−2}^{R}(L, M)

where the isomorphisms are the maps Ext_{κ}^{∗}(M, N) and Tor_{∗}^{κ}(L, M) induced by

Hom_{R}(F, J) ⟶^{≃} Hom_{K}(V, Y)  with  γ ↦ η^{Y} ∘γ∘ε_{V}

and

U ⊗_{K} V ⟶^{≃} E ⊗_{R} F with u ⊗_{K} v ↦ ε_{U}(u) ⊗_{R} ε_{V} (v) .

We call χ = (χ_{j})_{1}^{c} the family of cohomology operators defined by x.

For another construction of cohomology operators we study

2.4. DG-projectives. Let P be a DG-projective K–module.

The projectivity of K^{♮}–module P^{♮}, cf. 1.2, implies that of \overline{P} = P^{♮}/K_{+}^{♮}P^{♮} over Q, hence the epimorphism P^{♮} ⟶ \overline{P} is split by a homomorphism σ of graded Q– modules. The degree zero homomorphism of K^{♮}–modules π: K^{♮} ⊗_{Q} Im(σ) ⟶ P^{♮} given by π(a ⊗ p) = ap induces an isomorphism π ⊗_{K^{♮}} Q, so Coker(π) = 0. Thus, we have an exact sequence of graded K^{♮}–modules

0 ⟶ Ker π ⟶ K^{♮} ⊗_{Q} Im(σ) ⟶^{π} P^{♮} ⟶ 0

which splits because P^{♮} is projective. In the exact sequence obtained by applying −⊗_{K^{♮}} Q the map π ⊗_{K^{♮}} Q is bijective, so Ker(π)⊗_{K^{♮}} Q = 0. Due to the nilpotency of the ideal K_{+}^{♮} = (ξ_{1}, …, ξ_{c})K^{♮}, we have Ker(π) = 0, hence π is bijective.

For {h_{1}, …, h_{i}} = H ⊆ Z = {1, …, c} with h_{1} < ⋯ < h_{i}, set ξ_{H} = ξ_{h_{1}} ∧ ⋯ ∧ ξ_{h_{i}} ∈ K. Each v ∈ P can be written in the form v = ∑_{H⊆Z} ξ_{H}⟨v⟩_{H} with uniquely defined ⟨v⟩_{H} ∈ Im(σ), so for each H ⊆ Z we have a Q–linear endomorphism

∂_{H}: Im(σ) ⟶ Im(σ)  with  ∂_{H}(p) = (−1)^{|H|}⟨∂p⟩_{H}

of degree (−|H| − 1). The equation ∂^{2}(p) = 0 then yields

∂_{∅}^{2}(p) = ∑_{j=1}^{c} x_{j}∂_{j}(p) ;  (2.4.1)
∂_{∅}∂_{j}(p) ≡ ∂_{j}∂_{∅}(p) mod 𝔞 Im(σ)  for j = 1, …, c .  (2.4.2)

Note that D = R ⊗_{K} P ≅ Im(σ)/𝔞 Im(σ) = P/𝔄P is a complex of projective R–modules with differential ∂(1 ⊗ p) = 1 ⊗ ∂_{∅}(p), that the maps

τ_{j}^{D}: D ⟶ D  given by  1 ⊗ p ↦ 1 ⊗ ∂_{j}(p)  for j = 1, …, c

are degree −2 chain endomorphisms of D, and that the family τ^{D} = (τ_{j}^{D})_{1}^{c} is uniquely determined by x. It allows for chain level computations of the operators.

Proposition. Let L, M, N, be complexes of R–modules, and let ε_{L}^{R}: E ⟶ L, ε_{M}^{R}: F ⟶ M, η_{R}^{M}: M ⟶ I, be their DG resolutions over R described in 2.3.

If τ_{1}^{E}, …, τ_{c}^{E}: E ⟶ E and τ_{1}^{F} , …, τ_{c}^{F} : F ⟶ F are families of chain endomorphisms described above, then for j = 1, …, c there are equalities

χ_{j} =H Hom_{R}(τ_{j}^{F} , J): Ext_{R}^{∗}(M, N) ⟶ Ext_{R}^{∗}(M, N) ;
χ_{j} = H(τ_{j}^{E} ⊗_{R} F): Tor_{∗}^{R}(L, M) ⟶ Tor_{∗}^{R}(L, M) ;
χ_{j} = − H(E ⊗_{R} τ_{j}^{F}): Tor_{∗}^{R}(L, M) ⟶ Tor_{∗}^{R}(L, M) .

Proof. Along with the chosen in 2.3 DG-projective resolutions V and F of M over K and R, we consider the complex of R–modules F^{[j]} = R ⊗_{K^{[j]}} V. Identifying F^{[j]}^{♮} with F^{♮} ⊕ ξ_{j}F^{♮}, we see that its differential is given by

∂(f′ + ξ_{j}f′′) = ∂(f′) − ξ_{j}(τ_{j}^{F} (f′) + ∂(f′′)) for f′, f′′ ∈ F .

Due to 2.2 we have a commutative diagram of chain maps with exact rows

\begin{aligned}0 & ⟶ & V & ⟶^{α_{j}|_{KV}} & K ⊗_{K^{[j]}} V & ⟶^{β_{j}|_{KV}} & V & ⟶ & 0 \\ & ^{≃}↓ & & ↓^{≃} & & ↓^{≃} \\ 0 & ⟶ & F & ⟶^{α̈_{j}|_{RF}} & F^{[j]} & ⟶^{β̈_{j}|_{RF}} & F & ⟶ & 0 \end{aligned}  (2.4.3)

where the middle vertical arrow is κ ⊗_{K^{[j]}} V and the maps in the bottom row are α̈_{j}|_{RF} (f) = ξ_{j}f and β̈_{j}|_{RF} (f′ + ξ_{j}f′′) = f′.

Apply Hom_{K}(−, Y) to (2.4.3). As Y is DG-injective over K, this functor is exact. Furthermore, on complexes of R–modules it reduces to Hom_{R}(−, J), so we get a commutative diagram with exact rows

0 ⟶ Hom_{K}(V, Y) ⟶^{β_{j}|_{Y}^{V}} Hom_{K^{[j]}}(V, Y) ⟶^{α_{j}|_{Y}^{V}} Hom_{K}(V, Y) ⟶ 0
≃↑  ↑^{≃}  ↑^{≃}
0 ⟶ Hom_{R}(F, J) ⟶^{β̈_{j}|_{J}^{F}} Hom_{R}(F^{[j]}, J) ⟶^{α̈_{j}|_{J}^{F}} Hom_{R}(F, J) ⟶ 0

and chain maps β̈_{j}|_{J}^{F} (γ̃)(f′ + ξ_{j}f′′) = γ̃(f′) and α̈_{j}|_{J}^{F} (γ̃)(f) = (−1)^{|γ|}γ̃(ξ_{j}f). Its commutativity shows that the operator χ_{j} on Ext_{R}^{∗}(M, N) can be computed by the connecting homomorphism ð̈_{j} of the lower row. A direct computation shows that if γ is the class of a chain map γ̃: F ⟶ J, then ð̈_{j}(γ) = [γ̃∘τ_{j}^{F}]. In other words, χ_{j}(γ) = H Hom_{R}(τ_{j}^{F} , J)(γ).

Similar considerations applied to the DG-projective resolutions U and E of L over K and R produce a commutative diagram of chain maps with exact rows

\begin{aligned}0 & ⟶ & E & ⟶^{α̇_{j}|_{ER}} & E^{[j]} & ⟶^{β̇_{j}|_{ER}} & E & ⟶ & 0 \\ & ^{≃}↑ & & ↑^{≃} & & ↑^{≃} \\ 0 & ⟶ & U & ⟶^{α_{j}|_{UK}} & U ⊗_{K^{[j]}} K & ⟶^{β_{j}|_{UK}} & U & ⟶ & 0 \end{aligned}  (2.4.4)

with E^{[j]} = E ⊗_{K^{[j]}} R. In it the middle quasi-isomorphism is U ⊗_{K^{[j]}} κ, and the maps in the top row are α̇_{j}|_{ER}(e) = −ξ_{j}e and β̇_{j}|_{ER}(e′ + ξ_{j}e′′) = e′.

Applying (− ⊗_{K} V) to (2.4.4) and (U ⊗_{K} −) to (2.4.2), we arrive at diagram

0 ⟶ E ⊗_{R} F ⟶^{α̇_{j}|_{EF}} E^{[j]} ⊗_{R} F ⟶^{β̇_{j}|_{EF}} E ⊗_{R} F ⟶ 0
≃↑  ↑^{≃}  ↑^{≃}
0 ⟶ U ⊗_{R} V ⟶^{α_{j}|_{UV}} U ⊗_{K^{[j]}} V ⟶^{β_{j}|_{UV}} U ⊗_{R} V ⟶ 0
≃↓  ↓^{≃}  ↓^{≃}
0 ⟶ E ⊗_{R} F ⟶^{α̈_{j}|_{EF}} E ⊗_{R} F^{[j]} ⟶^{β̈_{j}|_{EF}} E ⊗_{R} F ⟶ 0

which is commutative with exact rows. By the bottom portion, χ_{j} on Tor_{∗}^{R}(L, M) is the connecting homomorphism ð̇_{j} of a short exact sequence of complexes in which α̈_{j}|_{EF} (e ⊗ f) = (−1)^{|e|}e ⊗ (ξ_{j}f) and β̈_{j}|_{EF} (e ⊗ (f′ + ξ_{j}f′′)) = e ⊗ f′. By an easy calculation ð̇_{j}([ ∑_{k} e_{k} ⊗ f_{k}]) = −[∑_{k} e_{k} ⊗ τ_{j}^{F} (f_{k})], hence χ_{j} = − H(F ⊗_{R} τ_{j}^{F}).

The top part shows that the action of χ_{j} on Tor_{∗}^{R}(L, M) is also given by the connecting homomorphism ð_{j} of a sequence with α̇_{j}|_{EF} (e ⊗ f) = −(ξ_{j}e) ⊗ f, and β̇_{j}|_{EF} ((e′+ξ_{j}e′′)⊗f) = e′⊗f. This time, ð_{j}([ ∑_{k} e_{k}⊗f_{k}]) = [ ∑_{k} τ_{j}^{E}(e_{k})⊗f_{k}]. □

Corollary. If η_{R}^{M}: M ⟶ I is a DG-injective resolution over R and 1_{M} = [η_{R}^{M}∘ε_{M}^{R}] ∈ Ext_{R}^{0}(M, M), then χ_{j}(1_{M}) = [η_{R}^{M}∘ε_{M}^{R}∘τ_{j}^{F}] ∈ Ext_{R}^{2}(M, M) for j = 1, …, c. □

An alternative approach to operators in cohomolgy is obtained by using

2.5. DG-injectives. In addition to the assumptions and notation of 2.1 and 2.3, let {ξ_{H}^{∗}}_{H⊆Z} be the Q–basis of Hom_{Q}(K^{♮}, Q) dual to the basis {ξ_{H}}_{H⊆Z} of K^{♮}. The action of K^{♮} on Hom_{Q}(K^{♮}, Q) is described by ξ_{j}ξ_{H}^{∗} = 0 if j ∉ H, and ξ_{j}ξ_{H}^{∗} = (−1)^{r−1}ξ_{H∖j}^{∗} if j = h_{r}, where H = {h_{1}, …, h_{i}} and h_{1} < ⋯ < h_{i}.

Let I be a DG-injective K–module.

The graded module \underline{I} = (0 : K_{+}^{♮})_{I^{♮}} is then injective over Q, due to the injectivity of I^{♮} over K^{♮}, cf. 1.2. Let ρ be a Q–linear splitting of \underline{I} ⊆ I^{♮}, and set ι(y)(a) = (−1)^{|y||a|}ρ(ay) for y ∈ I^{♮} and a ∈ K^{♮}. Therefore, ι: I^{♮} ⟶ Hom_{Q}(K^{♮}, \underline{I}) is a degree 0 homomorphism of graded K^{♮}–modules. As Hom_{K^{♮}}(Q, ι) is bijective, we see that Ker(ι) = 0, so ι is injective and yields an exact sequence of graded K^{♮}–modules

0 ⟶ I^{♮} ⟶^{ι} Hom_{Q}(K^{♮}, \underline{I}) ⟶ Coker ι ⟶ 0 .

The sequence splits, due to the injectivity of I^{♮}. Applying to it Hom_{K^{♮}}(Q, −) we get an exact sequence whose first map is bijective, so Hom_{K^{♮}}(Q, Coker ι) = 0. We remark that if X is a K^{♮}–module such that Hom_{K^{♮}}(Q, X) = 0, then the nilpotency of the graded ideal K_{+}^{♮} implies that X = 0. Thus, Coker ι = 0 and ι is bijective.

The map κ: K^{♮} ⊗_{Q} \underline{I} ⟶ Hom_{Q}(K^{♮}, Q) ⊗_{Q} \underline{I} with a ⊗ w ↦ (−1)^{c|a|}aξ_{Z}^{∗} ⊗ w is a bijective degree (−c) homomorphism of graded K^{♮}–modules. Thus, y ∈ I has a unique expression of the form y = ∑_{H⊆Z} ξ_{H}⟨y⟩_{H} with ⟨y⟩_{H} ∈ W = ι^{−1}κ(1 ⊗_{Q} \underline{I}). In particular, for each y ∈ \underline{I} there is a unique w ∈ W such that y = ξ_{Z}w. For H ⊆ Z we define a Q–linear degree (−|H| − 1) endomorphism

∂_{H}: \underline{I} ⟶ \underline{I}  with  ∂_{H}(y) = (−1)^{c−|H|}ξ_{Z}⟨∂w⟩_{H} .

By a direct computation, the equation ∂^{2}(w) = 0 now yields a congruence

⟨∂⟨∂w⟩_{∅}⟩_{∅} ≡ − ∑_{j=1}^{c} x_{j}⟨∂w⟩_{j} + ∑_{j=1}^{c} ξ_{j}(⟨∂⟨∂w⟩_{j}⟩_{∅} − ⟨∂⟨∂w⟩_{∅}⟩_{j})

mod ( ∑_{{h,j}⊆Z} ξ_{j}𝔞⟨∂w⟩_{{h,j}} + ∑_{|H| ⩾ 2} ξ_{H}W ) .

Multiplication on the left with ξ_{Z} gives ξ_{Z}⟨∂⟨∂w⟩_{∅}⟩_{∅} = −ξ_{Z} ∑_{j=1}^{c} x_{j}⟨∂w⟩_{j}. Thus,

∂_{∅}^{2}(y) = (−1)^{c} ∑_{j=1}^{c} x_{j}∂_{j}(y)

and the congruence above simplifies to

∑_{j=1}^{c} ξ_{j}(⟨∂⟨∂w⟩_{∅}⟩_{j} − ⟨∂⟨∂w⟩_{j}⟩_{∅}) ≡ 0 mod ( ∑_{{h,j}⊆Z} ξ_{j}𝔞⟨∂w⟩_{{h,j}} + ∑_{|H| ⩾ 2} ξ_{H}W ) .

Multiplying the new congruence on the left with ξ_{Z∖j} for j = 1, …, c, we get

∂_{j}∂_{∅}(y) ≡ ∂_{∅}∂_{j}(y) mod ( ∑_{h=1}^{c} 𝔞⟨∂w⟩_{{h,j}}) .

Remark that y is in B = (0 : 𝔞)\underline{I} = (0 : 𝔄)_{I} if and only if w is in (0 : 𝔞)_{W}, and that the restriction of ∂_{∅} to B is equal to the differential ∂ of B. Thus, setting υ_{j} = (−1)^{c}∂_{j} and denoting υ_{j}^{B} the restriction of υ_{j} to B, we obtain a family υ^{B} = (υ_{j}^{B})_{1}^{c} of chain endomorphisms of degree −2 of the complex of R–modules B.

Proposition. Let M and N be complexes of R–modules, let F = V/𝔄V and J = (0 : 𝔄)_{Y} be their DG resolutions as in 2.3, and let υ_{1}^{J}, …, υ_{c}^{J}: J ⟶ J be the family of chain endomorphisms described above. For j = 1, …, c there are equalities

χ_{j} = H Hom_{R}(F, υ_{j}^{J}): Ext_{R}^{∗}(M, N) ⟶ Ext_{R}^{∗}(M, N) .

Proof. Consider the complex of R–modules J^{[j]} = Hom_{K^{[j]}}(R, Y), and identify J^{[j]}^{♮} with ξ_{Z}(0 : 𝔞)_{W}^{♮} ⊕ ξ_{Z∖j}(0 : 𝔞)_{W}^{♮}. The differential of J^{[j]} is then given by

∂(ξ_{Z}w′ + ξ_{Z∖j}w′′) = ∂_{J}(ξ_{Z}w′) + (−1)^{j}υ_{j}(ξ_{Z}w′′) + (−1)^{c−1}ξ_{Z∖j}⟨∂w′′⟩_{∅}

for w′, w′′ ∈ (0 : 𝔞)_{W}^{♮}. Thus, from 2.2 we get a commutative diagram of chain maps

0 ⟶ J ⟶^{β̇_{j}|_{J}^{R}}  J^{[j]}  ⟶^{α̇_{j}|_{J}^{R}} J ⟶ 0
≃↓  ↓^{≃}  ↓^{≃}
0 ⟶ Y ⟶^{β_{j}|_{Y}^{K}} Hom_{K^{[j]}}(K, Y) ⟶^{α_{j}|_{Y}^{K}} Y ⟶ 0

in which the rows are exact, the middle vertical arrow is induced by Hom_{K^{[j]}}(κ, Y) and is a quasi-isomorphism because the external two arrows are, and where

β̇_{j}|_{J}^{R}(ξ_{Z}w) = ξ_{Z}w  and  α̇_{j}|_{J}^{R}(ξ_{Z}w′ + ξ_{Z∖j}w′′) = (−1)^{j}ξ_{Z}w′′ .

The rest of the argument is similar to that for Proposition 2.4. □

3. Properties

We derive the main formal properties of the cohomology operators.

3.1. Naturality. Given a commutative diagram of ring homomorphisms

Q′ ⟶^{ψ} Q
ρ′↓  ↓^{ρ}
R′ ⟶^{φ} R

with 𝔞′ = Ker(ρ′) generated by a Koszul-regular set x′ = x′_{1}, …, x′_{c′} ⊂ Q′, let χ′ = {χ′_{1}, …, χ′_{c′}} be the family of cohomology operators defined by x′, and let

ψ(x′_{i}) = ∑_{j=1}^{c} q_{ij}x_{j}  with  q_{ij} ∈ Q  for 1 ≤ i ≤ c′

be equalities resulting from the inclusion ψ(x′) ⊂ Ker(ρ).

We can now describe the functoriality of the cohomology operators.

Theorem. Let μ: M′ ⟶ M be φ-covariant chain map.

If λ: L′ ⟶ L is a φ-covariant chain map, then

χ_{j}∘ Tor_{∗}^{φ}(λ, μ) = ∑_{i=1}^{c′} q_{ij} Tor_{∗}^{φ}(λ, μ)∘χ′_{i} for j = 1, …, c .

If ν: N ⟶ N′ is a φ-contravariant chain map and q_{ij} = ψ(q′_{ij}), then

∑_{i=1}^{c′} q′_{ij}χ′_{i}∘ Ext_{φ}^{∗}(μ, ν) = Ext_{φ}^{∗}(μ, ν)∘χ_{j} for j = 1, …, c .

Proof. This is immediate from 1.2, Proposition 2.4, and the following lemma. □

Lemma. Let K′ be the Koszul complex K(x′, Q′), and let Ψ: K′ ⟶ K be the homomorphism of DG algebras defined by Ψ(ξ′_{i}) = ∑_{j=1}^{c} q_{ij}ξ_{j} for 1 ≤ i ≤ c′. Let P′ be a DG-projective module over K′, and let τ′ = (τ′_{i})_{1}^{c′} be the family of chain endomorphisms of the complex of R′–modules D′ = P′/𝔄′P′ defined by x′. Given a Ψ-covariant chain map γ: P′ ⟶ P, let 1 ⊗ γ: D′ ⟶ D be the induced covariant chain map of complexes over the ring homomorphism φ: R′ ⟶ R.

The chain maps τ_{j}^{D}∘(1 ⊗ γ) and ∑_{i=1}^{c′} q_{ij}(1 ⊗ γ)∘τ_{i}^{D′} are homotopic.

Remark. In view of Proposition 4.2 the Lemma is equivalent to [7, Proposition 1.7].

Proof. In the notation of 2.4, for each H ⊆ Z = {1, …, c} consider the degree (|γ| − |H|) homomorphisms of graded Q′–modules

γ_{H}: Im(σ′) ⟶ Im(σ)  with  γ_{H}(y′) = (−1)^{|H|}⟨γ(y′)⟩_{H} .

By definition, we have

∂γ(y′) ≡ ∂_{∅}γ_{∅}(y′) − ∑_{j=1}^{c} x_{j}γ_{j}(y′)

+ ∑_{j=1}^{c} ξ_{j}(∂_{∅}γ_{j}(y′) − ∂_{j}γ_{∅}(y′) + ∑_{{h,j}⊆Z} ±x_{h}γ_{{h,j}}(y′)) mod ( ∑_{|H| ⩾ 2} ξ_{H}P ) .

Due to the Ψ-covariance of γ, we get

γ∂′(y′) ≡ γ_{∅}∂′_{∅}(y′) − ∑_{j=1}^{c} ξ_{j}γ_{j}∂′_{∅}(y′) − (−1)^{|γ|} ∑_{i=1}^{c′} Ψ(ξ′_{i}) γ_{∅}∂′_{i}(y′)

= γ_{∅}∂′_{∅}(y′)+∑_{j=1}^{c} ξ_{j}((−1)^{|γ|−1} ∑_{i=1}^{c′} q_{ij}γ_{∅}∂′_{i}(y′)−γ_{j}∂′_{∅}(y′)) mod ( ∑_{|H| ⩾ 2} ξ_{H}P ).

Noting that ∂γ = (−1)^{|γ|}γ∂′, and that γ_{∅} induces the morphism 1 ⊗ γ: D′ ⟶ D, we conclude by comparing the coefficients of ξ_{j} that

τ_{j}^{D}∘(1 ⊗ γ) − ∑_{i=1}^{c′} q_{ij}(1 ⊗ γ)∘τ_{i}^{D′} = ∂γ_{j} − (−1)^{|γ_{j}|}γ_{j}∂′ for j = 1, …, c .

Thus, γ_{j} is a homotopy from τ_{j}^{D}∘(1 ⊗ γ) to ∑_{i=1}^{c′} q_{ij}(1 ⊗ γ)∘τ_{i}^{D′}. □

3.2. Centrality. Cohomology operators (anti)commute with products:

Theorem. For j = 1, …, c there are equalities

χ_{j}(γ) · δ = χ_{j}(γ · δ) = γ · χ_{j}(δ) ∈ Ext_{R}^{∗}(L, N)
χ_{j}(γ) · t = −χ_{j}(γ · t) = −γ · χ_{j}(t) ∈ Tor_{∗}^{R}(L, N)
χ_{j}(δ) · u = χ_{j}(δ · u) = δ · χ_{j}(u) ∈ Tor_{∗}^{R}(M, N)

for all γ ∈ Ext_{R}^{∗}(M, N), δ ∈ Ext_{R}^{∗}(L, M), t ∈ Tor_{∗}^{R}(L, M), u ∈ Tor_{∗}^{R}(L, N).

Proof. In addition to the resolutions F, E, J, described in 2.3, we consider DG resolutions M ⟶^{≃} I_{R}^{M} = I and G = P_{N}^{R} ⟶^{≃} N.

Choose chain maps γ̃: F ⟶ J and δ̃: E ⟶ I such that γ = [γ̃] and δ = [δ̃], cf. 1.2. Extend γ̃ to γ̃_{•} ∈ Hom_{R}(I, J) and lift δ̃ to δ̃^{•} ∈ Hom_{R}(E, F). Lemma 3.1 applied to id_{K} shows that τ_{j}^{F} ∘δ̃^{•} and δ̃^{•}∘τ_{j}^{E} are homotopic, so Proposition 2.4 gives

χ_{j}(γ) · δ = [γ̃∘τ_{j}^{F}] · δ = [γ̃∘(τ_{j}^{F} ∘δ̃^{•})] = [γ̃∘(δ̃^{•}∘τ_{j}^{E})] = [(γ̃∘δ̃^{•})∘τ_{j}^{E}] = χ_{j}(γ · δ) .

On the other hand, by [6, §7, no. 2] and Proposition 2.4 we have

χ_{j}(γ · δ) = χ_{j}([γ̃∘δ̃^{•}]) = [(γ̃∘δ̃^{•})∘τ_{j}^{E}] = [(γ̃_{•}∘δ̃)∘τ_{j}^{E}] = γ · [δ̃^{•}∘τ_{j}^{E}] = γ · χ_{j}(δ) .

With ∑_{k} e_{k} ⊗ f_{k} representing t and γ^{•}: F ⟶ G lifting γ̃, Proposition 2.4 yields

χ_{j}(γ) · t = [γ∘τ_{j}^{F}] · [ ∑_{k} e_{k} ⊗ f_{k}] = [ ∑_{k}(−1)^{(|γ|−2)|e_{k}|}e_{k} ⊗ γ^{•}(τ_{j}^{F} (f_{k}))];

χ_{j}(γ · t) = χ_{j}([∑_{k}(−1)^{|γ||e_{k}|}e_{k} ⊗ γ^{•}(f_{k})]) = −[∑_{k}(−1)^{|γ||e_{k}|}e_{k} ⊗ τ_{j}^{G}(γ^{•}(f_{k}))];

γ · χ_{j}(t) = γ · [ − ∑_{k} e_{k} ⊗ (τ_{j}^{F} (f_{k}))] = −[∑_{k}(−1)^{|γ||e_{k}|}e_{k} ⊗ γ^{•}(τ_{j}^{F} (e_{k}))] .

As τ_{j}^{G}∘γ^{•} and γ^{•}∘τ_{j}^{F} are homotopic by Lemma 3.1, we get the second statement. Finally, if ∑_{k} e_{k} ⊗ g_{k} represents u, then by Proposition 2.4 we have

χ_{j}(δ) · u = [δ̃∘τ_{j}^{E}] · [∑_{k} e_{k} ⊗ g_{k}] = [∑_{k} (δ̃^{•}∘τ_{j}^{E}(e_{k})) ⊗ g_{k}];

χ_{j}(δ · u) = χ_{j}([ ∑_{k} (δ̃^{•}(e_{k})) ⊗ g_{k}]) = [ ∑_{k} (τ_{j}^{F} ∘δ̃^{•}(e_{k})) ⊗ g_{k}];

δ · χ_{j}(u) = δ · [ ∑_{k} (τ_{j}^{E}(e_{k})) ⊗ g_{k}] = [ ∑_{k} (δ̃^{•}∘τ_{j}^{E}(e_{k})) ⊗ g_{k}] .

As τ_{j}^{F} ∘δ̃^{•} and δ̃^{•}∘τ_{j}^{E} are homotopic, we have the last statement. □

3.3. Characteristic homomorphism. Set C = 𝔞/𝔞^{2}, and let x_{1}^{∗}, …, x_{c}^{∗} denote the basis of C^{∗} = (𝔞/𝔞^{2})^{∗} dual to the basis \overline{x} = \{x_{1} + 𝔞^{2}, …, x_{c} + 𝔞^{2}\} of C.

Theorem. The unique homomorphism of R–modules

C^{∗} ⟶ Ext_{R}^{2}(M, M)  with  x_{j}^{∗} ↦ χ_{j}(1_{M})  for j = 1, …, c

is independent of the choice of the Koszul-regular set x. Its image lies in the center of Ext_{R}^{∗}(M, M), and so it extends canonically to a characteristic homomorphism of graded R–algebras from the symmetric algebra S = Sym_{R}(C^{∗}):

ζ_{M}: S ⟶ Ext_{R}^{∗}(M, M)

The maps ζ_{M} and ζ_{N} induce on Ext_{R}^{∗}(M, N) the same S–module structure:

ζ_{N}(x_{j}^{∗}) · γ = χ_{j}(γ) = γ · ζ_{M}(x_{j}^{∗}) for γ ∈ Ext_{R}^{∗}(M, N) ,

The maps ζ_{L} and ζ_{M} induce on Tor_{∗}^{R}(L, M) opposite S–module structures:

ζ_{L}(x_{j}^{∗}) · t = χ_{j}(t) = −ζ_{M}(x_{j}^{∗}) · t for t ∈ Tor_{∗}^{R}(L, M) .

Corollary. On Ext_{R}^{∗}(M, N) and Tor_{∗}^{R}(L, M) there are equalities

χ_{i}χ_{j} = χ_{j}χ_{i}  for i, j = 1, …, c .  □

Remark. When M is an R–module the characteristic homomorphism ζ_{M} coincides with one constructed by Mehta, cf. 4.4; his definition does not seem to generalize to complexes of R–modules. When c = 1, he proves in [9, Proposition 2.3] that the image of ζ_{M} is central and in [9, Proposition 2.4] that over complete local rings the actions on Tor coincide. The theorem extends (and corrects some signs of) loc. cit.

Proof of the Theorem. Theorem 3.2 yields the formulas for the action of ζ_{M}(x_{j}^{∗}), and the first one of them implies that the image of ζ_{M} is in the center of of Ext_{R}^{∗}(M, M).

Along with x, consider a Koszul-regular set x′ ⊂ Q such that (x′) = (x). As both \overline{x} and \overline{x}′ are bases of C, the set x′ also consists of c elements, say x′_{1}, …, x′_{c}. Let τ′_{1}^{F} , …, τ′_{c}^{F} be the chain endomorphisms of 2.4 defined by x′ on the chosen DG projective resolution F of M over R. Writing x′_{i} = ∑_{j=1}^{c} q_{ij}x_{j}, and applying Lemma 3.1 to the identity maps of K and of P = P_{M}^{K}, we get

τ_{j}^{F} = ∑_{i=1}^{c} q_{ij}τ′_{i}^{F} for j = 1, …, c .

It follows that the degree zero R–linear homomorphism (𝔞/𝔞^{2})^{∗} ⟶ Hom_{R}(F, F) with x_{j}^{∗} ↦ τ_{j}^{F} for j = 1, …, c does not depend on the choice of the Koszul-regular set x. In view of the definition of ζ_{M} and Corollary 2.4 we have

ζ_{M}(x_{j}^{∗}) = [η_{R}^{M}∘ε_{M}^{R}∘τ_{j}^{F}] = χ_{j}(1_{M}).

This establishes the independence of ζ_{M} from the choice of x. □

4. Comparison

We present the cohomology operators constructed by Gulliksen [8], Eisenbud [7], Mehta [9], Avramov [1], and compare them to the operators χ_{j} from 2.3.

Throughout this section ρ: Q ⟶ R is a surjective homomorphism of commutative rings with Ker(ρ) = 𝔞 generated by a Koszul-regular set x = x_{1}, …, x_{c}. We consider R–modules L, M, N; as a further adjustment, we assume that DGresolutions U, V, Y of L, M, N over K have been chosen with U_{i} = 0 and V_{i} = 0 for i < 0, respectively Y_{i} = 0 for i > 0. This can always be done, cf. [3], and has the effect that the chosen in 2.3 complexes F = U/𝔄U and E = V/𝔄V are classical projective resolutions of the R–modules L and M, while the complex J = (0: 𝔄)_{Y} is a classical injective resolution of the R–module module N. Finally, recall that

Ext_{R}^{∗}(M, N) = H Hom_{R}(F, J)  and  Tor_{∗}^{R}(L, M) = H(E ⊗_{R} F).

4.1. Gulliksen’s construction [8, p. 176] starts from short exact sequences

0 ⟶ E ⟶^{α′_{j}} E^{[j]} ⟶^{β′_{j}} E ⟶ 0

and

0 ⟶ F ⟶^{α′′_{j}} F^{[j]} ⟶^{β′′_{j}} F ⟶ 0

where E^{[j]} and F^{[j]} are as in 2.4; the maps in the first sequence are given by α′_{j}(e) = (−1)^{|e|}ξ_{j}e and β′_{j}(e′ + ξ_{j}e′′) = e′; the second sequence is defined similarly.

The preceding sequences yield exact sequences of (anti)chain^{1} maps

0 ⟶ Hom_{R}(F, N) ⟶^{Hom_{R}(β′′_{j},N)} Hom_{R}(F^{[j]}, N) ⟶^{Hom_{R}(α′′_{j},N)} Hom_{R}(F, N) ⟶ 0

and

0 ⟶ E ⊗_{R} M ⟶^{α′_{j}⊗_{R}M} E^{[j]} ⊗_{R} M ⟶^{β′_{j}⊗_{R}M} E ⊗_{R} M ⟶ 0

whose connecting homomorphisms we denote ð′_{j}.

1The relations α′_{j}∂ = ∂α′_{j} and α′′_{j}∂ = ∂α′′_{j} are opposite to those in 1.1 for degree 1 chain maps.

For j = 1, …, c, Gulliksen defines X_{j} by commutativity of the squares

Ext_{R}^{i}(M, N) ⟵^{≅} H_{−i} Hom_{R}(F, N)
X_{j} ↓  ↓^{(−1)^{i}ð′_{j}}
Ext_{R}^{i+2}(M, N) ⟵^{≅} H_{−i−2} Hom_{R}(F, N)

with isomorphism H Hom_{R}(F, η_{R}^{N}) and

H_{i}(E ⊗_{R} M) ⟵^{≅} Tor_{i}^{R}(L, M)
(−1)^{i}ð′_{j} ↓  ↓^{X_{j}}
H_{i−2}(E ⊗_{R} M) ⟵^{≅} Tor_{i−2}^{R}(L, M)

with isomorphism H(E ⊗_{R} ε_{M}^{R}). Proposition. There are equalities χ_{j} = X_{j} for j = 1, …, c. Proof. Set ω(f) = (−1)^{|f|}f, and paste (2.4.3) to the commutative diagram

0 ⟶ F ⟶^{α̈_{j}|_{RF}} F^{[j]} ⟶^{β̈_{j}|_{RF}} F ⟶ 0
ω↓  ∥  ∥
0 ⟶ F ⟶^{α′′_{j}} F^{[j]} ⟶^{β′′_{j}} F ⟶ 0

along their common row. The diagram induced by applying Hom_{R}(−, N) implies the desired assertion for Ext_{R}^{∗}(M, N). The argument for Tor_{∗}^{R}(L, M) is similar. □

4.2. Eisenbud’s construction [7, §1] begins with a lifting of a complex of free R–modules (F′, ∂) to a pair (F̃′, ˜∂) of a graded free Q–module F̃′ and its endomorphism ˜∂ of degree −1, such that (F̃′ ⊗_{Q} R, ˜∂ ⊗_{Q} R) ≅ (F′, ∂). It is easy to see that liftings always exist. Fixing one, note that ˜∂^{2} ⊗_{Q} R = 0, hence ˜∂^{2} = ∑_{j=1}^{c} x_{j}t̃_{j}(F̃′, ˜∂) for some family (t̃_{j}(F̃′, ˜∂))_{1}^{c} of degree −2 endomorphisms of the graded Q–module F̃′. As \overline{x} is a basis, t̃_{j}(F̃′, ˜∂) ⊗_{Q} R is a chain endomorphism of F̃′ ⊗_{Q} R. Let t_{j}^{F′} be the corresponding chain endomorphism of F′.

Eisenbud defines ϵ_{j}^{left} for j = 1, …, c by commutativity of the square

Ext_{R}^{i}(M, N) ⟵^{≅} H_{−i} Hom_{R}(F′, N)
ϵ_{j}^{left}↓  ↓^{H Hom_{R}(t_{j}^{F′} ,N)}
Ext_{R}^{i+2}(M, N) ⟵^{≅} H_{−i−2} Hom_{R}(F′, N)

and ϵ_{j}^{right} by that of

H_{i}(L ⊗_{R} F′) ⟵^{≅} Tor_{i}^{R}(L, M)
H(L⊗_{R}t_{j}^{F′})↓  ↓^{ϵ_{j}^{right}}
H_{i−2}(L ⊗_{R} F′) ⟵^{≅} Tor_{i−2}^{R}(L, M)

where F′ ⟶^{≃} M is a free resolution of the R–module M, ϑ: F ⟶ F′ is a comparison of resolutions, and the isomorphisms are H Hom_{R}(ϑ, η_{R}^{N}) and H(ε_{L}^{R}⊗_{R}ϑ). Operators ϵ_{j}^{left} are similarly defined on Tor_{i}^{R}(L, M), by means of a free resolution E′ ⟶^{≃} L.

Proposition. On Ext_{R}^{∗}(M, N) there are equalities χ_{j} = ϵ_{j}^{left} for j = 1, …, c. On Tor_{∗}^{R}(L, M) there are equalities χ_{j} = ϵ_{j}^{left} = −ϵ_{j}^{right} for j = 1, …, c.

Remark. This corrects a sign error in [7, Proposition 1.6], which asserts that ϵ_{j}^{left} = ϵ_{j}^{right}. The argument given there may not be used to compare the two maps: it assumes incorrectly that the canonical projections

E′ ⊗_{R} M ⟵ E′ ⊗_{R} F′ ⟶ L ⊗_{R} F′

commute with the chain endomorphisms

t_{j}^{E′} ⊗_{R} M , t_{j}^{E′} ⊗_{R} F′ + E′ ⊗_{R} t_{j}^{F′} , E′ ⊗ t_{j}^{F′} .

Proof. For an R–free resolution F′ ⟶^{≃} M, pick a homogeneous basis {f′_{λ}}_{λ∈Λ} of the graded R–module F′^{♮}, and let P^{♮} be a graded K^{♮}–module on a basis {g_{λ} | |g_{λ}| = |f′_{λ}|}_{λ∈Λ}. Set ω_{<0} = 0, and assume by induction that a κ-covariant morphism ω_{<h}: P_{<h} = ∐_{|g_{λ}|<h} K^{♮}g_{λ} ⟶ F′_{<h} has been constructed such that ω_{<h}(g_{λ}) = f′_{λ} when |g_{λ}| < h, and ω_{<h} is a surjective quasi-isomorphism.

If |f′_{λ}| = h, then ∂(f′_{λ}) is a cycle in Im(ω_{<h}), hence ∂(f′_{λ}) = ω_{<h}(z_{λ}) for some cycle z_{λ} ∈ P_{<h}. Extend the differential from P_{<h} to P_{≤h} by setting ∂(g_{λ}) = z_{λ}, and extend ω_{<h} to ω_{≤h}: P_{≤h} ⟶ F′_{≤} by ω_{≤h}(g_{λ}) = f′_{λ} for all λ with |g_{λ}| ≤ h. By the choices made ω_{≤h} is a morphism and by the Five–Lemma it is a quasi-isomorphism onto its image. The induction step is complete.

Set P = \underrightarrow{lim}_{h} P_{<h} and ω = \underrightarrow{lim}_{h} ω_{<h}: P ⟶ F′.

Clearly, ω is a κ-covariant surjective morphism, and the induced map R⊗_{K} P ⟶ F′ is an isomorphism of complexes of R–modules. Using the notation of 2.4, we see that the pair (P^{♮}/K_{+}^{♮}P^{♮}, ∂_{∅}) is a lifting of the complex F′. In view of (2.4.1) we can choose ∂_{1}, …, ∂_{c} to be the chain endomorphisms associated with this lifting. This choice leads to equalities of Eisenbud’s endomorphisms with those from 2.4:

t_{j}^{F′} = τ_{j}^{F′}  for j = 1, …, c .  (4.2.1)

Lemma 3.1 with φ = id_{R} and γ = ϑ shows that ϑτ_{j}^{F} and τ_{j}^{F′}ϑ are homotopic, hence

H Hom_{R}(ϑ, η_{R}^{N})∘ H Hom_{R}(t_{j}^{F′}, N) = H Hom_{R}(τ_{j}^{F} , J)∘ H Hom_{R}(ϑ, η_{R}^{N}) .

As H Hom_{R}(τ_{j}^{F} , J) = χ_{j}, we get χ_{j} = ϵ_{j}^{left}. The argument for Tor is similar. □

4.3. Avramov’s construction [1, §1] dualizes that of Eisenbud described in 4.2.

Recall that J is the chosen injective resolution of the R–module N. Choose a graded Q–module J̃ such that the Q-module J̃_{i} is an injective envelope of the Q–module J_{i} for each i. Clearly, Hom_{Q}(R, J̃) ≅ J. Since J̃ is a graded injective Q–module, there exits an endomorphism ˜∂ of degree −1 on J̃ such that ˜∂|_{J} = ∂. By [1, Proposition 1.2] there are degree −2 endomorphisms ũ_{j}(J̃, x): J̃ ⟶ J̃, which satisfy ˜∂^{2} = ∑_{j=1}^{c} x_{j}ũ_{j}(J̃, x), and which restrict to degree −2 chain endomorphisms u_{j}^{J}: J ⟶ J.

The operators ϵ_{j}^{right} studied in [1] are defined by commutativity of the square

Ext_{R}^{i}(M, N) ⟵^{≅} H_{−i} Hom_{R}(M, J)
ϵ_{j}^{right}↓  ↓^{H Hom_{R}(M,u_{j}^{J})}
Ext_{R}^{i+2}(M, N) ⟵^{≅} H_{−i−2} Hom_{R}(M, J)

with isomorphisms H Hom_{R}(ε_{M}^{R}, J).

Proposition. On Ext_{R}^{∗}(M, N) there are equalities χ_{j} = ϵ_{j}^{right} for j = 1, …, c.

Remark. Comparison with Proposition 4.2 shows that ϵ_{j}^{left} = ϵ_{j}^{right}. This is stated in [1, (1.5)], as a consequence of [1, Proposition 1.4]. However, that proposition is not correct: the morphisms Hom_{R}(F, η_{R}^{N}) and Hom_{R}(ε_{M}^{R}, J) do not commute with the actions induced by the chain endomorphisms t_{j}^{F} and u_{j}^{J}.

Proof. With the notation of 2.5, we set \underline{Y} = (0: K_{+}^{♮})_{Y} and consider the endomorphisms ∂_{H}: \underline{Y} → \underline{Y} constructed in 2.5. Because J = (0 : 𝔞)\underline{Y} = (0 : 𝔄)_{Y} and ∂_{J} = ∂_{∅}|_{J} = ∂_{Y} |_{J}, there is an injective homomorphism of graded Q–modules ι̃: J̃ ⟶ \underline{Y} with ι̃|_{J} = id_{J} and ι̃∘˜∂ − ∂_{∅}∘ι̃ = ∑_{j=1}^{c} x_{j}h_{j} for appropriate degree −1 homomorphisms of graded Q–modules h_{j}: J̃ ⟶ \underline{Y}. A direct computation yields

∑_{j=1}^{c} x_{j}(ι̃∘u_{j}^{J}(J, x) − υ_{j}^{J}∘ι̃) = ∑_{j=1}^{c} x_{j}(h_{j}∘˜∂ + ∂_{∅}∘h_{j}) .

As the Koszul complex is exact, u_{j}^{J} is homotopic to υ_{j}^{J}, yielding the middle equality

ϵ_{j}^{right} = H Hom_{R}(F, u_{j}^{J}) = H Hom_{R}(F, υ_{j}^{J}) = χ_{j} ;

the last one comes from Proposition 2.5. □

4.4. Mehta’s construction [9, §1] proceeds from a presentation

0 ⟶ B ⟶ G_{0} ⟶^{ε} M ⟶ 0

with a projective Q–module G_{0}.

In view of the isomorphisms Tor_{1}^{Q}(R, G_{0}) = 0 and Tor_{1}^{Q}(R, M) ≅ C ⊗ M, where C = 𝔞/𝔞^{2} is the cotangent module, it induces an exact sequence of R–modules

E^{M}: 0 ⟶ C ⊗ M ⟶^{ι} B′ ⟶ G′_{0} ⟶^{ε′} M ⟶ 0 .

where −′ = R ⊗_{Q} −. In the notation of cf. 2.1, ι(∑_{k} \overline{x}_{k} ⊗ m_{k}) = 1 ⊗ ∑_{k} x_{k}m′_{k}, where m′_{k} ∈ G_{0} satisfy ε(m′_{k}) = m_{k}. For each j we get an exact sequence

E_{j}^{M}: 0 ⟶ M ⟶ M_{j} ⟶ G′_{0} ⟶^{ε′} M ⟶ 0 .

as the pushout of E^{M} along x_{j}^{∗} ⊗ M: C ⊗_{R} M ⟶ R ⊗_{R} M = M, where x_{1}^{∗}, …, x_{c}^{∗} is the basis of C^{∗} dual to \overline{x}_{1}, …, \overline{x}_{c}.

The iterated connecting homomorphisms induced by E_{j}^{M} yield operators δ_{j}^{right} on Ext_{R}^{∗}(M, N) and δ_{j}^{left} on Tor_{∗}^{R}(L, M) studied by Mehta. Furthermore, the exact sequences E_{j}^{N} and E_{j}^{L} yield operators δ_{j}^{left} on Ext_{R}^{∗}(M, N) and δ_{j}^{right} on Tor_{∗}^{R}(L, M).

Proposition. On Ext_{R}^{∗}(M, N) there are equalities χ_{j} = δ_{j}^{left} = −δ_{j}^{right} for j = 1, …, c.

On Tor_{∗}^{R}(L, M) there are equalities χ_{j} = −δ_{j}^{left} = δ_{j}^{right} for j = 1, …, c.

Remarks. (1) The last statement corrects a sign error in [9, Proposition 2.4], which asserts that δ_{j}^{left} = δ_{j}^{right} on Tor_{∗}^{R}(L, M).

(2) Mehta also considers an analogous exact sequence obtained as the pullback of an injective copresentation of M; when c = 1 he proves that these sequences are congruent with E_{j}^{M}, and hence define the same cohomology operators. His argument in [9, Proposition 2.5] can be extended to arbitrary codimension.

Proof. Let (G, ∂) be a free resolution of M over Q. As x_{j}M = 0, for each j there is a homotopy σ^{j} from x_{j} id_{G} to 0_{G}. Consider the diagram of R–modules

G′′_{1} ⊕ G′_{3} ⟶^{∂′′_{3}} G′′_{0} ⊕ G′_{2} ⟶^{∂′′_{2}} G′_{1} ⟶^{∂′_{1}} G′_{0} ⟶^{ε′} M ⟶ 0
↓  ^{[C⊗ε′ 0]}↓  ↓^{π}  ∥  ∥
0  ⟶ C ⊗ M ⟶^{ι} B′ ⟶ G′_{0} ⟶^{ε′} M ⟶ 0
x_{j}^{∗}⊗M↓  ↓  ∥  ∥
0  ⟶ M ⟶ M_{j} ⟶ G′_{0} ⟶^{ε′} M ⟶ 0

where −′′ = C ⊗_{Q} −, ∂′′_{3} = [ \begin{matrix}C⊗∂_{1} & 0 \\ σ_{1} & ∂′_{3} \end{matrix} ] and ∂′′_{2} = [σ_{0} ∂′_{2}], the middle and lower rows

are the exact sequences E^{M} and E_{j}^{M} above, π comes from the canonical factorization of ∂′_{1} and σ_{i}(∑_{k} \overline{x}_{k} ⊗ g_{k}) = 1 ⊗ ∑_{k} σ_{i}^{k}(g_{k}) for i = 0, 1.

The commutativity of the upper left hand square is clear. The one to its right commutes because ∂_{1}σ_{0}^{j} = x_{j} id_{G}. Thus, the diagram is commutative. It is checked directly that the first row is exact^{2}. We extend it to a free resolution F′ of M over R, and identify Ext_{R}^{∗}(M, M) with H Hom_{R}(F′, I) via the canonical isomorphism.

Let μ_{j}^{M}: F′ ⟶ I be the chain map which on F′_{2} is the composition of x_{j}^{∗} ⊗ \overline{ε} with the embedding η: M ⟶ I_{0}, and is trivial elsewhere. The commutativity of the diagram implies that −[μ_{j}^{M}] is the extension class associated with the exact sequence E_{j}^{M}, cf. [6, §7, no. 3, Definition 1]. Now [6, §7, no. 6, Corollary 3] yields

δ_{j}^{left}(γ) = −γ · [−μ_{j}^{M}] and δ_{j}^{right}(γ) = [−μ_{j}^{N}] · γ for γ ∈ Ext_{R}^{∗}(M, N) .

while [6, §7, no. 8, Corollaries 2 and 4] show that

δ_{j}^{right}(t) = [−μ_{j}^{M}] · t and δ_{j}^{left}(t) = [−μ_{j}^{L}] · t for t ∈ Tor_{∗}^{R}(L, M) .

On the other hand, lifting F′_{2} ⟶ F′_{1} ⟶ F′_{0} to homomorphisms of free Q–modules

G_{0}^{c} ⊕ G_{2} \underrightarrow{˜∂_{2}} G_{1} \underrightarrow{˜∂_{1}} G_{0}

with ˜∂_{1} = ∂_{1} and ˜∂_{2}((g_{1}, …, g_{c}) + g) = ∑_{k} σ_{0}^{k}(g_{k}) + ∂_{2}(g), we see that Eisenbud’s operator t_{j}^{F′} from 4.2 acts on F′_{2} by the formula t_{j}^{F′}( ∑_{k} \overline{x}_{k} ⊗ g_{k} + 1 ⊗ g) = 1 ⊗ g_{j}. This implies [μ_{j}^{M}] = [η_{R}^{M}∘ε_{M}^{R}∘t_{j}^{F′}]. As t_{j}^{F′} = τ_{j}^{F′} by (4.2.1), Corollary 2.4 and Theorem 3.3 yield χ_{j} = δ_{j}^{left} = −δ_{j}^{right} on Ext_{R}^{∗}(M, N) and χ_{j} = δ_{j}^{right} = −δ_{j}^{left} on Tor_{∗}^{R}(L, M). □

5. Applications

We extend the important finiteness result discussed in the introduction and its recent converse to Ext’s of complexes of R–modules. We also establish a new property of the cohomology operators—their primitivity.

For the first two subsections we keep the notation and hypotheses of 2.1, and as in Proposition 3.3 denote S the symmetric algebra of the R–module Hom_{R}(𝔞/𝔞^{2}, R), graded by assigning to its generators cohomological degree 2.

2In fact, it represents the beginning of a free resolution of M over R, constructed from G by Eisenbud [7], but we shall not need the rest of it.

5.1. Finiteness. The R–module Ext_{Q}^{∗}(M, N) is noetherian when Ext_{Q}^{n}(M, N) is noetherian over R (or, equivalently, over Q) for each n, and vanishes for n ≫ 0.

Theorem. Let M and N be complexes of R–modules. The R–module Ext_{Q}^{∗}(M, N) is noetherian if and only if the S–module Ext_{R}^{∗}(M, N) is noetherian.

Comments in place of proof. Eisenbud’s constructions of operators ϵ_{j}^{left}, recalled in 4.2, can easily be extended to complexes, and the proof that they coincide with the cohomology operators χ_{j} carries over. Thus, by Theorem 3.3, we may consider Ext_{R}^{∗}(M, N) as a module over the ring S of Eisenbud operators.

If Ext_{Q}^{∗}(M, N) is noetherian over Q, then the spectral sequence argument given in [2] for finiteness over S shows that Ext_{R}^{∗}(M, N) is noetherian over S. (If H(M) is bounded below and H(N) is bounded above, then Gulliksen’s original proof of [8, Theorem 3.1] can also be applied, in view of Proposition 4.1.)

If M and N are Q–modules and Ext_{R}^{∗}(M, N) is noetherian over S, then it is proved in [4, Theorem 4.2] that Ext_{Q}^{∗}(M, N) is noetherian over Q. That argument only depends on the formal properties of the spectral sequence of [4, Theorem 4.4], whose construction works equally well for complexes. □

Let Z_{R}^{∗}(M) denote the subalgebra of Ext_{R}^{∗}(M, M) generated over R/ ann(M) by the central elements of degree 2. From Theorems 5.1 and 3.3 we get

Corollary. If the Q–module Ext_{Q}^{∗}(M, N) is noetherian, then the Z_{R}^{∗}(M)–Z_{R}^{∗}(N)– bimodule Ext_{R}^{∗}(M, N) is noetherian over each ring. □

5.2. Primitivity. A DG algebra with divided powers (or DGΓ-algebra) over R is a graded commutative DG R–algebra A with A_{i} = 0 for i ≤ 0, equipped with maps A_{2n} ∋ a ↦ a^{(i)} ∈ A_{2ni}, defined for all n ≥ 1 and i ≥ 0, and satisfying some standard conditions, cf. e.g. [5, (1.3)]. A Γ -derivation is an R–linear map θ: A ⟶ A such that

θ(a′a′′) = θ(a′)a′′ + (−1)^{|θ||a′|}a′θ(a′′)  and  θ(a^{(i)}) = θ(a)a^{(i−1)} .

Let S and T be DGΓ algebras over R. An extension of Tate’s procedure [10] of adjunction of Γ-variables to R factors the structure homomorphism R ⟶ S as the of composition R ⟶ E, where E^{♮} = R⟨X⟩ is a free Γ-algebra over R, with a surjective quasi-isomorphism E ⟶^{≃} S. By [5, (2.16)], Tor_{∗}^{R}(S, T) ≅ H_{∗}(E ⊗_{R} T) inherits a structure of Γ-algebra, which is independent of the choice of E.

Theorem. On Tor_{∗}^{R}(S, T) the operator χ_{j} is a Γ -derivation for 1 ≤ j ≤ c.

Proof. Consider a graded Γ-algebra Ẽ = Q⟨X̃⟩, with X̃ a new set of divided power variables in bijective, degree preserving correspondence x̃ ↔ x with those of X. This correspondence extends to an isomorphism of Γ-algebras Ẽ ⊗_{Q} R = E^{♮}. For each x ∈ X, choose in Ẽ an element x′ such that x′ ⊗ 1 = ∂(x), and let ˜∂ be the unique Q–linear Γ-derivation of Ẽ with ˜∂(x̃) = x′. It is easy to see that ˜∂^{2} is a Γ-derivation.

For each x̃ ∈ X̃ there are e_{j}(x̃) ∈ Ẽ such that ˜∂^{2}(x̃) = ∑_{j} x_{j}e_{j}(x̃). Let t̃_{j} be the unique Q–linear Γ-derivation of Ẽ such that t̃_{j}(x̃) = e_{j}(x̃) for all x̃ ∈ X̃. As ˜∂^{2} and ∑_{j=1}^{c} x_{j}t̃_{j} agree on the Γ-generators of of the Q–algebra Ẽ, these two endomorphisms of Ẽ coincide. Thus, t_{j}^{E} = t̃_{j} ⊗_{Q} R is a Γ-derivation of Ẽ ⊗_{Q} R = E, and hence H(t_{j}^{E} ⊗_{R} T) is a Γ-derivation of Tor_{∗}^{R}(S, T).

As χ_{j} = H(t_{j}^{E} ⊗_{R} T) by Proposition 4.2, this proves our assertion. □

We say that γ ∈ Ext_{R}^{n}(S, S) is primitive if γ(tu) = 0 whenever t ∈ Tor_{i}^{R}(S, S) and u ∈ Tor_{n−i}^{R}(S, S) with 0 < i < n, and γ(t^{(i)}) = 0 for all t ∈ Tor_{2n}^{R}(S, S) with n > 0 and i ≥ 2. Primitives form a graded R-submodule Pr_{R}^{∗}(S) ⊆ Ext_{R}^{∗}(S, S), and Pr_{R}^{2}(S) = Ker(π_{R}^{S}), where π_{R}^{S} is the homomorphism of R–modules

π_{R}^{S}: Ext_{R}^{2}(S, S) ⟶ Hom_{R} ( Tor_{1}^{R}(S, S)^{2} , Tor_{0}^{R}(S, S)) , π_{R}^{S}(γ)(t) = γ · t .

We denote P_{R}^{∗}(S) the subalgebra of Ext_{R}^{∗}(S, S) generated over R/ ann(S) by the central primitive elements of degree 2. The last theorem and Corollary 5.1 yield

Corollary. If T is a DG Γ -algebra over R and M is a complex of R–modules such Ext_{Q}^{∗}(M, T) is noetherian over Q, then Ext_{R}^{∗}(M, T) is noetherian over P_{R}^{∗}(T). □

5.3. CI-dimension. Let R be a noetherian local ring with residue field k.

A quasi-deformation of R is a diagram of homomorphisms R ⟶ R′ ⟵ Q of local rings, with R ⟶ R′ faithfully flat and R′ ⟵ Q surjective with kernel generated by a regular sequence. An R–module M ≠ 0 has finite CI-dimension, cf. [4], if R has a quasi-deformation such that M ⊗_{R} R′ has finite projective dimension over Q.

Theorem. If R is a local ring with residue field k and M is a finite R–module of finite CI-dimension, then the left module Ext_{R}^{∗}(M, k) is finite over the k-subalgebra P_{R}^{∗}(k) ⊆ Ext_{R}^{∗}(k, k) generated by the central primitive elements of degree 2.

Proof. We set S = k ⊗_{R} R′ and identify Ext_{R}^{∗}(k, k) ⊗_{k} S with Ext_{R′}^{∗}(S, S) through the canonical isomorphism due to the flatness of R′ over R. We have

Z_{R}^{∗}(k) ⊗_{k} S = Z_{R′}^{∗}(S)

since the center of a tensor product of algebras over a field is equal to the tensor product of the centers of the factors. The flatness of R′ also identifies Tor_{∗}^{R}(k, k)⊗_{k}S and Tor_{∗}^{R′}(S, S) in a way compatible with products in Tor and the action of Ext, so

Pr_{R}(k) ⊗_{R} S = Ker(π_{R}^{k}) ⊗_{R} S = Ker(π_{R′}^{S}) = Pr_{R′}(S)

and we conclude that

P_{R}^{∗}(k) ⊗_{k} S = P_{R′}^{∗}(S) .

Identifying Ext_{R}^{∗}(M, k)⊗_{k} S with Ext_{R′}^{∗}(M ⊗_{R} R′, S), and noting that by Corollary 5.2 the latter is a noetherian module over P_{R′}^{∗}(S), we conclude by flat descent that Ext_{R}^{∗}(M, k) is noetherian over P_{R}^{∗}(k). □

References

[1] L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71–101.

[2] L. L. Avramov, R.-O Buchweitz, Homological algebra modulo a regular sequence, with special attention to codimension two, preprint, 1998.

[3] L. L. Avramov, H.-B. Foxby, S. Halperin, Differential graded homological algebra, in preparation.

[4] L. L. Avramov, V. G. Gasharov, I. V. Peeva, Complete intersection dimension, Publ. Math. I.H.E.S. 86 (1997), 67-114.

[5] L. L. Avramov, S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, Algebra, Algebraic Topology, and their Interactions, (J.- E. Roos, ed.), Lecture Notes in Math. 1183, Springer-Verlag, Berlin, 1986, pp. 1–27.

[6] N. Bourbaki, Algèbre, Chapitre X. Algèbre Homologique, Masson, Paris, 1980.

[7] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.

[8] T. H. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183.

[9] V. Mehta, Endomorphisms of complexes and modules over Golod rings, Thesis, University of California, Berkeley, Calif., 1976.

[10] J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–25.

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U.S.A.

E-mail address: avramov@math.purdue.edu

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U.S.A.

E-mail address: sunlc@puma.mt.att.com