MODULES WITH PRESCRIBED COHOMOLOGICAL SUPPORT

LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR

To Phil Griffith, algebraist and friend.

Abstract. A cohomological support, Supp_{A}^{∗}(M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the A-module Ext_{R}^{∗}(M, M) is noetherian and Ext_{R}^{i}(M, R) = 0 for i ≫ 0, then every closed subset of Supp_{A}^{∗}(M) is the support of some finitely generated R-module. This theorem is shown to specialize to realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. It is also used to produce large families of finitely generated modules of finite projective dimension over commutative local noetherian rings.

Contents

Introduction  1
1. Graded rings  3
2. Cohomological supports  5
3. Realizability by modules  8
4. Bialgebras and Hopf algebras  10
5. Associative algebras  12
6. Commutative local rings  13
References  15

Introduction

In a seminal paper, [21], Quillen used methods from algebraic geometry to study cohomology rings of finite groups. His ideas and techniques have led to the appearance of a number of highly developed theories, which provide insight into the structure of an algebraic object through some geometric ‘variety’ attached to it. Use of these invariants has been crucial to progress on a number of difficult problems.

For applications, a ‘variety theory’ needs to posess certain formal properties. One of them guarantees that ‘homologically similar’ modules, like all syzygy modules of a given module, share the same variety. Thus, modules with different varieties are expected to exhibit quantifiable differences in homological behavior, so a description of all varieties produced by a given theory is important for classification purposes.

Date: February 22, 2007. 2000 Mathematics Subject Classification. 13D03, 13D05, 13H10, 13D40, 16E40, 20C05, 20J06. Key words and phrases. support varieties, realizability, complete intersections, Hopf algebras. Research partly supported by NSF grants DMS 0201904 (L.L.A), DMS 0602498 (S.I.).

The prototype theory applies to all finite dimensional represenations of a finite group; see [8] for a detailed exposition. It has been extended to representations of finite dimensional cocommutative Hopf algebras, [13, 24]. Parallel theories have been constructed for finitely generated modules over finite dimensional self-injective algebras, [12, 22], and over local complete intersection rings [1, 2]. Historically, in each concrete case the proofs of the formal properties of a theory and of the relevant realizability theorem have involved delicate arguments specific to that context.

We are interested in modules over a fixed associative ring R. The vehicle for passing from algebra to geometry is provided by a choice of commutative graded ring A of central cohomology operations on the derived category of R. In the examples above there are natural candidates for A: the even cohomology ring of the group (or Hopf algebra); the even subalgebra of the Hochschild cohomology of the associative algebras; the polynomial ring of Gulliksen operators over the complete intersection. However, other choices are possible, and sometimes are desirable.

For each pair (M, N) of R-modules the graded group Ext_{R}^{∗}(M, N) has a natural structure of graded A-module. The cohomological support of (M, N) is the set

Supp_{A}^{∗}(M, N) = {𝔭 ∈ Proj A | Ext_{R}^{∗}(M, N)_{𝔭} ≠ 0}

where Proj A is the space of all essential homogeneous prime ideals in A with the Zariski topology. The set Supp_{A}^{∗}(M, M) is called the cohomological support of M.

The principal contribution of this work is a method for constructing modules with prescribed cohomological supports. Part of our main result reads as follows:

Theorem 1. Let R be a noetherian ring and let M and N be finite R-modules, such that the graded A-module Ext_{R}^{∗}(M, N) is noetherian.

If Ext_{R}^{i}(M, R) = 0 for all i ≫ 0, then for every closed subset X ⊆ Supp_{A}^{∗}(M, N) there exist finite R-modules M_{X} and N_{X} (with N_{X} = M_{X} when N = M) such that

Supp_{A}^{∗}(M_{X}, N) = X = Supp_{A}^{∗}(M, N_{X}) .

Suitable specializations of Theorem 1 yield known realizibility results over finite dimensional algebras: See Section 4 for Hopf algebras and Section 5 for associative algebras. Earlier proofs are modeled on Carlson’s Tensor Product Theorem [11] for varieties over group algebras; they rely heavily on the nature of R (in the first case) or on that of A (in the second). Our argument in Section 3, based on work in Section 2, requires few structural restrictions on R and none on A.

Another application of Theorem 1 goes into a completely different direction:

Theorem 2. Let (Q, 𝔮, k) be a commutative noetherian local ring and f a Q-regular sequence of length c contained in 𝔮^{2}.

For R = Q/Qf and k̄ an algebraic closure of k there exists a map

V : \{ \begin{matrix}isomorphism classes [M] \\ of finite R-modules with \\ proj dim_{Q} M < ∞ \end{matrix}\} ⟶ \{ \begin{matrix}closed algebraic \\ sets X ⊆ ℙc−1 ; k̄ \\ defined over k \end{matrix}\}

with the following properties:

(1) V is surjective.

(2) V ([M]) = ∅ if and only if proj dim_{R} M < ∞.

(3) V ([M]) = V ([Ω_{n}^{R}(M)]) for every syzygy module Ω_{n}^{R}(M).

(4) V ([M]) = V ([M/xM]) for every M-regular sequence x in R.

This result is surprising. Indeed, it exhibits large families of modules of finite projective dimension over any ring Q with depth Q ≥ 2, contrary to a commonly held perception that finite projective dimension is ‘rare’ over singular commutative rings. Furthermore, the remaining statements ascertain that modules mapping to distinct closed cones in k̄^{c} cannot be linked by any sequence of standard operations known to preserve finite projective dimension.

Theorem 2 is proved in Section 6. With a little extra input: a descent result of independent interest, we deduce from it a recent result on the existence of cohomological varieties for modules over complete intersection local rings.

In this paper varieties of modules are discussed in the broader context of varieties of complexes. The resulting marginal technical complications are easily offset by a gain in flexibility: We realize a given set first as the cohomological support of a bounded complex, using constructions whose effect is easy to track. To shown that this set is also the support of a module we use ‘syzygy complexes’, a new notion.

This paper is part of an ongoing study of cohomological supports of modules over general associative rings. In [5] we focus on proving existence of variety theories with desirable properties under a small set of conditions on a ring, its module(s), and a ring of central cohomological operators. The properties that have to be established are clarified in [9] by Benson, Iyengar, and Krause, who investigate a notion of support for compactly generated triangulated categories.

1. Graded rings

In this section we describe notation and terminology used throughout the paper.

1.1. Graded modules. Let A be a commutative ring that is non-negatively graded: A = ⨁_{i∈ℤ} A^{i} with A^{i}A^{j} ⊆ A^{i+j} and A^{i} = 0 for i < 0.

Modules over A are ℤ-graded: M = ⨁_{j∈ℤ} M^{j} with A^{i}M^{j} ⊆ M^{i+j}. We say that an A-module M is finite if it is finitely generated, and that it is eventually zero if M^{j} = 0 holds for all j ≫ 0. The annihilator of M is the set

ann_{A} M = {a ∈ A | aM = 0} .

This is a homogeneous ideal in A, so A/ ann_{A} M is naturally a graded ring, and M is a graded module over it. When M is noetherian so is A/ ann_{A} M, so modulo ann_{A} M every ideal in A is generated by finitely many homogeneous elements.

1.2. Supports. Let Spec A be the space of prime ideals of A, with the Zariski topology. For an A-module M, set Supp_{A} M = {𝔭 ∈ Spec A | M_{𝔭} ≠ 0}. We set

Proj A = {𝔭 ∈ Spec A | 𝔭 homogeneous and 𝔭 ⊉ A^{⩾1}};
SuppA^{+} M = Supp_{A} M ∩ Proj A .

The following properties of graded A-modules L, M, and N follow from the definition of support and the exactness of localization.

(1) If M is eventually zero, then SuppA^{+} M = ∅.

(2) If L ⟶^{ι} M ⟶^{ε} N is an exact sequence, then

SuppA^{+} M ⊆ SuppA^{+} L ∪ SuppA^{+} N ;

equality holds when ι is injective and ε is surjective.

(3) For each i ∈ ℤ, one has SuppA^{+} (M^{⩾i}) = SuppA^{+} M .

(4) If some M^{⩾n} is finite, say, if M is eventually noetherian, then

SuppA^{+} M = {𝔭 ∈ Proj A | 𝔭 ⊇ ann_{A}(M^{⩾i})}

holds for every i ≥ n; in particular, SuppA^{+} M is a closed subset of Proj A.

(5) If the A-modules M and N are finite, then

SuppA^{+}(M ⊗_{A} N) = SuppA^{+} M ∩ SuppA^{+} N .

In some cases, supports have a natural geometric interpretation.

1.3. Varieties. Let k be a field. Assume that the commutative graded ring A has A^{0} = k and is generated over k by finitely many homogeneous elements of positive degree. Let k̄ be an algebraic closure of k, and for each graded A-module M set

V_{A}(M) = (SuppĀ (M ⊗_{k} k̄) ∩ Max Ā) ∪ {Ā^{⩾1}}

where Ā denotes the ring A ⊗_{k} k̄ and Max Ā the set of its maximal ideals.

Let M be a finite graded A-module. The subset V_{A}(M) of Max Ā then is closed in the Zariski topology; it is also k-rational and conical, in the sense that it can be defined by homogeneous elements in A. Furthermore, the Nullstellensatz implies that each one of the sets V_{A}(M) and SuppA^{+} M determines the other.

1.4. Derived categories. Let R be an associative ring. We let D(R) denote the (full) derived category of left R-modules. We write ≃ to indicate a quasi-isomorphism of complexes; these are the isomorphisms in D(R). The symbol ≅ is reserved for isomorphisms of complexes, and ≡ for homotopy equivalences.

We recall, mostly following [4], basic concepts from DG homological algebra.

1.5. Semiprojective complexes. A complex P of R-modules is called semiprojective if Hom_{R}(P, −) preserves surjective quasi-isomorphisms; equivalently, if P is a complex of projective modules and Hom_{R}(P, −) preserves quasi-isomorphisms. Every complex P of projective R-modules with P_{i} = 0 for all i ≪ 0 is semiprojective.

Every quasi-isomorphism of semiprojective complexes is a homotopy equivalence. Every surjective quasi-isomorphism to a semiprojective complex has a left inverse. Every semiprojective complex with zero homology has the form cone(id^{B}) where B is a complex of projective modules with zero differential.

Let M be a complex of R-modules. A semiprojective resolution of M is a quasiisomorphism P → M from a semiprojective complex P. Every complex M has one, and it is unique up to homotopy equivalence.

1.6. Cohomology. Let M and N be complexes of R-modules, and let P → M and Q → N be semiprojective resolutions. For each i ∈ ℤ the abelian group

Ext_{R}^{i}(M, N) = H_{−i}(Hom_{R}(P, Q)) = H^{−i}(Hom_{R}(P, Q))

is independent of the choice of semiprojective resolutions P and Q. We set

Ext_{R}^{∗}(M, N) = ⨁_{i∈ℤ} Ext_{R}^{i}(M, N) .

This is a graded module over R^{c}, the center of the ring R.

Composition of homomorphisms turns Hom_{R}(Q, Q) and Hom_{R}(P, P) into DG algebras over R^{c}, and Hom_{R}(P, Q) is a left DG module over the first and a right DG module over the second. These structures are compatible, so Ext_{R}^{∗}(N, N) and Ext_{R}^{∗}(M, M) become graded R^{c}-algebras and Ext_{R}^{∗}(M, N) a left-right graded module over them. Again, these structures do not depend on choices of resolutions.

2. Cohomological supports

In this section R is an associative ring.

2.1. Cohomology operations. A ring of central cohomology operations is a commutative graded ring A equipped for each M ∈ D(R) with a homomorphism

ζ_{M}: A ⟶ Ext_{R}^{∗}(M, M)

of graded rings, such that for all N ∈ D(R), ξ ∈ Ext_{R}^{∗}(M, N), and a ∈ A one has

(2.1.1)  ξ · ζ_{M}(a) = ζ_{N}(a) · ξ .

For N = M this formula implies that ζ_{M}(A) is in the center of Ext_{R}^{∗}(M, M).

We assume that A is non-negatively graded and, in addition, A^{i} = 0 for i odd or 2A = 0. This hypothesis covers existing examples and avoids sign trouble.

2.2. Scalars. Using the standard identifications of rings

Ext_{R}^{∗}(R, R) = Hom_{R}(R, R) = R^{o} ,

where R^{o} denotes the opposite ring of R, one sees from (2.1.1) that the homomorphism of rings ζ_{R}: A → R^{o} maps every element a ∈ A^{0} to the center of R^{o}. Identifying that center and R^{c}, formula (2.1.1) shows that the action of a on Ext_{R}^{∗}(M, N) coincides with the maps induced by left multiplication with ζ_{R}(a) on M or on N.

For the next definition we use the notion of support introduced in (1.2).

2.3. Cohomological supports. Let A be a graded ring of central cohomology operations, as above. For each pair (M, N) in D(R) we call the subset

Supp_{A}^{∗}(M, N) = SuppA^{+} (Ext_{R}^{∗}(M, N)) ⊆ Proj A .

the cohomological support of the pair. The cohomological support of M is the set

Supp_{A}^{∗}(M) = Supp_{A}^{∗}(M, M) .

Given a class of complexes C of D(R), we write Thick_{R}(C) for its thick closure, that is to say, the intersection of the thick subcategories of D(R) containing C.

Theorem 2.4. Let M and N be complexes of R-modules.

If the graded A-module Ext_{R}^{∗}(M, N) is noetherian, then for every closed subset X ⊆ Supp_{A}^{∗}(M, N) there exist complexes M_{X} ∈ Thick_{R}(M) and N_{X} ∈ Thick_{R}(N), such that the following equalities hold:

X = Supp_{A}^{∗}(M_{X}, N) = Supp_{A}^{∗}(M_{X}, N_{X}) = Supp_{A}^{∗}(M, N_{X}) .

When N = M one can take N_{X} = M_{X}.

The proof appears at the end of the section. Some of the preparatory material is used repeatedly throughout the paper.

2.5. Functoriality. Let M, M′, M′′ and N, N′, N′′ be complexes of R-modules.

We let Σ: D(R) → D(R) denote the shift functor, defined by (ΣM)_{n} = M_{n−1} and ∂^{ΣM} = −∂^{M}. On the other hand, the dth twist of a graded A-module M is the graded module M(d) having M(d)^{j} = M^{d+j} for all j.

There are then canonical isomorphisms of graded A-modules:

Ext_{R}^{∗}(ΣM, N) ≅ Ext_{R}^{∗}(M, N)(−1) ;
Ext_{R}^{∗}(M, ΣN) ≅ Ext_{R}^{∗}(M, N)(1) ; (2.5.1)
Ext_{R}^{∗}(M′ ⊕ M′′, N) ≅ Ext_{R}^{∗}(M′, N) ⊕ Ext_{R}^{∗}(M′′, N) ;
Ext_{R}^{∗}(M, N′ ⊕ N′′) ≅ Ext_{R}^{∗}(M, N′) ⊕ Ext_{R}^{∗}(M, N′′) . (2.5.2)

Exact triangles M′ → M → M′′ → and N′ → N → N′′ → in D(R) induce exact sequences of graded A-modules

(2.5.3)  [figure: Ext_{R}^{∗}(M′′, N); Ext_{R}^{∗}(M, N); Ext_{R}^{∗}(M′, N); Ext_{R}^{∗}(M′′, N)(1); Ext_{R}^{∗}(M, N)(1); Ext_{R}^{∗}(M, N′); Ext_{R}^{∗}(M, N); Ext_{R}^{∗}(M, N′′); Ext_{R}^{∗}(M, N′)(1); Ext_{R}^{∗}(M, N)(1)]

Indeed, basic properties of the functor Hom_{D(R)}(−, −) show that when N (respectively, M) is fixed the canonical isomorphisms of graded R^{c}-modules are linear for the action of Ext_{R}^{∗}(N, N) on the left (respectively, of Ext_{R}^{∗}(M, M) on the right). All of them are then A-linear because of the centrality of A, see (2.1.1).

Putting together the remarks in 1.2 and 2.5, one gets:

Lemma 2.6. In the notation of 2.5 the following statements hold.

(2.6.0) Supp_{A}^{∗}(M, N) = ∅ when Ext_{R}^{∗}(M, N) is eventually zero .

(2.6.1) Supp_{A}^{∗}(ΣM, N) = Supp_{A}^{∗}(M, N) = Supp_{A}^{∗}(N, ΣN) .

Supp_{A}^{∗}(M′ ⊕ M′′, N) = Supp_{A}^{∗}(M′, N) ∪ Supp_{A}^{∗}(M′′, N) .
Supp_{A}^{∗}(M, N′ ⊕ N′′) = Supp_{A}^{∗}(M, N′) ∪ Supp_{A}^{∗}(M, N′′) .

(2.6.2)

Supp_{A}^{∗}(M, N) ⊆ Supp_{A}^{∗}(M′, N) ∪ Supp_{A}^{∗}(M′′, N) .
Supp_{A}^{∗}(M, N) ⊆ Supp_{A}^{∗}(M, N′) ∪ Supp_{A}^{∗}(M, N′′) .

(2.6.3)

The exact sequences (2.5.3) imply the following statement:

Lemma 2.7. Let M be a complex of R-modules.

The full subcategory of D(R) whose objects are the complexes L with Ext_{R}^{∗}(M, L) (respectively, Ext_{R}^{∗}(L, M)) noetherian over A is thick. □

2.8. Mapping cone. Let M be a complex of R-modules. For each φ ∈ A^{d} the morphism ζ_{M}(φ): M → Σ^{d}M defines a unique up to isomorphism exact triangle

(2.8.1)  M ⟶^{ζ_{M} (φ)} Σ^{d}M ⟶ M//φ ⟶ .

Let N be a complex of R-modules and set M = Ext_{R}^{∗}(M, N). By (2.5.3) and (2.5.1), the triangle above yields an exact sequence of graded A-modules

(2.8.2)  M(−d − 1) ⟶ M(−1) ⟶ Ext_{R}^{∗}(M//φ, N) ⟶ M(−d) ⟶ M .

The definition of composition products shows that the maps on both ends are given by multiplication with φ, so from (2.8.2) one can extract an exact sequence

(2.8.3) 0 ⟶ (M/Mφ)(−1) ⟶ Ext_{R}^{∗}(M//φ, N) ⟶ (0 :_{M} φ)(−d) ⟶ 0 .

For a sequence φ = φ_{1}, …, φ_{n} of homogeneous elements in A, the complex

M//φ = (M//φ′)//φ_{n} , where φ′ = φ_{1}, …, φ_{n−1} ,

is defined uniquely up to isomorphism in D(R). Iterated references to (2.8.1) yield

(2.8.4)  M//φ ∈ Thick_{R}(M) .

Example 2.9. If φ_{1}, …, φ_{n} are in A^{0}, then ζ_{M}(φ_{i}) is the homothety M → M defined by the central element z_{i} = ζ_{R}(φ_{i}) ∈ R; see (2.2). Thus, in D(R) one has M//φ ≃ M ⊗_{R^{c}} K(z), where K(z) is the classical Koszul complex on z = z_{1}, …, z_{n}.

Proposition 2.10. Let M and N be complexes of R-modules and φ = φ_{1}, …, φ_{n} a sequence of homogeneous elements in A.

If the graded A-module Ext_{R}^{∗}(M, N) is eventually noetherian, then so are the A-modules Ext_{R}^{∗}(M//φ, N), Ext_{R}^{∗}(M, N//φ), and Ext_{R}^{∗}(M//φ, N//φ), and one has

Supp_{A}^{∗}(M//φ, N) = Supp_{A}^{∗}(M//φ, N//φ) = Supp_{A}^{∗}(M, N//φ)

= Supp_{A}^{∗}(M, N) ∩ SuppA^{+}(A/Aφ) .

Proof. Clearly, it suffices to treat the case when φ has a single element, φ. From the exact sequence (2.8.3), one sees that Ext_{R}^{∗}(M//φ, N) is eventually noetherian.

Set M = Ext_{R}^{∗}(M, N). The inclusion below holds because (0 :_{M} φ) is a submodule of M and a module over A/Aφ, while the equality comes from 1.2(5):

SuppA^{+}(0 :_{M} φ) ⊆ SuppA^{+} M ∩ SuppA^{+}(A/Aφ) ;
SuppA^{+} (M/Mφ) = SuppA^{+} M ∩ SuppA^{+}(A/Aφ) .

The exact sequence (2.8.3) and 1.2(2) now imply an equality

Supp_{A}^{∗}(M//φ, N) = SuppA^{+} M ∩ SuppA^{+}(A/Aφ) .

A similar argument shows that Ext_{R}^{∗}(M, N//φ) is eventually noetherian and yields

Supp_{A}^{∗}(M, N//φ) = SuppA^{+} M ∩ SuppA^{+}(A/Aφ) .

The last equality to be proved is a formal consequence of those already available. □

Proof of Theorem 2.4. Set I = ann_{A} M. From 1.2(3) and 1.2(4) we get

Supp_{A}^{∗}(M, N) = SuppA^{+} M = SuppA^{+}(A/I)

As M is a noetherian the closed subset X in SuppA^{+}(A/I) has the form

X = SuppA^{+}(A/I) ∩ SuppA^{+}(A/Aφ) .

for some finite set φ of homogeneous elements of A; see 1.1. These equalities yield

X = Supp_{A}^{∗}(M, N) ∩ SuppA^{+}(A/Aφ) .

By (2.8.4), the complexes M_{X} = M//φ and N_{X} = N//φ are in Thick_{R}(M ⊕ R) and Thick_{R}(N ⊕ R), respectively. Proposition 2.10 yields equalities

X = Supp_{A}^{∗}(M_{X}, N) = Supp_{A}^{∗}(M_{X}, N_{X}) = Supp_{A}^{∗}(M, N_{X}) .

Evidently, when M = N our constructions give N_{X} = M_{X}. □

3. Realizability by modules

In this section R is an associative ring and A is a ring of central cohomology operations on D(R), see 2.1. The principal result in this section is a partial enhancement of Theorem 2.4. It contains Theorem 1 from the introduction.

Existence Theorem 3.1. Let R be a left noetherian ring, M and N complexes of R-modules with H(M) and H(N) finite, and X ⊆ Supp_{A}^{∗}(M, N) a closed set.

(1) If the graded A-module Ext_{R}^{∗}(M, N) is eventually noetherian (and X is irreducible), then there exists a finite (and indecomposable) R-module M_{X} with

X = Supp_{A}^{∗}(M_{X}, N) and M_{X} ∈ Thick_{R}(M ⊕ R) .

(2) If, furthermore, Ext_{R}^{∗}(M, R) is eventually zero (and X is irreducible), then there exists a finite (and indecomposable) R-module N_{X} with

X = Supp_{A}^{∗}(M, N_{X}) and N_{X} ∈ Thick_{R}(N ⊕ R) .

When N = M one may choose N_{X} = M_{X}.

In the next four items R may be any associative ring. The arguments below use properties of semiprojective complexes recalled in 1.5.

Lemma 3.2. If π: P → Q be a quasiisomorphism of semiprojective complexes of R-modules, then for each integer n there is a homotopy equivalence

P_{⩾n} ⊕ Σ^{n}Q′_{n−1} ≡ Q_{⩾n} ⊕ Σ^{n}P′_{n−1}

where P′_{n−1} and Q′_{n−1} are projective R-modules.

Proof. Assume first that π is surjective. It has then a left inverse, hence P ≅ Q⊕E with E = Ker(π). This implies E is semiprojective and H(E) = 0, and hence E = cone(id^{P′}) for some complex of projectives P′ with ∂^{P′} = 0. Thus, one gets

P_{⩾n} ≅ Q_{⩾n} ⊕ cone(id^{P′_{⩾n}}) ⊕ Σ^{n}P′_{n−1} .

As H(id^{P′_{⩾n}}) is zero, the canonical map P_{⩾n} → Q_{⩾n} ⊕ Σ^{n}P′_{n−1} is a quasiisomorphism, and so a homotopy equivalence. This proves the assertion for π surjective.

In general, π factors as P → F ⟶^{ψ} Q with F = P ⊕Σ^{−1}cone(id^{Q}) and ψ the sum of π and the canonical surjection Σ^{−1}cone(id^{Q}) → Q. Thus, ψ is a surjective quasiisomorphism of semi-projective complexes, and so is the canonical map F → P. The already settled case yields projective modules P′ and Q′ and homotopy equivalences

P_{⩾n} ⊕ Σ^{n}Q′ ← F → Q_{⩾n} ⊕ Σ^{n}P′ .  □

3.3. Syzygy complexes. Let M be a complex of R-modules and P a semiprojective resolution. For each integer n the complex Σ^{−n}(P_{⩾n}) is independent of the choice of P, in a sense explained by the preceding lemma. We call any such complex an nth syzygy complex of M over R, and let Ω_{n}^{R}(M) denote any one of them.

The nth syzygy complex of an R-module is isomorphic, in D(R), to an nth syzygy module in some projective resolution; this follows part (1) of the next lemma.

Lemma 3.4. Let M be a complex of R-modules and set s = sup{i| H_{i}(M) ≠ 0}. For an integer n the following assertions hold.

(1) If n ≥ s then Ω_{n}^{R}(M) is quasiisomorphic to H_{0}(Ω_{n}^{R}(M)).

(2) When R is noetherian and the R-module H(M) is finite one can find

Ω_{n}^{R}(M) ∈ Thick_{R}(M ⊕ R) .

Proof. Let P → N be a semiprojective resolution, such that Ω_{n}^{R}(M) = Σ^{−n}(P_{⩾n}).

(1) For i ≥ n + 1 one has isomorphisms H_{i}(P) ≅ H_{i}(P_{⩾n}) ≅ H_{i}(M) = 0, the first one of which comes from the exact sequence of complexes

0 → P_{<n} → P → P_{⩾n} → 0 .

(2) One may choose P with each P_{i} finite and P_{i} = 0 for i ≪ 0. One then has P_{<n} ∈ Thick_{R}(R) and P ≃ M, so we conclude from the exact sequence above. □

Lemma 3.5. Let M and N be complexes of R-modules with bounded homology. For each integer m one has Supp_{A}^{∗}(M, N) = Supp_{A}^{∗}(Ω_{m}^{R}(M), N). If Ext_{R}^{∗}(M, F) is eventually zero for each free module F, then one has

Supp_{A}^{∗}(M, N) = Supp_{A}^{∗}(M, Ω_{n}^{R}(N)) for each n ∈ ℤ .

Proof. We prove the second assertion; the argument for the first one is similar.

Replacing N with a semiprojective resolution, we may assume that each N_{j} is projective and N_{j} = 0 for all j ≪ 0. An elementary argument using the exact sequence (2.5.3) shows that the class of complexes of R-modules N for which Ext_{R}^{∗}(M, N) is eventually zero is a thick subcategory of D(R). It contains the free modules, by hypothesis, and hence it contains all bounded complexes of projective modules. Therefore, one gets Ext_{R}^{i}(M, N_{<n}) = 0 for all i ≫ 0.

The inclusion N_{<n} ⊆ N gives rise to an exact triangle N_{<n} → N → N_{≥n} → in D(R). Again by (2.5.3), it induces an exact sequence of graded A-modules

Ext_{R}^{∗}(M, N_{<n}) ⟶ Ext_{R}^{∗}(M, N) ⟶ Ext_{R}^{∗}(M, N_{≥n}) ⟶ Ext_{R}^{∗}(M, N_{<n})(1)

which yields Ext_{R}^{⩾i}(M, N) ≅ Ext_{R}^{⩾i}(M, N_{⩾n}) in view of the preceding discussion. It remains to remark that one has N_{⩾n} = Σ^{−n}Ω_{n}^{R}(N), since N is semiprojective. □

In the proof of Theorem 3.1 we use also the following elementary result.

Lemma 3.6. Let R be a noetherian ring and M a complex such that the R-module H(M) is finite. Let C be a complex in Thick_{R}(M ⊕ R).

(1) The R-module H(C) is noetherian.

(2) If N is a bounded complex, then Ext_{R}^{≫0}(M, N) = 0 implies Ext_{R}^{∗}(C, N) = 0.

(3) When F is a projective R-module Ext_{R}^{≫0}(M, R)=0 implies Ext_{R}^{≫0}(C, F)=0.

Proof. Complexes L with H(L) finite form a thick subcategory of D(R). As it contains M and R, it does Thick_{R}(M ⊕ R) as well. This proves (1). A similar argument settles (2). For M and F as in (3) there is an isomorphism

Ext_{R}^{∗}(M, F) ≅ Ext_{R}^{∗}(M, R) ⊗_{R} F ,

because M has a degreewise finite semiprojective resolution. Thus, Ext_{R}^{≫0}(M, R) = 0 implies Ext_{R}^{≫0}(M, F) = 0. Now part (2) yields Ext_{R}^{≫0}(C, F) = 0, as desired. □

Proof of Theorem 3.1. Using Theorem 2.4, choose complexes C ∈ Thick_{R}(M) and D ∈ Thick_{R}(N), with Supp_{A}^{∗}(C, N) = X = Supp_{A}^{∗}(M, D).

By Lemma 3.6(1), the R-modules H(C) and H(D) are noetherian, so one has H_{⩾n}(C) = 0 = H_{⩾n}(D) for some integer n. Moreover, Lemma 3.4(2) yields syzygy complexes Ω_{n}^{R}(C) ∈ Thick_{R}(M ⊕ R) and Ω_{n}^{R}(D) ∈ Thick_{R}(N ⊕ R). Another application of Lemma 3.6(1) shows that the following R-modules are finite:

M_{X} = H_{0}(Ω_{n}^{R}(C)) and N_{X} = H_{0}(Ω_{n}^{R}(D))

Lemma 3.4(1) yields Ω_{n}^{R}(C) ≃ M_{X} and Ω_{n}^{R}(D) ≃ N_{X}. Being a noetherian module, M_{X} is a finite direct sum of indecomposables. When X is irreducible one can replace M_{X} by an appropriate indecomposable direct summand without changing the set Supp_{A}^{∗}(M_{X}, N); see (2.6.2). A similar argument applies to N_{X}.

(1) follows from the equalities below, the second one coming from Lemma 3.5:

X = Supp_{A}^{∗}(C, N) = Supp_{A}^{∗}(Ω_{n}^{R}(C), N) = Supp_{A}^{∗}(M_{X}, N) .

(2) As Ext_{R}^{∗}(M, R) is eventually zero, so is Ext_{R}^{∗}(M, F); see Lemma 3.6(3). Thus, one obtains equalities, once again referring to Lemma 3.5 for the second one:

X = Supp_{A}^{∗}(M, D) = Supp_{A}^{∗}(M, Ω_{n}^{R}(D)) = Supp_{A}^{∗}(M, N_{X}) .

When N = M one can choose D = C by Theorem 2.4 and get

N_{X} = M_{X} ≃ Ω_{n}^{R}(C) ∈ Thick_{R}(M ⊕ R) .

Lemma 3.6(3) now shows that Ext_{R}^{∗}(M_{X}, F) is eventually zero when F is free. Using Lemma 3.5, one obtains a final sequence of equalities

X = Supp_{A}^{∗}(C, C) = Supp_{A}^{∗}(C, M_{X}) = Supp_{A}^{∗}(M_{X}, M_{X}) .  □

4. Bialgebras and Hopf algebras

In this section k denotes a field. We recall some notions concerning bialgebras and Hopf algebras, referring to [19] for details.

A bialgebra over k is a k-algebra R with structure map η: k → R and product μ: R⊗_{k} R → k, equipped with homomorphism of rings ε: R → k, the augmentation and Δ: R → R ⊗_{k} R, the co-product, which satisfy the equalities

εη = id^{k} , (Δ ⊗ id^{R})Δ = Δ(Δ ⊗ id^{R}) ,
μ(id^{R} ⊗ηε)Δ = id^{R} = μ(ηε ⊗ id^{R})Δ .

Given R-modules M and N over a bialgebra R, the natural R⊗_{k}R-module structure on M ⊗_{k} N restricts along Δ to produce a canonical R-module structure:

r · (m ⊗ n) = ∑_{i=1}^{n}(r′_{i}m ⊗_{k} r′′_{i}n) when Δ(r) = ∑_{i=1}^{n}(r′_{i} ⊗_{k} r′′_{i}) .

This extends to a tensor product of complexes of R-modules. Let M be a complex of R-modules. The canonical isomorphisms below are easily seen to be R-linear:

(4.0.1)  k ⊗_{k} M ≅ M and M ⊗_{k} k → M .

4.1. Cohomology operations. Let R be a bialgebra over k, and view k as an Rmodule via the augmentation ε. The ring Ext_{R}^{∗}(k, k) has Ext_{R}^{0}(k, k) = k and is graded-commutative: for all α ∈ Ext_{R}^{i}(k, k) and β ∈ Ext_{R}^{j}(k, k) one has

α · β = (−1)^{ij}β · α;

see [17, (VIII.4.7), (VIII.4.3)] or [16, (5.5)]. Thus, every graded k-subalgebra

(a)  A ⊆ ExtR^{•}(k, k) = \begin{cases}⨁_{i⩾0} Ext_{R}^{2i}(k, k) & if char(k) ≠ 2 ; \\ ⨁_{i⩾0} Ext_{R}^{i}(k, k) & if char(k) = 2 . \end{cases}

is commutative. The functor − ⊗_{k} M preserves quasi-isomorphisms of complexes of R-modules, so it induces a functor − ⊗_{k} M: D(R) → D(R). In view of the isomorphism k ⊗_{k} M ≅ M, see (4.0.1), for each M one obtains a map

(b)  ζ_{M}: ExtR^{•}(k, k) → Ext_{R}^{∗}(M, M) .

It is readily verified to be a central homomorphism of graded k-algebras.

The results in Section 2 apply to any algebra A as above. More comprehensive information is available for special classes of bialgebras.

A Hopf algebra is a bialgebra R equipped with a k-linear map σ: R → R, the antipode, satisfying εσ = ε and μ(1 ⊗ σ)Δ = μ(σ ⊗ 1)Δ. Quantum groups offer prime examples. A Hopf algebra is cocommutative if it satisfies τΔ = Δ, where τ(r ⊗ s) = s ⊗ r. For instance, when G is a group the k-linear maps defined by

ε(g) = 1 , Δ(g) = g ⊗ g , and σ(g) = g^{−1} for g ∈ G

turn the group algebra kG into a cocommutative Hopf algebra. Other classical examples are given by universal enveloping algebras of Lie algebras and by restricted universal enveloping algebras of p-Lie algebras, where p = char(k) > 0.

4.2. Finiteness. Let R be a Hopf algebra with rank_{k} R finite.

When R is cocommutative ExtR^{•}(k, k) is finite as a k-algebra, and Ext_{R}^{∗}(M, N) is a finite ExtR^{•}(k, k)-module for all R-modules M, N of finite k-rank by a celebrated theorem of Friedlander and Suslin [13, (1.5.2)]. It extends results for group algebras (Golod, Evens, Venkov) and for restricted Lie algebras (Friedlander and Parshall).

It is an open problem whether similar finiteness properties for cohomology hold when R is not cocommutative; for positive solutions in interesting classes of such Hopf algebras see Pevtsova and Witherspoon [20], and the bibliography there.

4.3. Cohomological varieties. Let R be a Hopf algebra with rank_{k} R finite, set A = ExtR^{•}(k, k), see 4.1(a), and let k̄ be an algebraic closure of k.

For a complex M with Ext_{R}^{∗}(M, M) noetherian over A, define the cohomological variety of M to be the set

V_{R}^{∗}(M) = V_{A}(Ext_{R}^{∗}(M, M)) ⊆ Max (A ⊗_{k} k̄) ,

see 1.3. For instance, when the ring A itself is noetherian and rank_{k} R is finite, cohomological varieties are defined for each finite R-module M.

Existence Theorem 4.4. Let k be a field, R a Hopf algebra over k with rank_{k} R finite, and set A = ExtR^{•}(k, k). Let M be a complex with H(M) finite over R.

If the A-module Ext_{R}^{∗}(M, M) is eventually noetherian, then for each closed conical k-rational subset X of V_{A}^{∗}(M) there is a finite R-module M_{X}, such that

X = V_{A}^{∗}(M_{X}) and M_{X} ∈ Thick_{R}(M ⊕ R) .

Proof. Hopf algebras of finite rank are self-injective, see [19, (2.1.3)(4)], so one one has Ext_{R}^{⩾1}(−, R) = 0. It remains to invoke Theorem 3.1 and refer to 1.3. □

4.5. Applications. In view of 4.2, the theorem for M = k specializes to results of Carlson [11], Suslin, Friedlander, and Bendel [24, (7.5)], Pevtsova and Witherspoon [20, (4.5)], among others. It is clear that there is also a version dealing with the support of a pair (M, N) of modules, and for complexes.

5. Associative algebras

In k is a field and R is a k-algebra. Let R^{o} denote the opposite algebra of R, set R^{e} = R ⊗_{k} R^{o}, and give R a structure of left R^{e}-module by (r ⊗ r′) · s = rsr′.

5.1. Cohomology operations. The Hochschild cohomology of R is the k-algebra

H^{∗}(R|k) = ⨁_{i⩾0} Ext_{R^{e}}^{i}(R, R) .

Gerstenhaber [14, Corollary 1] proved that it is graded-commutative, so any subring

(a)  A ⊆ H^{•}(R|k) = ⨁_{i⩾0} H^{2i}(R|k)

is commutative. The map r ↦ 1⊗r is a homomorphism of rings R^{o} → R^{e}. It turns each complex of R^{e}-modules into one of right R-modules. Thus, − ⊗_{R} M is an additive functor from complexes of R^{e}-modules to complexes of R-modules, where R acts on the target via the homomorphism of rings R → R^{e} given by r ↦ r ⊗ 1. It induces an exact functor − ⊗_{R}^{L} M: D(R^{e}) → D(R) of derived categories, which yields a homomorphism of graded rings

Ext_{R^{e}}^{∗}(R, R) → Ext_{R}^{∗}(R ⊗_{R}^{L} M, R ⊗_{R}^{L} M) .

Noting the quasi-isomorphism R⊗_{R}^{L} M ≃ M, one obtains a natural homomorphism

(b)  ζ_{M}: H^{•}(R|k) → Ext_{R}^{∗}(M, M)

of graded rings. These maps satisfy condition (2.1.1), see [23, (10.1)].

The results in Sections 2 and 3 apply to any algebra A as above. Once again, we focus on a special case to relate them to available literature.

5.2. Finiteness. Let k be a field, R a k-algebra with rank_{k} R finite, J the Jacobson radical of R, and set K = R/J. It is rarely the case that the H^{∗}(R|k)-module Ext_{R}^{∗}(M, K) is noetherian for every finite R-module M; see [12, §1]. Examples when this property holds include the Hopf algebras in (4.2), exterior algebras, and commutative complete intersections rings, see (6.1).

5.3. Cohomological varieties. For R as in (5.2), let A be a subring of H^{•}(R|k) with A^{0} = k, see 5.1, and let k̄ be an algebraic closure of k.

For a complex of R-modules M, such that the A-module Ext_{R}^{∗}(M, K) is noetherian, define the cohomological variety of M to be the set

V_{A}^{∗}(M) = V_{A}(Ext_{R}^{∗}(M, K)) ⊆ Max (A ⊗_{k} k̄) .

As A acts on Ext_{R}^{∗}(M, K) through Ext_{R}^{∗}(K, K), one has V_{A}^{∗}(M) ⊆ V_{A}^{∗}(K).

Existence Theorem 5.4. Let R and A be as in 5.3, and let M be a complex of R-modules with H(M) finite over R.

If the A-module Ext_{R}^{∗}(M, K) is eventually noetherian, then for each closed conical k-rational subset X of V_{A}^{∗}(M) there is a finite R-module M_{X}, such that

X = V_{A}^{∗}(M_{X}) and M_{X} ∈ Thick_{R}(M ⊕ R) .

Proof. The R-module R admits a finite filtration with subquotients isomorphic to direct summands of K. Thus, when the A-module Ext_{R}^{∗}(M, K) is noetherian, so is Ext_{R}^{∗}(M, R). On this module A acts through Ext_{R}^{∗}(R, R), that is to say, through R^{c}. This means Ext_{R}^{≫0}(M, R) = 0, so we may use Theorem 3.1, then 1.3. □

5.5. Applications. When the A-module Ext_{R}^{∗}(K, K) is noetherian, applying Theorem 5.4 with M = K recovers a result of Erdmann et al; see [12, (3.4)].

6. Commutative local rings

Following standard usage in commutative algebra, we say (R, 𝔪, k) is a local ring if R is a commutative noetherian ring with unique maximal ideal 𝔪, and k = R/𝔪.

An embedded deformation of codimension c of R is an isomorphism R ≅ Q/I, with (Q, 𝔮, k) a local ring and I an ideal generated by a Q-regular sequence in 𝔮^{2}.

6.1. Cohomology operations. Let (R, 𝔪, k) be a local ring with an embedded deformation R ≅ Q/I. Set c = dim Q − dim R and

(a)  A = R[χ_{1}, …, χ_{c}]

where χ_{1}, …, χ_{c} are indeterminates of degree 2. For each M ∈ D(R) Avramov and Sun [7, (2.7), p. 700] construct a natural homomorphism of graded rings

(b)  ζ_{M}: A → Ext_{R}^{∗}(M, M)

satisfying condition (2.1.1); when M and N are R-modules the resulting structure of graded A-module on Ext_{R}^{∗}(M, N) coincides with that defined by Gulliksen [15]. For complexes M and N, the A-module Ext_{R}^{∗}(M, N) is finite if and only if the Q-module Ext_{Q}^{∗}(M, N) is finite; see [7, (5.1)] and [3, (4.2)].

When N = k, the action of A on Ext_{R}^{∗}(M, k) factors through the graded ring

(c)  R = A ⊗_{R} k = k[χ_{1}, …, χ_{c}] .

Recall that a complex of Q-modules is said to be perfect if it is isomorphic, in D(Q), to a bounded complex of finite free Q-modules.

Lemma 6.2. Let Q, R, and R be the rings in 6.1. The following conditions are equivalent for each complex M of R-modules:

(i) M is perfect over Q.

(ii) H(M) is finite over R and Ext_{R}^{∗}(M, k) is finite over R.

Proof. When M is perfect, the Q-module Ext_{Q}^{∗}(M, k) is finite, and hence the Rmodule Ext_{R}^{∗}(M, k) is finite, by 6.1. This proves (i) ⟹ (ii).

(ii) ⟹ (i) It from 6.1 that Ext_{Q}^{∗}(M, k) is finite over Q, and hence eventually zero. Since H(M) is finite over Q, the complex M admits a semiprojective resolution F with each F_{i} finite, F_{i} = 0 for i ≪ 0, and ∂(F) ⊆ 𝔪F; see, for example, [4]. Thus, Ext_{Q}^{i}(M, k) ≅ Hom_{Q}(F_{i}, k), and hence, Ext_{Q}^{i}(M, k) = 0 for i ≥ n implies F_{i} = 0 for i ≥ n, so F is a perfect complex which is quasi-isomorphic to M. □

6.3. Cohomological varieties. Let (R, 𝔪, k) be a local ring as in 6.1 and let M be a complex of R-modules with Ext_{R}^{∗}(M, k) noetherian over R. Let k̄ be an algebraic closure of k. The cohomological variety of M is the set

V_{R}^{∗}(M) = V_{R}(Ext_{R}^{∗}(M, k)) ⊆ Max (R ⊗_{k} k̄) = k̄^{c};

the last equality comes from Hilbert’s Nullstellensatz. When M is a module the construction above yields the cone V_{R}^{∗}(M) defined in [1].

A Q-module is perfect when viewed as a complex if and only if proj dim_{Q} M is finite. Thus, Theorem 2 from the introduction is obtained from the next result by replacing affine cones by their projectivizations.

Existence Theorem 6.4. Let (Q, 𝔮, k) be a local ring, f ⊂ 𝔮^{2} a Q-regular sequence of length c, and set R = Q/Qf.

The assignment M ↦ V_{R}^{∗}(M), which maps complexes in D(R) that are perfect over Q to closed k-rational cones in k̄^{c}, has the following properties:

(1) It is surjective, even when restricted to modules.

(2) V_{R}^{∗}(M) = {0} if and only if M is perfect over R.

(3) V_{R}^{∗}(M) = V_{R}^{∗}(Ω_{n}^{R}(M)) for every syzygy module Ω_{n}^{R}(M).

(4) V_{R}^{∗}(M) = V_{R}^{∗}(M/xM) for every M-regular sequence x in R.

We import some material for the proof of part (1) of the theorem.

6.5. A module with full support. Under the hypotheses above Avramov, Gasharov, and Peeva [3, (3.3), (3.11), and (6.2)] construct a finite R-module G with

Ext_{R}^{∗}(G, k) ≅ R ⊗_{k} Ext_{Q}^{∗}(G, k) and proj dim_{Q} G < ∞ .

Proof of Theorem 6.4. (1) As k̄^{c} = V_{R}(R) by the Nullstellensatz, in view of 1.3 it suffices to find a module G with Spec R = Supp_{R}^{∗}(G, k), and to show that any closed subset X of Supp_{R}^{∗}(G, k) is realizable by a module M_{X} with proj dim_{Q} M_{X} < ∞.

The R-module G from 6.5 has the necessary property, as Ext_{R}^{∗}(G, k) is a graded free R-module. Theorem 3.1 yields a module M_{X} with the desired cohomological support and M_{X} ∈ Thick_{R}(G ⊕ R); the last condition implies proj dim_{Q} M_{X} < ∞.

(2) Note that V_{R}^{∗}(M) = {0} if and only if Supp_{R} Ext_{R}^{∗}(M, k) = ∅. The latter condition holds if and only if Ext_{R}^{∗}(M, k) is eventually zero, for it is finite over R. Thus, Lemma 6.2, applied with Q = R, yields the desired equivalence.

(3) This follows from Lemma 3.5.

(4) By Example 2.9, the complex M//x is quasi-isomorphic to the Koszul complex on x, and hence to the R-module M/xM. Thus, Proposition 2.10 and 1.3 imply V_{R}^{∗}(M/xM) = V_{R}^{∗}(M), as desired. □

In Theorem 6.4 the hypothesis that R has an embedded deformation can be weakened in a useful way. The non-trivial property is (1), so we focus on it.

6.6. Completions. The 𝔪-adic completion of a local ring (R, 𝔪, k) is then a local ring, (R̂, 𝔪̂, k). The homomorphisms R → R̂ and M → M̂ = R̂ ⊗_{R} M induce maps

(d)  Ext∗ ; R̂(k, k) ⟶ Ext_{R}^{∗}(k, k) and Ext∗ ; R̂(M̂, k) ⟶ Ext_{R}^{∗}(M, k) ;

the first one is an isomorphism of graded k-algebras, the second an equivariant isomorphism of graded modules over it. Thus, when R̂ has an embedded deformation of codimension c the ring R from 6.1 acts on Ext_{R}^{∗}(M, k) for each M ∈ D(R). As in 6.3, when Ext_{R}^{∗}(M, k) is noetherian over R we define a cohomological variety by:

V_{R}^{∗}(M) = V_{R}(Ext_{R}^{∗}(M, k)) ⊆ k̄^{c} .

Observe that if R ≅ Q/I is an embedded deformation of codimension c, R, then R̂ ≅ Q̂/Q̂I is one of R̂, and the isomorphisms in (d) above yield V_{R}^{∗}(M) = V_{R}^{∗}(M̂).

Lemma 6.7. Let (R, 𝔪, k) be a local ring and R̂ ≅ Q/I an embedded deformation. For every perfect over Q complex L ∈ D(R̂) there is a finite R-module M with

V_{R}^{∗}(M) = V_{R}^{∗}(L) and proj dim_{Q} M̂ < ∞ .

Proof. By 1.3, it suffices to prove SuppA^{+} Ext_{R}^{∗}(M, k) = SuppA^{+} Ext_{R}^{∗}(L, k).

Choose a set of generators of the ideal 𝔪 and let x be its image under the composition R → R̂ → A. Since x Ext∗ ; R̂(L, k) = 0, one has an equality

SuppA^{+} Ext∗ ; R̂(L, k) ⊆ SuppA^{+} (A/(x)A) .

This yields the second equality below; the first one holds by Lemma 2.7:

SuppA^{+} Ext∗ ; R̂(L//x, k) = SuppA^{+} Ext∗ ; R̂(L, k) ∩ SuppA^{+} (A/(x)A)

= SuppA^{+} Ext∗ ; R̂(L, k) .

The complex L//x is quasi-isomorphic to the Koszul complex on x with coefficients in L; see 2.9. Thus, H(L//x) has finite length over R̂, hence also over R. Let F → L//x be a semi-projective resolution with F a complex of finite free R-modules. One then has quasi-somorphisms

R̂ ⊗_{R} F ≃ R̂ ⊗_{R} (L//x) ≃ L//x

due to the flatness of R̂ over R and, for the second one, also to the finiteness of the length of H(L//x) over R. Fix an integer n so that H_{i>n}(L//x) = 0 and set M = H_{n}(F_{⩾}n). One then has quasi-isomorphisms of complexes of R̂-modules

M̂ ≅ R̂ ⊗_{R} M ≃ R̂ ⊗_{R} (F_{⩾n}) ,

which imply M̂ ≅ ΩR̂ ; i(L//x). It yields the first equality below:

SuppA^{+} Ext∗ ; R̂(M̂, k) = SuppA^{+} Ext∗ ; R̂(Ω_{i}^{R}(L//x), k)

= SuppA^{+} Ext∗ ; R̂(L//x, k) .

Lemma 3.4 gives the second one. It remains to note that since L and R̂ are both perfect over Q, so is any complex in ThickR̂(L ⊕ R̂). Thus, L//x is perfect over Q, by (2.8.4), and hence so is M̂ = ΩR̂ ; i(L//x), by Lemma 3.4(3). □

Existence Theorem 6.8. Let (R, 𝔪, k) be a local ring.

If R̂ has an embedded deformation Q of codimension c, then for each closed k-rational cone X ⊆ k̄^{c} there exists a finite R-module M with

X = V_{R}^{∗}(M) and proj dim_{Q} M̂ < ∞ .

Proof. Theorem 6.4(1) provides a finite R̂-module L with X = V_{R}^{∗}(L), and of finite projective dimension over Q. Now apply Lemma 6.7. □

6.9. Application. A local ring R is complete intersection if R̂ admits a deformation R̂ ≅ Q/I where Q is a regular local ring; see [18, §29]. For such an R Theorem 6.8 specializes to a result proved by Bergh [10, (2.3)] by using Tate cohomology, and by Avramov and Jorgensen [6] who establish an existence theorem for cohomology modules by using equivalences of triangulated categories and Koszul duality.

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Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

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

Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

E-mail address: iyengar@math.unl.edu