Cohomological Support and the Geometric Join

Hailong Dao, William T. Sanders

Received: December 3, 2016 Revised: August 29, 2017

Communicated by Henning Krause

Abstract. Let M, N be finitely generated modules over a local complete intersection R. Assume that all the modules Tor_{i}^{R}(M, N) are zero for i > 0. We prove that the cohomological support of M ⊗_{R} N (in the sense of Avramov-Buchweitz) is equal to the geometric join of the cohomological supports of M, N. This result gives a new connection between two active areas or research, and immediately produces several interesting corollaries. Naturally, it also raises many intriguing new questions about the homological properties of modules over a complete intersection, some of which are investigated in this work.

2010 MSC Classification: 13D07

1 Introduction

Let (R, 𝔪, k) be a local complete intersection and M a finitely generated Rmodule. Inspired by the ideas of Quillen in modular representations context, Avramov and Buchweitz in [6] defined a geometric object attached to the total Ext module ⨁ Ext_{R}^{i}(M, k). It was originally called the support variety, or cohomological support of M, and denoted by V^{∗}(M) (see Section 2 for details). In this paper, we shall refer to this object as the cohomological support. It is a closed subscheme of ℙ_{k}^{c−1}, where c is the codimension of R. The following is an immediate consequence of the theory of cohomological supports developed in [6]: if M, N are non-zero and Tor-independent, i.e. Tor_{i}^{R}(M, N) = 0 for i > 0, then V^{∗}(M), V^{∗}(N) are disjoint and

dim V^{∗}(M ⊗_{R} N) = dim V^{∗}(M) + dim V^{∗}(N) + 1.

One of our main results in this paper gives a geometric clarification of this formula, by proving that under the above hypothesis, the cohomological support of M ⊗_{R} N is actually the join of V^{∗}(M), V^{∗}(N). Here, the join of two disjoint closed subschemes is the closure of the union of all the lines joining two points, one from each subscheme.

Theorem 1.1 (Theorem 3.1,Theorem 4.7). Let R be a local complete intersection, and M, N ∈ mod(R) with M, N ≠ 0.

1. If Tor_{>0}(M, N) = 0, then

V^{∗}(M ⊗ N) = Join(V^{∗}(M), V^{∗}(N)).

2. If Ext^{>0}(M, N) = 0, then

V^{∗}(Hom(M, N)) = Join(V^{∗}(M), V^{∗}(N)).

Thus, we provide a link between the theory of cohomological support to a very classical concept of algebraic geometry. Unsurprisingly, this immediately leads to many corollaries and interesting questions, some of them we also address in this work. The proof of the first part of Theorem 1.1 occupies Section 3. It combines homological and geometric techniques (the preparatory materials are collected in Section 2). In Section 4, we give some quick corollaries including the Theorem 1.1 (2). In Section 5, we collect some examples regarding the case when Tor_{i}(M, N) do not vanish. While computing these examples, we noticed certain patterns involving the asymptotic behaviour of V^{∗}(Tor_{i}(M, N)), motivating us to pose a couple of questions. Furthermore, in Section 5, we list a few situations which give us positive answers to our questions. In these investigations, we show the following.

Theorem 1.2 (Corollary 5.10). Let (R, 𝔪, k) be a local complete intersection of codimension c and M and N finitely generated R-modules. Fix m ∈ ℕ. If Tor_{i}(M, N) eventually has finite length, then there exists a ν ∈ ℕ such that for any n ≥ ν

V^{∗}(Tor_{n}(N, M)) ∪ V^{∗}(Tor_{n+2m}(N, M)) ∪ ⋯ ∪ V^{∗}( Tor_{n+2mc}(N, M))

= ⋃_{i=ν}^{∞} V^{∗}(Tor_{i}(M, N)).

In particular, there exists an l ∈ ℕ such that

⋃_{i=0}^{∞} V^{∗}(Tor_{i}(M, N)) = ⋃_{i=0}^{l} V^{∗}(Tor_{i}(M, N)).

2 Background

2.1 The ring of cohomological operators

The study of cohomological supports over complete intersection rings was initiated by Avramov and Buchweitz in [6]. For the entirety of this section, (R, 𝔪, k) will be a local complete intersection of codimension c such that R̂ = Q/(f_{1}, …, f_{c}) where Q is a regular local ring and \underline{f} = f_{1}, …, f_{c} a regular sequence not contained in the square of the maximal ideal of Q. Let k̃ be the algebraic closure of k. The cohomological support of a finitely generated R-module M is essentially the support of ExtR̂(M̂, k) as a module over the polynomial ring S = k[χ_{1}, …, χ_{c}]. We will now elaborate on this defintion. Let X be a finitely generated R̂-module. We recall a construction from [13] which gives an action of S on ExtR̂(X, k). Let (F_{•}, ∂) be a free resolution of X over R̂. Each F_{n} = R̂^{i_{n}} and we may view ∂ as a sequence of matrices with entries in R̂. Let F̃_{n} = Q^{i_{n}} and ˜∂ be the lift of ∂ to F̃_{•}. Since ∂^{2} = 0, we know that ˜∂^{2} is a sequence of matrices whose entries are in the ideal (f_{1}, …, f_{c}). Thus we may write

˜∂^{2} = ∑_{i=1}^{c} f_{i}Φ̃_{i}

where Φ̃_{i} is a sequence of matrices with entries in Q. Set Φ_{i} = Φ̃_{i}⊗R̂. Eisenbud shows in [13] that multiplication by Φ_{i} is a degree -2 chain map from F_{•} → F_{•}. It follows that Φ induces a degree 2 operator χ_{i} on

ExtR̂(X, k) = ⨁_{i=0}^{∞} Exti ; R̂(X, k).

It is also shown in [13] that the oeprators χ_{i} commute with eachother, i.e. χ_{i}χ_{j} = χ_{j}χ_{i}. Thus we can extend the action of the χ_{i} linearly to an action of the polynomial ring S = k[χ_{1}, …, χ_{c}], turning Ext(X, k) into a graded S-module, where each χ_{i} is degree 2. Furthermore, Eisenbud shows in [13] that this action is independent of our choices of F_{•} and Φ̃_{i} and that ExtR̂(X, k) is actually a finitely generated S-module. The ring S is known as the ring of cohomological operators and has been the focus of much study including [3, 5, 7, 13, 18]. In fact, there are several equivalent methods for constructing the action of S on Ext(X, k), the first of which was given in [14]: see [9] for a detailed discussion. The following result shows that the action is actually invariant up to a change of coordinates of the ideal generated by the regular sequence.

Theorem 2.1 ([13, Proposition 1.7], cf. [9, (3.1)]). Let f_{1}, …, f_{c} and f′_{1}, …, f′_{c} be two regular sequences of a regular local ring Q which generate the same ideal. Write

f_{i} = ∑_{j=1}^{c} q_{i,j}f′_{j}

with each q_{i,j} ∈ Q. Letting χ_{1}, …, χ_{c} and χ′_{1}, …, χ′_{c} be the cohomological operators associated to f_{1}, …, f_{c} and f′_{1}, …, f′_{c} respectively, we have

χ′_{j} = ∑_{i=1}^{c} q_{i,j}χ_{i}

Thus the matrix (q_{i,j}) acts as a change of basis matrix, changing the coordinates of ℙc−1 ; k̃. When k = k̃, any change of coordinates of ℙc−1 ; k̃ corresponds to choosing a different regular sequence which generates the ideal (f_{1}, …, f_{c}). This important fact is critical to several proofs in this document, thus we state it precisely.

Proposition 2.2. Assume that k is algebraically closed and set I = (f_{1}, …, f_{c}) where f_{1}, …, f_{c} is a regular sequence of a regular local ring Q. Let

φ : ℙ_{k}^{c−1} → ℙ_{k}^{c−1}

be an automorphism, i.e. a change of coordinates. Then there exists a regular sequence f′_{1}, …, f′_{c} generating I such that φ_{∗}(χ_{i}) = χ′_{i} where χ_{1}, …, χ_{c} and χ′_{1}, …, χ′_{c} are the cohomological operators associated to f_{1}, …, f_{c} and f′_{1}, …, f′_{c} respectively.

Proof. Set ψ = φ^{−1}, and let φ̃ and ψ̃ be the lifts of φ and ψ in Q such that ϕ̃ = φ̃^{−1}. We can regard φ̃ and ψ̃ as a matrices whose entries are q_{i,j} ∈ Q and p_{i,j} ∈ Q respectively. Set

f′_{i} = ∑_{i=1}^{c} p_{i,j}f_{j}.

By Nakayama’s lemma, f′_{1}, …, f′_{c} generates I, and since there are c elements, f′_{1}, …, f′_{c} is necessarily a regular sequence. However since φ̃ψ̃ is the identity, we also have

f_{i} = ∑_{j=1}^{c} q_{i,j}f′_{j}.

It follows from Theorem 2.1 that

χ′_{j} = ∑_{i=1}^{c} q_{i,j}χ_{i} = φ_{∗}(χ_{j}).

2.2 Cohomological supports

With the machinery of the cohomological operators in place, we may now discuss cohomological supports. Using the notation of the last subsection, we define

V (Q, \underline{f}; X) = \{ā ∈ 𝔸c ; k̃ | g(ā) = 0 ∀g ∈ ann_{S} Ext(X, k)\}

where k̃ is the algebraic closure of k.

Definition 2.3. Let R be a complete intersection ring. Following [6], for a finitely generated R-module M, the cohomological support, denoted V^{∗}(M), is the projectivization in ℙc−1 ; k̃ of the cone V (Q, \underline{f}; M̂). Occasionally, V_{R}^{∗}(M) will be used to indicate which ring is used to compute the cohomological support.

Proposition 2.4 ([6, Theorem 5.3]). For any finitely generated R-module M, V^{∗}(M) is independent of the choices of Q, and \underline{f} up to a change of coordinates.

Remark 2.5. What we call the cohomological support is referred to as the support variety in [6] and other works. In [8], the terminology cohomological support and cohomological variety are both used. Since geometers generally require varieties to be irreducible closed subsets and since V^{∗}(M) is generally not irreducible, we have decided to use the term cohomological support.

Remark 2.6. In [6] and in other works, the authors consider V^{∗}(M) as a cone in 𝔸c ; k̃. To facilitate the statement of certain results, we have found it easiest to work in projective space.

The following is a combination of the results [6, Theorem 5.6,Theorem 6.1].

Theorem 2.7. For finitely generated R-modules M and N, the following are equivalent.

1. V^{∗}(M) ∩ V^{∗}(N) = ∅

2. Tor_{≫0}(M, N) = 0

3. Ext^{≫0}(M, N) = 0

4. Ext^{≫0}(N, M) = 0

Hence cohomological supports encode homological information about a module. The following result gives another description of cohomological supports.

Theorem 2.8 ([6, Theorem 2.5],[3, Corollary 3.11]). Suppose that the residue field k is algebraically closed. For any module M ∈ mod(R), we have

V^{∗}(M) = {(a_{1}, …, a_{c}) ∈ ℙ_{k}^{c−1} | pd_{Q/(ã_{1}f_{1}+···+ã_{c}f_{c})} M̂ = ∞}

where ã_{i} is a lift in Q of a_{i}.

From this result and Lemma 2.17, we can easily deduce these corollaries.

Corollary 2.9. For a finitely generated R-module M, V^{∗}(M) = ∅ if and only if pd M < ∞. Also V^{∗}(k) = ℙc−1 ; k̃.

Corollary 2.10. Let f_{1}, …, f_{c} be a regular sequence of a regular local ring Q, and let k[χ_{1}, …, χ_{c}] be the ring of cohomological operators for Q/(f_{1}, …, f_{c}). Take n such that 1 ≤ n ≤ c. Let H be the linear space defined by

χ_{n+1} = ⋯ = χ_{c} = 0.

For any module M over Q/(f_{1}, …, f_{c}),

V_{Q/(f_{1},...,f_{n})}^{∗}(M) = V_{Q/(f_{1},...,f_{c})}^{∗}(M) ∩ H.

Corollary 2.11. Suppose M ∈ mod(R) and x ∈ R is regular on M. Then V^{∗}(M) = V^{∗}(M/xM).

Proof. Let x̃ ∈ Q be a lift of x. Then x̃ is still regular on M and so M has finite projective dimension over Q/a if and only if M/x̃M = M/xM does too. The result now follows from Theorem 2.8.

A generalization of Corollary 2.9 exists involving complexity.

Definition 2.12. For a sequence (a_{n})_{n≥0} of nonnegative integers, we can define the complexity

cx(a_{n})_{n≥0} = min{deg f | f ∈ ℚ[t] a_{n} ≤ f(n) ∀n ≫ 0} + 1.

For a module M, we set cx M = cx β_{n}(M).

A module has finite projective dimension if and only if cx M = 0. Since R is a complete intersection of codimension c, cx k equals c.

Proposition 2.13 ([6, Theorem 5.6]). For any R-module, we have dim V^{∗}(M) = cx M − 1.

Remark 2.14. Note that in the previous result, we consider V^{∗}(M) as a closed set of projective space instead of a cone in affine space.

The following are useful results on cohomological supports.

Theorem 2.15 ([10, Corollary 2.3],[8, Theorem 7.8]). If k is algebraically closed, for each closed set V ⊆ ℙ_{k}^{c−1} there is a maximal Cohen-Macaulay module M such that V^{∗}(M) = V.

When working with cohomological supports, it is important to be able to reduce to the case where R is complete and k is algebraically closed. These two results which allow us to do this.

Lemma 2.16. For any R-module M, we have V_{R}^{∗}(M) = V ∗ ; R̂(M̂).

Lemma 2.17 ([6, Lemma 2.2],[11, App., Théorèm 1, Corollaire])). There exists a local complete intersection ring (R̃, 𝔪̃, k̃) of codimension c such that R̃ is a flat extension of R, 𝔪R̃ = 𝔪̃, and the induced map k → k̃ is the inclusion of k into its algebraic closure. Furthermore, we have V_{R}^{∗}(M) = V ∗ ; R̃(M ⊗ R̃).

2.3 Thick subcategories

There is a deep connection between cohomological supports and the thick subcategories of mod(R). This connection begins with the following result.

Proposition 2.18 ([6, Theorem 5.6]). If

0 → X_{1} → X_{2} → X_{3} → 0

is exact, then

V^{∗}(X_{i}) ⊆ V^{∗}(X_{j}) ∪ V^{∗}(X_{l})

with {i, j, l} = {1, 2, 3}. In particular, V^{∗}(M) = V^{∗}(Ω M). Furthermore

V^{∗}(X ⊕ Y) = V^{∗}(X) ∪ V^{∗}(Y)

Definition 2.19. A subcategory C ⊆ mod(R) is thick if

1. R ∈ C

2. C is closed under direct summands, that is if X ⊕ Y ∈ C then X, Y ∈ C

3. C has the two out of three property, that is if 0 → X_{1} → X_{2} → X_{3} → 0 and X_{i}, X_{j} ∈ C, then X_{l} ∈ C with {i, j, l} = {1, 2, 3}.

Let Thick M denote the smallest thick category containing M.

The thick subcategories of R are in bijection with the thick subcategories of the triangulated category \underline{MCM(R)}, the stable category of maximal Cohen-Macaulay modules. The category of modules of finite projective dimension is thick. We can generalize this example: by Proposition 2.18, for any V ⊆ ℙc−1 ; k̃, the category

{M ∈ mod(R) | V^{∗}(M) ⊆ V }

is thick. In fact, cohomological supports are used to classify the thick subcategories in the complete intersection in [20], and, in the zero dimensional case, [12, Remark 5.12]. The proceeding result follows from the classification of thick subcategories [20, Remark 10.7].

Theorem 2.20. For two modules M, N ∈ mod(R), V_{R_{p}}^{∗}(M_{p}) ⊆ V_{R_{p}}^{∗}(N_{p}) for every p ∈ Sing R if and only if Thick M ⊆ Thick N where Thick M and Thick N are the smallest thick subcategories containing M and N.

Remark 2.21. The full classification theorem in [20] utilises the full scheme structure of ℙ_{k}^{c−1} instead of the closed points of ℙc−1 ; k̃, a context considered in other works such as [6]. The difference between such approaches are subtle and beyond the scope of this article.

2.4 Geometric join

In this subsection we give attention to another construction central to this paper.

Definition 2.22. Let U, V ⊆ ℙ_{k}^{n} be Zariski closed sets. We define the join of U and V to be

Join(U, V) = \overline{⋃_{\begin{matrix}u∈U v∈V \\ u≠v \end{matrix}} line(u, v)}

where line(u, v) is the projective line containing u and v.

Documenta Mathematica 22 (2017) 1593–1614

Remark 2.23. When U and V are disjoint Zariski closed sets, we may simplify this definition to

Join(U, V) = ⋃_{u∈U v∈V} line(u, v).

and we still obtain a closed set, see [15, Example 6.17]. In most contexts in this paper, we will be taking the join of disjoint sets.

Remark 2.24. There is some ambiguity with this definition when V is empty. To that end, we establish the following convention:

Join(U, ∅) = U.

This is the convention followed in [1]. We justify this convention by considering the join in affine space: the empty set corresponds to the the zero point and the join of the cone and the zero point is simply the original cone.

The following in anopther interesting fact about joins.

Lemma 2.25 ([15, Proposition 6.13,Example 6.14]). If U and V are irreducible closed sets, then Join(U, V) is also irreducible.

To visualize the join, consider the following easy examples. The join of two distinct points is a projective line, and the join of two skew lines is a three dimensional projective linear space. In fact, the join of any two linear spaces is the smallest linear space containing both of them.

Theorem 2.26 ([1, 1.1]). For two closed sets U, V ⊆ ℙ_{k}^{n}, we have

dim Join(U, V) ≤ dim U + dim V + 1

and if U ∩ V = ∅, then

dim Join(U, V) = dim U + dim V + 1.

The converse is not true, and, in fact, it is not known in general when dim Join(U, V) = dim U + dim V + 1 in the case U ∩ V ≠ ∅ . In particular, an active topic of research is understanding when Join(V, V) ≠ 2 dim V + 1.

3 Joins of cohomological supports

In this section, let (R, 𝔪, k) be a local complete intersection of codimension c. The goal of this section is to prove our first main result of this paper, namely the following theorem.

Theorem 3.1. Let R be a local complete intersection, and M, N ∈ mod(R) with M, N ≠ 0. If Tor_{>0}(M, N) = 0, then V^{∗}(M ⊗ N) = Join(V^{∗}(M), V^{∗}(N)).

We prove Theorem 3.1 in a few steps. First we cite a critical fact.

Lemma 3.2 ([19, Proposition 2.1]). If Tor_{>0}(M, N) = 0, then cx(M ⊗ N) = cx M + cx N.

We now prove a special case of the main result.

Lemma 3.3. If R has codimension two and Tor_{>0}(M, N) = 0 with M, N ≠ 0, then

V^{∗}(M ⊗ N) = Join(V^{∗}(M), V^{∗}(N)).

Proof. By Lemma 2.16 and Lemma 2.17, we may assume that R is complete and k is algebraically closed. If pd M, pd N < ∞, then pd M ⊗ N < ∞ and the conclusion is clear. Assume that pd M = pd N = ∞. Since V^{∗}(M) and V^{∗}(N) are disjoint, nonempty, and lie in ℙ_{k}^{1}, we know that dim V^{∗}(M) = dim V^{∗}(N) = 0. Therefore, the complexity of both M and N is one. Since Tor_{>0}(M, N) = 0, the complexity of M ⊗ N is two, and thus dim V^{∗}(M ⊗ N) = 1. This means that

V^{∗}(M ⊗ N) = ℙ_{k}^{1} = Join(V^{∗}(M), V^{∗}(N)).

We now assume that pd M < ∞ and pd N = ∞. Using the conventions in Remark 2.11, we may assume that M ≠ 0. We wish to show that V^{∗}(M ⊗N) = V^{∗}(N). Letting

0 → R^{n_{t}} → ⋯ → R^{n_{0}} → M → 0

be a free resolution, the sequence

0 → N^{n_{t}} → ⋯ → N^{n_{0}} → M ⊗ N → 0

is exact, implying V^{∗}(M ⊗ N) ⊆ V^{∗}(N). By Lemma 3.2, we have

cx(M ⊗ N) = cx M + cx N = cx N.

Therefore dim V^{∗}(M ⊗N) is the same as dim V^{∗}(N). So if V^{∗}(N) is irreducible, we are done. In particular, if dim V^{∗}(N) = 1, then V^{∗}(N) = ℙ_{k}^{1} and we are done. So suppose dim V^{∗}(N) = 0, that is V^{∗}(N) = {p_{1}, …, p_{n}} where p_{i} are points. The short exact sequence 0 → Ω N → R^{m} → N → 0 yields the short exact sequence

0 → M ⊗ Ω N → M^{m} → M ⊗ N → 0.

But since V^{∗}(M) is empty, we have V^{∗}(M ⊗ Ω N) = V^{∗}(M ⊗ N) by Proposition 2.18. Thus, since V^{∗}(N) = V^{∗}(Ω N) we may assume that N is maximal Cohen-Macaulay by replacing N with a sufficiently high syzygy. Therefore, by Theorem 3.1 of [10], we may write N = N_{1} ⊕ ⋯ ⊕ N_{n} with V^{∗}(N_{i}) = {p_{i}}. Since each V^{∗}(N_{i}) is irreducible, we have

V^{∗}(M ⊗ N) = V^{∗}(M ⊗ N_{1} ⊕ ⋯ ⊕ M ⊗ N_{n})

= V^{∗}(M ⊗ N_{1}) ∪ ⋯ ∪ V^{∗}(M ⊗ N_{n}) = {p_{1}} ∪ ⋯ ∪ {p_{n}} = V^{∗}(N)

which completes the proof.

Proposition 3.4. If Tor_{>0}(M, N) = 0 with M, N ≠ 0, then

Join(V^{∗}(M), V^{∗}(N)) ⊆ V^{∗}(M ⊗ N).

Proof. By Lemma 2.16 and Lemma 2.17, we may assume that R is complete and k is algebraically closed. We proceed by induction on the codimension of R, which we will denote by c. When c = 1, the cohomological supports lie in ℙ_{k}^{0}, a point. Thus the cohomological support of a module is either that point or empty, depending on whether or not the module has finite projective dimension. Since Tor_{>0}(M, N) = 0, the result follows from the equality

cx M + cx N = cx M ⊗ N.

If c = 2, the statement is true by Lemma 3.3. Now suppose that c ≥ 3. It suffices to show that for any hyperplane H ⊆ ℙ_{k}^{c−1} we have

Join(V^{∗}(M), V^{∗}(N)) ∩ H ⊆ V^{∗}(M ⊗ N) ∩ H

Since c ≥ 2, any hyperplane is a linear space with dimension at least one. Therefore, for any x ∈ V^{∗}(M) ∩ H and y ∈ V^{∗}(N), the projective line between x and y is also in H. Because of this, we have

Join(V^{∗}(M), V^{∗}(N)) ∩ H = Join(V^{∗}(M) ∩ H, V^{∗}(N) ∩ H).

Thus we need to show that

Join(V^{∗}(M) ∩ H, V^{∗}(N) ∩ H) ⊆ V^{∗}(M ⊗ N) ∩ H.

To that end, we fix a hyperplane H. Now may write R = Q/(f_{1}, …, f_{c}) where Q is a regular local ring and f_{1}, …, f_{c} is a regular sequence. By Proposition 2.2, we may change our coordinate system and assume that H = V (χ_{1}) where k[χ_{1}, …, χ_{c}] is the ring of cohomological operators. Set T = Q/(f_{2}, …, f_{c}) and f = f_{1}. Note that T is a complete intersection with codim T = c−1, f is regular on T, and R = T/(f). For any module X ∈ mod(R), V_{R}^{∗}(X) ∩ H = V_{T}^{∗}(X) by Corollary 2.10. Therefore we need to show that

Join(V_{T}^{∗}(M), V_{T}^{∗}(N)) ⊆ V_{T}^{∗}(M ⊗ N).

Since Tor_{>0}^{R}(M, N) = 0, by [4, Lemma 9.3.8], we have Tor_{1}^{T} (M, N) = M ⊗ N and Tor_{>1}^{T}(M, N) = 0. It follows that Tor_{>0}^{T}(M, Ω_{T} N) = 0. After tensoring 0 → Ω_{T} N → T^{t} → N → 0 with M, we get the exact sequence

0 → M ⊗ N → M ⊗ Ω_{T} N → M^{t} → M ⊗ N → 0.

Thus, by induction and the above exact sequence, we have

Join(V_{T}^{∗}(M), V_{T}^{∗}(N)) = Join(V_{T}^{∗}(M), V_{T}^{∗}(Ω_{T} N))

⊆ V_{T}^{∗}(M ⊗ Ω_{T} N) ⊆ V_{T}^{∗}(M ⊗ N) ∪ V_{T}^{∗}(M).

Note, that we can only choose Ω_{T} N to not be zero. A similar argument using Ω_{T} M gives us

Join(V_{T}^{∗}(M), V_{T}^{∗}(N)) ⊆ V_{T}^{∗}(M ⊗ N) ∪ V_{T}^{∗}(N).

This implies that

Join(V_{T}^{∗}(M), V_{T}^{∗}(N)) ⊆ V_{T}^{∗}(M ⊗ N) ∪ (V_{T}^{∗}(M) ∩ V_{T}^{∗}(N)) = V_{T}^{∗}(M ⊗ N)

proving the claim.

Lemma 3.5. Suppose that c = codim R ≥ 2, R is complete, and k is algebraically closed. Fix M ∈ mod(R) such that V_{R}^{∗}(M) = q for some point q ∈ ℙ_{k}^{c−1}. For any p ∈ ℙ_{k}^{c−1} distinct from q, there exists an L ∈ MCM such that V_{R}^{∗}(L) = p and V_{R}^{∗}(M ⊗ L) = Join(p, q).

Proof. As R is complete, we write R = Q/(f_{1}, …, f_{c}) where Q is a regular local ring and \underline{f} is a regular sequence. After a change of coordinates, we may assume that p = (1, 0, 0, …, 0). Set T = Q/(f_{1}), X = Ω_{T}^{d−1} k where d = dim Q, and L = X/(f_{2}, …, f_{c})X. We prove that L is our desired module. First, we show that V^{∗}(L) = p. Since pd_{T} L = ∞, it follows from Theorem 2.8 that p ∈ V^{∗}(L). Take any point p′ ∈ ℙ_{k}^{c−1} such that p′ ≠ p. By Theorem 2.15, there exists a Y ∈ mod(R) such that p′ = V^{∗}(Y). It follows from Theorem 2.8 that pd_{T} Y < ∞. Furthermore, we have Tor_{i}^{R}(Y, L) ≅ Tor_{i}^{T} (Y, X) for all i > 0, and thus Tor_{≫0}^{R}(Y, L) = 0. Therefore V^{∗}(Y) ∩ V^{∗}(L) = ∅ and so p′ ∉ V^{∗}(L). We now have V^{∗}(X) = {p} as claimed. Set l = Join(p, q). We now show that V^{∗}(M ⊗ L) = l. By proposition 3.4, we have l ⊆ V^{∗}(M ⊗ L). Take any point r ∉ l. We claim that r is not in V^{∗}(M ⊗L). After changing coordinates, we may assume that r = (0, 1, 0, …, 0). Set S = Q/(f_{1}, f_{2}) and X′ = X/f_{2}X. Let l′ be the projective line defined by p and r. Since r ∉ l, we have q ∉ l′. Hence Corollary 2.10 implies that V_{S}^{∗}(M) = V_{R}^{∗}(M) ∩ l′ = ∅. Therefore, Tor_{≫0}^{R′}(M, X′) = 0. However, since X′ is maximal Cohen-Macaulay over S, Lemma 4.4 implies that Tor_{>0}^{S}(M, X′) = 0. Since M ⊗ L ≅ M ⊗ X′, and since codim S = 2, Lemma 3.3 implies that

V_{S}^{∗}(M ⊗ L) = V_{S}^{∗}(M ⊗ X′) = Join(V_{S}^{∗}(M), V_{S}^{∗}(X′)) = V_{S}^{∗}(X′).

By Corollary 2.10 and Corollary 2.11, we have

V_{S}^{∗}(X′) = V_{S}^{∗}(L) = V_{R}^{∗}(L) ∩ l′ = p.

Therefore, we have p = V_{S}^{∗}(M ⊗ L) = V_{R}^{∗}(M ⊗ L) ∩ l′ by Corollary 2.10. Since r is in l′, r is not in V^{∗}(M ⊗ L), as desired.

We now proceed with the proof of the main result of this paper.

Proof of Theorem 3.1. By Lemma 2.16 and Lemma 2.17, we may assume that R is complete and k is algebraically closed. Proposition 3.4 gives us one containment, which leaves us to show the reverse containment:

V^{∗}(M ⊗ N) ⊆ Join(V^{∗}(M), V^{∗}(N)).

We will proceed by induction using induction on

α(M, N) = 2 depth R − depth M − depth N.

Assume for the moment that we have shown the base case, i.e. the theorem is true when when α(M, N) = 0, which is precisely the case when both M and N are maximal Cohen-Macaulay. Suppose that α(M, N) > 0. Then one of the modules, say M, is not maximal Cohen-Macaulay, and

α(Ω M, N) = α(M, N) − 1.

Tensoring the short exact sequence 0 → Ω M → R^{s} → M → 0 with N yields

0 → Ω M ⊗ N → N^{s} → M ⊗ N → 0.

By induction, we have the following

V^{∗}(M ⊗ N) ⊆ V^{∗}(N) ∪ V^{∗}(Ω M ⊗ N)

⊆ V^{∗}(N) ∪ Join(V^{∗}(Ω M), V^{∗}(N)) = Join(V^{∗}(M), V^{∗}(N))

which yields the desired inclusion. We now prove the base case. Assume that α(M, N) = 0 or, equivalently, that M and N are maximal Cohen-Macaulay modules. First we show the theorem when V^{∗}(M) is simply a single point, say q. Take any p ∉ Join(V^{∗}(M), V^{∗}(N)). By Lemma 3.5, there exists maximal Cohen-Macaulay module L such that V^{∗}(L) = p and V^{∗}(M ⊗ L) = Join(p, q) = Join(V^{∗}(M), V^{∗}(L)). However, since p ∉ Join(V^{∗}(M), V^{∗}(N)), there are no lines containing p, q and a point in V^{∗}(N). Therefore V^{∗}(M ⊗ L) = Join(p, q) and V^{∗}(N) are disjoint. Since M, N, L are all maximal Cohen-Macaulay, this shows that Tor_{>0}(M, L) = 0 and also Tor_{>0}(M ⊗ L, N) = 0. Now let A_{•}, B_{•}, C_{•} be free resolutions of L, M, N respectively. Then, (M ⊗ L) ⊗ C_{•} is quasi-isomorphic to Tot_{•}(A_{•} ⊗ B_{•} ⊗ C_{•}) which is quasi-isomorphic to A_{•} ⊗ (M ⊗ N). Therefore

Tor_{i}(L, M ⊗ N) ≅ Tor_{i}(M ⊗ L, N) = 0

for all i ≫ 0. This implies that V^{∗}(M ⊗ N) does not contain p = V^{∗}(L), yielding the desired containment. Now we show the general case. Again, take a point p ∉ Join(V^{∗}(M), V^{∗}(N)). By Theorem 2.15, there exists an L ∈ mod(R) with V^{∗}(L) = p. In the previous paragraph, we have shown that V^{∗}(M ⊗ L) = Join(V^{∗}(M), V^{∗}(L)). Thus the argument in the previous paragraph still applies, completing the proof.

4 Corollaries of Theorem 3.1

We now state some interesting corollaries of Theorem 3.1. The following is immediate.

Corollary 4.1. If N is not zero and Tor_{>0}(M, N) = 0, then

V^{∗}(M) ⊆ V^{∗}(M ⊗ N).

From this we derive a plethora of other corollaries.

Corollary 4.2. If N ≠ 0 and Tor_{>0}(M, N) = 0, then the following hold.

1. Ext^{≫0}(M ⊗ N, L) = 0 ⇒ Ext^{≫0}(M, L) = 0

2. Ext^{≫0}(L, M ⊗ N) = 0 ⇒ Ext^{≫0}(L, M) = 0

3. Tor_{≫0}(M ⊗ N, L) = 0 ⇒ Tor_{≫0}(M, L) = 0

Proof. The previous corollary shows that V^{∗}(M ⊗ N, L) = ∅ implies that V^{∗}(M, L) = ∅.

Corollary 4.3. Suppose Tor_{>0}(M, N) = 0. If

Sing R ⊆ supp N ∪ (spec R\ supp M)

then M is in Thick M ⊗ N. In particular, if R is an isolated singularity, then Tor_{>0}(M, N) = 0 implies that M, N ⊆ Thick M ⊗ N when M, N ≠ 0.

Proof. First note that Tor_{>0}(M_{p}, N_{p}) = 0 for every p ∈ spec R. Let p ∈ Sing R. Then either p ∈ supp N or p ∉ supp M. Then Corollary 4.1 implies that V_{R_{p}}^{∗}(M_{p}) ⊆ V_{R_{p}}^{∗}((M ⊗ N)_{p}) for all p ∈ Sing R. The result then follows by Theorem 2.20.

The next results use the following lemma.

Lemma 4.4. Suppose R is a complete intersection ring and either M or N is maximal Cohen-Macaualay. If Tor_{≫0}(M, N) = 0, then Tor_{>0}(M, N) = 0.

Proof. If not, then for every n ∈ ℕ, there exists a cosyzygy M′ of M such that Tor_{n}(M′, N) ≠ 0 and Tor_{≫0}(M′, N) = 0. But this contradicts the fact that complete intersections are AB; see [17, Corollary 3.4].

We may also use Theorem 3.1 to construct modules with linear cohomological supports.

Corollary 4.5. Assume that k is algebraically closed and R is complete. Set p_{i} = (0, …, 1, …, 0) ∈ ℙ_{k}^{c−1} be the point that is one in the ith position and zeros elsewhere. Let L be the affine span of p_{1}, …, p_{n}. Set

X_{i} = \frac{Ω_{Q/(f_{i})}^{d−1} k}{(Ω_{Q/(f_{i})}^{d−1} k)(f_{1}, …, f̂_{i}, …, f_{c})}

Documenta Mathematica 22 (2017) 1593–1614

where d = dim Q. Then X_{1} ⊗ ⋯ ⊗ X_{n} is maximal Cohen-Macaulay and L = V^{∗}(X_{1} ⊗ ⋯ ⊗ X_{n}).

Note that by Proposition 2.2, after changing coordinate any linear space of ℙ_{k}^{c−1} is of the form of

L = V^{∗}(X_{1} ⊗ ⋯ ⊗ X_{n}).

Proof. By Theorem 2.8, V^{∗}(X_{i}) = {p_{i}}. We work by induction on n. When n = 1, we are done. So assume the statement is true for n − 1. Let L′ be the affine span of p_{1}, …, p_{n−1}. The induction hypothesis implies that V^{∗}(X_{n−1}) = L′. Since X_{n} is maximal Cohen-Macaulay , Tor_{>0}(X_{1} ⊗⋯⊗X_{n−1}, X_{n}) = 0 by Lemma 4.4. Then X_{1} ⊗ ⋯ ⊗ X_{n} is maximal Cohen-Macaulay and by Theorem 3.1, we have

V^{∗}(X_{1} ⊗ ⋯ ⊗ X_{n}) = Join(V^{∗}(X_{1} ⊗ ⋯ ⊗ X_{n−1}), V^{∗}(X_{n})) = Join(L′, p_{n}) = L

proving the claim.

The main result of this paper also prevents certain tor modules from vanishing.

Corollary 4.6. Suppose M_{1}, …, M_{c+1} are nonfree maximal Cohen-Macaulay modules. Then for some i ∈ {1, …, c},

Tor_{n}(M_{1} ⊗ ⋯ ⊗ M_{i}, M_{i+1}) ≠ 0

for infinitely many n.

Proof. Proceeding by contradiction, suppose that

Tor_{≫0}(M_{1} ⊗ ⋯ ⊗ M_{i}, M_{i+1}) = 0

for each 1 ≤ i ≤ c. Inducting on i, we will show two statements: first that

V^{∗}(M_{1} ⊗ ⋯ ⊗ M_{i}) = Join(V^{∗}(M_{1}), …, V^{∗}(M_{i}))

for each i in {1, …, c}, and second that V^{∗}(M_{1} ⊗ … ⊗ M_{i}) contains a linear space of dimension i − 1. When i = 1, the statement is trivial. So suppose the statement is true for i. Since R is a complete intersection and each M_{i+1} is maximal Cohen-Macaulay, Lemma 4.4 implies that Tor_{>0}(M_{1}⊗⋯⊗M_{i}, M_{i+1}) = 0. By Theorem 3.1, it follows that

V^{∗}(M_{1} ⊗ ⋯ ⊗ M_{i} ⊗ M_{i+1}) = Join((V^{∗}(M_{1}), …, V^{∗}(M_{i})), V^{∗}(M_{i+1}))
= Join(V^{∗}(M_{1}), …, V^{∗}(M_{i+1}))

Furthermore, let L be the dimension i linear space in V^{∗}(M_{1} ⊗ ⋯ ⊗ M_{i}) guaranteed by the induction hypothesis. Take x ∈ V^{∗}(M_{i+1}) which exists since M_{i+1} is not free. Now x is not in L and so

Join(L, x) ⊆ V^{∗}(M_{1} ⊗ ⋯ ⊗ M_{i+1}).

But Join(L, x) is a linear space of dimension i + 1, proving the claim. Now the contradiction is clear, for there is a c-dimensional linear space contained in V^{∗}(M_{1} ⊗ ⋯ ⊗ M_{c+1}) which is a closed subset of ℙ^{c−1}.

We now prove our second main result, Theorem 1.1 (2), which is the analogue of Theorem 3.1 for Ext.

Theorem 4.7. Suppose R is a complete intersection ring and M, N ∈ mod(R). If Ext^{>0}(M, N) = 0, then V^{∗}(Hom(M, N)) = Join(V^{∗}(M), V^{∗}(N)).

Proof. Since Ext^{>0}(M, N) = 0, [2, Lemma 2.5] implies that M is maximal Cohen-Macaulay. By [16, Proposition 3.6], we have the following exact sequence

Tor_{2}(Tr M, N) → Hom(M, R) ⊗ N → Hom(M, N) → Tor_{1}(Tr M, N) → 0

where Tr M is the Auslander-Bridger transpose of M. However, since R is Gorenstein, Tr M is an inverse syzygy of M, i.e. there exists an exact sequence

0 → M → F → G → Tr M → 0

with F, G free. So Tor_{≫0}(Tr M, N) = 0. Since Tr M is again Cohen-Macaulay, it follows from Lemma 4.4 that Tor_{>0}(Tr M, N) = 0. Therefore, the above short exact sequence gives us the isomorphism M^{∗} ⊗ N ≅ Hom(M, N). Since V^{∗}(M) = V^{∗}(M^{∗}), we know that Tor_{≫0}(M^{∗}, N) = 0. Since M is maximal Cohen-Macaulay, Lemma 4.4 implies Tor_{>0}(M^{∗}, N) = 0. Therefore, by Theorem 3.1 we have

V^{∗}(Hom(M, N)) = V^{∗}(M^{∗} ⊗ N)
= Join(V^{∗}(M^{∗}), V^{∗}(N)) = Join(V^{∗}(M), V^{∗}(N))

This result provides an elementary proof of the following result.

Corollary 4.8. If Ext^{>0}(M, N) = 0, then cx Hom(M, N) = cx M + cx N.

Proof. It follows from the previous theorem, Theorem 2.26, and Proposition 2.13 that

cx Hom(M, N) = dim Join(V^{∗}(M), V^{∗}(N)) + 1
= dim V^{∗}(M) + dim V^{∗}(N) + 2 = cx M + cx N

5 Questions and examples

What happens to Theorem 3.1 if we remove the assumption that all the Tor modules vanish? The following two examples show that in general neither containment holds.

Example 5.1. Let k be a field and set R = k[x, y]/(xy). Now the modules M = R/(x + y) and N = R/(x − y) have finite projective dimension. However, we have

Join(V^{∗}(M), V^{∗}(N)) = ∅ ⊉ V^{∗}(M ⊗ N) = ℙ0 ; k̃

showing that V^{∗}(M ⊗ N) is not always contained in Join(V^{∗}(M), V^{∗}(N), even if Tor_{≫0}(M, N) = 0.

Example 5.2. Set R = ℚ[a, b, c]/(a^{2} − b^{2}, b^{3} − c^{3}) and

M = coker [\begin{matrix}8ab^{2}c^{2} + 4abc^{3} + 6b^{2}c^{3} + 8ac^{4} + 6bc^{4} + c^{5} & 3ab + 4b^{2} + 7ac + 7bc + 4c^{2} \\ 4ab^{2}c^{2} + 6abc^{3} + 9b^{2}c^{3} + ac^{4} + 9bc^{4} + 4c^{5} & 4ab + 5b^{2} + 3ac + 5bc + 5c^{2} \end{matrix}]

N = \frac{R}{(8ab^{2}c + 8b^{2}c^{2} + 6ac^{3} + 5bc^{3} + c^{4}, 3ab + 2b^{2} + 3ac + 2bc + 9c^{2})}.

An easy computation in Macaulay2 shows that

cx M = 0  cx N = 2  cx M ⊗ N = 1.

This shows that Join(V^{∗}(M), V^{∗}(N)) = V^{∗}(N) ⊈ V^{∗}(M ⊗ N).

We now give an example where none of the modules involved have finite projective dimension.

Example 5.3. Set R = ℚ[a, b, c, d]/(a^{2} − b^{2}, b^{2} − c^{2}, d^{2}) and define the ideal

I = (\frac{3}{5}a + \frac{8}{7}b + \frac{5}{2}c, 2a + \frac{1}{2}b + 3c, d) .

A simple computation in Macaulay2 shows that

V^{∗}(I) = V (3740x_{1} + 477x_{2})

V^{∗}(I ⊗ I) = V (0) = ℙ2 ; ℚ̃

Where ℚ̃[x_{1}, x_{2}, x_{3}] is the ring of cohomological operators over the algebraic closure of ℚ. Since V^{∗}(I) is linear, we have

Join(V^{∗}(I), V^{∗}(I)) = V^{∗}(I) ⊉ V^{∗}(I ⊗ I).

Example 5.4. Let R = ℚ[a, b, c]/(a^{2}, b^{2}, c^{2}) and I = (b) and J = (ab). An easy computation yields V^{∗}(R/I) = V (x_{1}, x_{3}) and V^{∗}(R/J) = V (x_{1}) where ℚ̃[x_{1}, x_{2}, x_{3}] is the ring of cohomological operators over the algebraic closure of ℚ. Now because V^{∗}(R/J) is a linear space containing V^{∗}(R/I), we have

Join(V^{∗}(R/I), V^{∗}(R/J)) = V^{∗}(R/J) ⊈ ⊈ V^{∗}(R/J ⊗ R/I) = V^{∗}(R/(I + J)) = V^{∗}(I).

The authors wondered if there was a relation between the asymptotic behaviour of V^{∗}(Tor_{i}(M, N)) and Join(V^{∗}(M), V^{∗}(N)). Investigations using Macaulay2 compelled them to ask the following questions.

Question 1. Does

⋃_{i=0}^{n} V^{∗}(Tor_{i}(M, N))

stabilize as n tends to infinity?

Question 2. For any modules M and N, do we have the following?

Join(V^{∗}(M), V^{∗}(N)) ⊆ ⋃_{i=0}^{∞} V^{∗}(Tor_{i}(M, N))

It seems that these statements cannot be made any stronger. For instance, Example 5.3 shows that we cannot hope to replace the containment in Question 2 with equality. Similarly, the following example shows that V^{∗}(Tor_{i}(M, N)) need not stabelize.

Example 5.5. Let k be a field and set R = k[x, y]/(xy). It is easy to show that Torodd(R/(x), R/(y)) = 0 and Toreven(R/(x), R/(y)) ≅ k. The projective dimension of the former is obviously finite, and the projective dimension of the latter is infinite. Thus V^{∗}(Tor_{i}(R/(x), R/(y))) cannot stabilize.

For closed sets U, V ∈ ℙ_{k}^{n}, it is known that dim Join(U, V) ≤ dim U + dim V + 1. It is not known when precisely this equality is strict. This question is particularly interesting when U = V and has been the subject of much research. As the following shows, Question 1 and Question 2 are actually related to this topic.

Proposition 5.6. Suppose Question 1 and Question 2 have positive answers. Then for any modules M and N over a complete intersection ring,

dim Join(V^{∗}(M), V^{∗}(N)) ≤ max_{i∈ℕ} cx Tor_{i}(M, N) − 1.

Proof. The result is obvious after recalling that dim V^{∗}(Tor_{i}(M, N)) = cx Tor_{i}(M, N) − 1.

The rest of this paper will discuss special cases where we know Question 1 and 2 to be true. First, we note that Question 1 has a positive answer when R is a hypersurface, because over such rings Tor_{i}(M, N) is eventually periodic.

Proposition 5.7. Questions 1 and 2 have positive answers when Tor_{≫0}(M, N) = 0.

Proof. The first question is trivially true in this case. We prove that the second question is true using induction on the smallest n such that Tor_{>n}(M, N) = 0. When n = 0, the the statement follows from Theorem 3.1. So suppose n > 0. Then we have Tor_{>n−1}(Ω M, N) = 0 and so by induction, we have

Join(V^{∗}(M), V^{∗}(N)) = Join(V^{∗}(Ω M), V^{∗}(N))

⊆ ⋃_{i=0}^{∞} V^{∗}(Tor_{i}(Ω M, N))

= ⋃_{i=2}^{∞} V^{∗}(Tor_{i}(M, N)) ∪ V^{∗}(Ω M ⊗ N).

Note that M is not free and so Ω M is not zero. The short exact sequence

0 → Ω M → R^{m} → M → 0

yields

0 → Tor_{1}(M, N) → Ω M ⊗ N → N^{m} → M ⊗ N → 0.

It follows that V^{∗}(Ω M ⊗ N) ⊆ V^{∗}(N) ∪ V^{∗}(M ⊗ N) ∪ V^{∗}(Tor_{1}(M, N)) and hence

Join(V^{∗}(M), V^{∗}(N)) ⊆ ⋃_{i=0}^{∞} V^{∗}(Tor_{i}(M, N)) ∪ V^{∗}(N).

Similarly, we have

Join(V^{∗}(M), V^{∗}(N)) ⊆ ⋃_{i=0}^{∞} V^{∗}(Tor_{i}(M, N)) ∪ V^{∗}(M)

but since V^{∗}(M) ∩ V^{∗}(N) = ∅, this shows the desired result.

It is interesting that the corresponding statement in Question 1 holds for Ext. We thank Mark Walker for bringing the following argument to our attention.

Proposition 5.8. Let (R, 𝔪, k) be a local complete intersection of codimension c and M and N finitely generated R modules. Fix m ∈ ℕ. There exists a ν ∈ ℕ such that for any n ≥ ν

V^{∗}(Ext^{n}(N, M)) ∪ V^{∗}(Ext^{n+2m}(N, M)) ∪ ⋯ ∪ V^{∗}( Ext^{n+2mc}(N, M))

= ⋃_{i=ν}^{∞} V^{∗}(Ext^{i}(M, N)).

In particular, there exists an l ∈ ℕ such that

⋃_{i=0}^{∞} V^{∗}(Ext^{i}(M, N)) = ⋃_{i=0}^{l} V^{∗}(Ext^{i}(M, N)).

When m = 1, the first statement shows that the union of c terms of the sequences V^{∗}(Ext^{2i}(M, N)) and V^{∗}(Ext^{2i+1}(M, N)) stabilises.

Proof. Let S = R[χ_{1}, …, χ_{c}] be the ring of cohomological operators. Recall that Ext(M, N) is a graded S-module and that each χ_{i} has degree 2. Take any m ∈ ℕ, and let

K_{•} = K_{•}(χ_{1}^{m}, …, χ_{c}^{m}; Ext(M, N))

be the Koszul complex of powers of the cohomological operators on the Smodule Ext(M, N). Note that this is a complex of graded S-modules. Therefore, in degree n, we get a complex of R-modules

0 → Ext^{n+(c+1)m}(M, N) →Ext^{n+(c)m}(M, N)^{a_{1}} → Ext^{n+(c−1)m}(M, N)^{a_{2}}
→ ⋯ → Ext^{n+m}(M, N)^{a_{c−1}} → Ext^{n}(M, N) → 0

where a_{i} = \binom{c}{i}. Since Ext(M, N) is finitely generated over R[χ_{1}, …, χ_{n}], we know that the homology of K_{•} is annihilated by some power of the ideal (χ_{1}, …, χ_{c}), and thus is concentrated in low degrees. In particular, the complex K_{•} is exact in all degrees larger than some constant ν ∈ ℕ. Since the differentials of K_{•} have degree 2m, this means that for any n ≥ ν, the above complex is exact. We can conclude two statements:

V^{∗}(Ext^{n+(c+1)m}(M, N)) ⊆ ⋃_{i=0}^{c} V^{∗}(Ext^{n+im}(M, N))

V^{∗}(Ext^{n}(M, N)) ⊆ ⋃_{i=1}^{c+1} V^{∗}(Ext^{n+im}(M, N)).

Working inductively, and shifting n, we can use these two statements to prove the first claim. The second claim follows by taking l = ν + n + (c + 1)m.

The idea of taking the Koszul complex over the ring of cohomological operators originates in [7]

Remark 5.9. The same argument works in a more general situation. Let R be Noetherian ring which is not necessarily commutative or local. Suppose a Noetherian graded commutative ring S concentrated in non-negative degrees acts on D(R) (see [8] for the relevant definitions and examples). Suppose further that for complexes M, N, Ext(M, N) is finitely generated as an S module. Letting x_{1}, …, x_{c} be a generating set of ideal S^{>0} ⊆ S, the Koszul complex

K_{•}(x_{1}^{m}, …, x_{c}^{m}; Ext(M, N))

yields a similar conclusion regarding the objects supp_{S} Ext(Ext^{i}(M, N), L) where L is an object in D(R).

Under certain conditions, we prove a dual version of Proposition 5.8.

Corollary 5.10. Let (R, 𝔪, k) be a local complete intersection of codimension c and M and N finitely generated R-modules. Fix m ∈ ℕ. If Tor_{i}(M, N) eventually has finite length, then there exists a ν ∈ ℕ such that for any n ≥ ν

V^{∗}(Tor_{n}(N, M)) ∪ V^{∗}(Tor_{n+2m}(N, M)) ∪ ⋯ ∪ V^{∗}( Tor_{n+2mc}(N, M))

= ⋃_{i=ν}^{∞} V^{∗}(Tor_{i}(M, N)).

In particular, there exists an l ∈ ℕ such that

⋃_{i=0}^{∞} V^{∗}(Tor_{i}(M, N)) = ⋃_{i=0}^{l} V^{∗}(Tor_{i}(M, N)).

This gives an affirmative answer to Question 1 in the isolated singularity case.

Proof. By replacing M and N with high enough syzygies, we may assume that M and N are maximal Cohen-Macaulay. We claim two things. First, we claim that Ext^{i}(M, N^{∗}) also eventually has finite length where N^{∗} = Hom(N, R). Second, we claim that eventually

Ext^{d}(Ext^{i}(M, N^{∗}), R) ≅ Tor_{i+d}(M, N).

By [6, Theorem 5.6], it follows that eventually

V^{∗}(Ext^{i}(M, N^{∗})) = V^{∗}(Ext^{d}(Ext^{i}(M, N^{∗}), R)),

and the corollary is now clear. To prove the first claim, note that Tor_{i}^{R_{𝔭}}(M_{𝔭}, N_{𝔭}) ≅ Tor_{i}(M, N)_{𝔭} eventually vanishes for every 𝔭 ∈ spec R\𝔪. Since R_{𝔭} is also complete intersection, [17, Theorem 2.1, Proposition 4.3] and implies that Ext_{R_{𝔭}}^{i}(M_{𝔭}, N_{𝔭}^{∗}) vanishes for all i > dim R_{𝔭}. Therefore, Ext^{i}(M, N^{∗}) is eventually finite length. Now we prove the second claim. Let F_{•} and G_{•} be resolutions of M and N respectively, and let I^{•} be an injective resolution of R. We have the following quasi-isomorphism

F_{•} ⊗ G_{•} ≅ Hom(Hom(F_{•} ⊗ G_{•}, R), R) ≃ Hom(Hom(F_{•}, Hom(G_{•}, R)), I_{•})
≃ Hom(Hom(F_{•}, N^{∗}), I^{•})

where the second to last quasi-isomorphism is because R is Gorenstein and N is maximal Cohen-Macaulay. Letting

E_{i,j}^{0} = Hom(Hom(F_{i}, N^{∗}), I^{−j})

we get a spectral sequence converging to Tor_{i−j}(M, N). Note that since I^{•} is a bounded complex, the sequence does indeed converge. Furthermore, we have

E_{i,j}^{2} = Ext^{−j}(Ext^{i}(M, N^{∗}), R).

However, since eventually Ext^{i}(M, N^{∗}) has finite length, for i ≫ 0, E_{i,j}^{2} = 0 for all j ≠ −d. Furthermore, since E_{i,j}^{0} = 0 for all j not in [−d, 0], it follows that eventually

Tor_{i+d}(M, N) = E_{i,−d}^{2} = Ext^{d}(Ext^{i}(M, N^{∗}), R).

Acknowledgements

The authors would like to thank Srikanth Iyengar and Greg Stevenson for their helpful insights. The authors would like to thank Mark Walker for an inspiring conversation which leads to the proof of Proposition 5.8. We would also like to thank the anonymous referee for his comments.

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Hailong Dao

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