On support varieties for modules over complete intersections

We show that, over a local complete intersection, every possible variety is realized as the cohomological support variety of some module. Moreover, we show that the projective variety of a complete indecomposable maximal Cohen-Macaulay module is connected.

wherek is the algebraic closure of k. This is equal to the algebraic set defined by the annihilator in H of E ( M , k).
For an ideal a of H we denote by V H (a) the algebraic set ink c defined by a, i.e.

Realizing support varieties
Before proving the main results we need some notation. Let R be a commutative Noetherian local ring and X an R-module with minimal free resolution · · · → P 2 → P 1 → P 0 → X → 0, and denote by Ω n R (X) the n'th syzygy of X. For an R-module Y , a homogeneous element η ∈ Ext * R (X, Y ) can be represented by a map f η : Ω |η| R (X) → Y , giving the pushout diagram with exact rows. Note that the module K η is independent, up to isomorphism, of the map f η chosen as a representative for η. The construction of this module first appeared in the paper [AGP] by L. Avramov, V. Gasharov and I. Peeva, where it is used in the proof of Theorem 7.8. If θ ∈ Ext * R (X, X) is another homogeneous element, then the Yoneda product ηθ ∈ Ext * R (X, Y ) is a homogeneous element of degree |η|+ |θ|. The following lemma links K η and K θ to K ηθ via a short exact sequence, and will be a key ingredient in the proof of the decomposition theorem in the next section.
Lemma 2.1 ( [Ber,Lemma 2.3]). If θ ∈ Ext * R (X, X) and η ∈ Ext * R (X, Y ) are two homogeneous elements, then there exists an exact sequence Now suppose R is Gorenstein and X is a maximal Cohen-Macaulay (or "MCM" from now on) module. Then there exists a complete resolution of X, i.e. a doubly infinite exact sequence of free modules in which Im d is isomorphic to X. For an integer n ∈ Z the stable cohomology module Ext n R (X, Y ) is defined as the n'th homology of the complex Hom R (P, Y ). If X and Y are A-modules and X is MCM, then Ext * b is a module over the ring A[χ] of cohomology operators, and the exact same proof as the one used to prove [EHSST,Lemma 4.2] shows that for any prime ideal q = (χ) of We are now ready to prove the first result, which shows that when "cutting down" the variety of an MCM A-module by a homogeneous element, the resulting homogeneous algebraic set is also the variety of an A-module.
Theorem 2.2. Let η ∈ H + = (χ) be a homogeneous element, and let η ∈ A[χ] be a homogeneous element such that η ⊗ 1 corresponds to η under the isomorphism and equality holds whenever X is MCM.
Proof. Consider the exact sequence representing θ. Since varieties are invariant under syzygies we have V(K θ ) ⊆ V(X), and so the first half of the theorem will follow if we can establish the inclusion The exact sequence induces a long exact sequence in cohomology, from which we obtain the short exact sequence Now for any A-modules W and Z the left and right scalar actions from A[χ] on Ext * b A (W, Z), through the ring homomorphisms φ Z and φ W , respectively, are actually equal (see [AvB,1.1 (X, k), and the above short exact sequence is a sequence of A[χ]-modules and maps. Moreover, the end terms are both annihilated by the element η. To see this, note that since A (X, X)-module annihilated by θ, and with the property that each graded part G i is finitely generated over A, then This implies that G i · φ X (η) vanishes itself, hence η annihilates G * . In particular, the element η annihilates the end terms in the above short exact sequence. Now for any i ≥ 0, let w be an element of Ext i b A (K θ , k), and consider the element thereby establishing the first half of the theorem.
Next suppose that X is MCM, and let p = H + be a prime ideal of H containing η and Ann H E(X, k). Choose a prime ideal p = (χ) of A[χ] corresponding to p and containing η and the annihilator of Ext * b A (X, k), and suppose p does not contain the annihilator of Ext * b A (K θ , k). The exact sequence from the beginning of the proof induces a long exact sequence in stable cohomology, which in turn gives the exact sequence in which the index * ranges over all the integers. Now let be a complete resolution of X, and consider the group Ext A (X, k) from both sides, and these actions coincide.
(X, k), and the above short exact sequence is a sequence of A[χ]-modules. Now recall from the discussion prior to this theorem that Ext * b A (W, Z) p for any A-modules W and Z with W MCM. As p does not contain the annihilator of Ext * b A (K θ , k), we see by localizing the above short exact sequence at p that Ext * b A (X, X), and the ideal p contains both η and the m i (it contains the element η by assumption, and contains the ideal m because it corresponds to the ideal p ⊆ H under the isomorphism A A (X, k) p = 0. This contradicts the assumption that p contains the annihilator of Ext * b A (X, k), and therefore p must contain the annihilator of Ext * b A (K θ , k). But then Ann H E(K θ , k) ⊆ p, giving the inclusion of ideals in H, and consequently we get V(X) ∩ V H (η) ⊆ V(K θ ).
Suppose now that we start with an A-module M , and consider its completion M . Let η ∈ H + be a homogeneous element, let η ∈ A[χ] be a corresponding element, and consider the element φ c whose middle term is the completion of an A-module. By Theorem 2.2 the inclusion holds, with equality holding whenever M is MCM, and consequently we obtain the following corollary, showing that every homogeneous algebraic set ink c is the variety of an MCM A-module.

Corollary 2.3. Every closed homogeneous variety ink c is the variety of some MCM A-module.
Proof. Let η 1 , . . . , η t be homogeneous elements in H + , and denote by M the MCM module Ω dim A A (k). Then V(M ) = V(k) =k c , and from the theorem and the above discussion we see that there exists a homogeneous element θ 1 ∈ Ext (ii) In [EHSST] a realization theorem is proved for finite dimensional algebras, and this result applies to complete intersections containing a field (see also [SnS,Section 7]).

Decomposition
Before proving the next result, recall that an MCM-approximation of an Amodule X is an exact sequence where C X is MCM and Y X has finite injective dimension. The approximation is minimal if the map f is right minimal, that is, if every map C X g − → C X satisfying f = f g is an isomorphism. This notion was introduced in [AuB], where it was shown that every finitely generated module over a commutative Noetherian ring admitting a dualizing module has an MCM-approximation. Moreover, it follows from the remark following [Mar,Theorem 18] that every finitely generated module over a commutative local Gorenstein ring has a minimal MCM-approximation, which is unique up to isomorphism. In particular this applies to our setting, where A is a local complete intersection. Furthermore, since A → A is a faithfully flat local homomorphism, an A-module Z has finite projective dimension if and only if the A-module Z has finite projective dimension, and it follows from [Mat,Theorem 23.3] that Z is MCM if and only if Z is MCM. Therefore, by [Mar,Proposition 19] and the fact that over a Gorenstein ring the modules having finite injective dimension are precisely those having finite projective dimension, we see that We are now ready to prove the second main result. It is the commutative complete intersection version of J. Carlson's famous theorem (see [Car]) from modular representation theory; if the variety V of a kG-module L (where k is an algebraically closed field and G is a finite group) decomposes as V = V 1 ∪ V 2 , where V 1 and V 2 are closed varieties having trivial intersection, then L decomposes as L = L 1 ⊕ L 2 where the variety of L i is V i . Our proof follows closely that of J. Carlson, but with some adjustments.
be the minimal MCM-approximation of M . Since Y has finite injective dimension (or equivalently, finite projective dimension), it follows from [AvB,Theorem 5.6] that V(Y ) is trivial and that we therefore have V(M ) = V(C). Moreover, by definition the equality V(X) = V( X) holds for every A-module X, and therefore we may suppose that A is complete. We argue by induction on the integer dim V 1 + dim V 2 . If one of V 1 and V 2 , say V 2 , is zero dimensional, then V 2 is trivial, and the decomposition C = C ′ ⊕ P , with P being the maximal projective summand of C, satisfies the conclusion of the theorem. Suppose therefore that dim V i is nonzero for i = 1, 2.
Let a 1 and a 2 be homogeneous ideals of H = k[χ] defining the varieties V 1 and V 2 , i.e. V i is the algebraic set V H (a i ) ink c defined by a i for i = 1, 2. We then have equalities and so it follows from Hilbert's Nullstellensatz that for each 1 ≤ i ≤ c we have χ i ∈ √ a 1 + a 2 . Therefore √ a 1 + a 2 is the graded maximal ideal H + of H, i.e. √ a 1 + a 2 = (χ). Pick a homogeneous element θ ∈ H + with the property that dim H/(a 2 , θ) < dim H/ a 2 (this is possible since dim H/ a 2 = dim V 2 > 0). By the above there is an integer n ≥ 1 such that θ n belongs to a 1 + a 2 , i.e. θ n = θ 1 + η where θ 1 ∈ a 1 and η ∈ a 2 . Then dim H/(a 2 , θ 1 ) < dim H/ a 2 , which translates to the language of varieties as dim (V H (a 2 ) ∩ V H (θ 1 )) = dim V H (a 2 +(θ 1 )) < dim V H (a 2 ). Similarly we can find an element θ 2 ∈ a 2 having the property that it "cuts down" the variety defined by a 1 . Hence the two homogeneous elements θ 1 and θ 2 satisfy , it follows once more from Hilbert's Nullstellensatz that θ 1 θ 2 ∈ Ann H E(C, C), where E(C, C) = Ext * A (C, C) ⊗ A k. Replacing θ 1 and θ 2 by suitable powers, we may assume that θ 1 θ 2 ∈ Ann H E(C, C). Viewed as elements in A[χ] ⊗ A k we have θ i = θ i ⊗ 1, where θ 1 and θ 2 are homogeneous elements of positive degrees in A[χ] with the property that θ 1 θ 2 ∈ Ann A[χ] Ext * A (C, C). To see the latter, note that 0 = θ 1 θ 2 Ext i A (C, C) ⊗ A k = θ 1 θ 2 Ext i A (C, C) ⊗ A k for every i ≥ 0, and since θ 1 θ 2 Ext i A (C, C) is a finitely generated A-module (θ 1 θ 2 commutes with elements in A), the claim follows. Now consider the images θ C 1 and θ C 2 of θ 1 and θ 2 in Ext * A (C, C). Since θ C 1 θ C 2 = 0, the bottom exact sequence in the exact commutative diagram / / 0 splits, where Q n denotes the n'th module in the minimal free resolution of C.
, and from Lemma 2.1 we see that there exists an exact sequence ) . By induction there exist A-modules X 1 , X 2 , Y 1 and Y 2 such that K θ C 1 = X 1 ⊕ X 2 and Ω |θ C 1 | A (K θ C 2 ) = Y 1 ⊕ Y 2 , and such that V(X 1 ) = V 1 , Now since V(X 1 ) ∩ V(Y 2 ) and V(X 2 ) ∩ V(Y 1 ) are contained in V 1 ∩ V 2 , which is trivial, we see from [AvB,Theorem 5.6] that Ext i A (X 1 , Y 2 ) and Ext i A (X 2 , Y 1 ) vanish for i ≫ 0. But K θ C 1 is MCM, implying X 1 and X 2 are both MCM, and so it follows from [ArY,Theorem 4.2] that Ext i A (X 1 , Y 2 ) and Ext i A (X 2 , Y 1 ) vanish for i ≥ 1. Therefore , and this implies that the exact sequence ( †) is equivalent to the direct sum of two sequences of the form 0 → Y i → Z i → X i → 0 for i = 1, 2, where Z i is an A-module. Then C ⊕ Ω |θ C 1 |+|θ C 2 |−1 A (C) ⊕ F must be isomorphic to Z 1 ⊕ Z 2 , and since V(Z i ) ⊆ V(X i ) ∪ V(Y i ) ⊆ V i and the Krull-Schmidt property holds for the category of (finitely generated) modules over a complete local ring, there must exist A-modules C 1 and C 2 such that C = C 1 ⊕ C 2 and V(C i ) = V(Z i ). Since V = V(C 1 ) ∪ V(C 2 ) ⊆ V 1 ∪ V 2 = V we must have V(C i ) = V i , and the proof is complete.
Corollary 3.2. The projective variety of a complete indecomposable MCM Amodule is connected.