ON SUPPORT VARIETIES FOR MODULES OVER COMPLETE INTERSECTIONS

PETTER ANDREAS BERGH

Abstract. Let (A, 𝔪, k) be a complete intersection of codimension c, and k̃ the algebraic closure of k. We show that every homogeneous algebraic subset of k̃^{c} is the cohomological support variety of an A-module, and that the projective variety of a complete indecomposable maximal Cohen-Macaulay A-module is connected.

1. Introduction

Support varieties for modules over complete intersections were defined by L. Avramov in [Avr], and L. Avramov and R.-O. Buchweitz showed in [AvB] that these varieties to a large extent behave precisely like the cohomological varieties of modules over group algebras of finite groups. Further illustrating this are the two main results in this paper, the first of which says that every homogeneous variety is realized as the variety of some module. The second is a version of J. Carlson’s result [Car, Theorem 1’] on varieties for modules over group algebras of finite groups. Namely, we prove that if the variety of a module decomposes as the union of two closed subvarieties having trivial intersection, then the (completion of the) minimal maximal Cohen-Macaulay approximation of the module decomposes accordingly.

Throughout this paper we let (A, 𝔪, k) be a commutative Noetherian local complete intersection, i.e. the completion Â of A with respect to the 𝔪-adic topology is the residue ring of a regular local ring modulo an ideal generated by a regular sequence. We denote by c the codimension of A, that is, the integer μ(𝔪) − dim A, where μ(𝔪) is the minimal number of generators for 𝔪. All modules are assumed to be finitely generated.

We now recall the definition of support varieties for modules over complete intersections; details can be found in [Avr, Section 1] and [AvB, Section 2]. Let Â[χ_{1}, …, χ_{c}] be the polynomial ring in the c commuting Eisenbud operators of cohomological degree 2. For every Â-module X there is a homomorphism Â[χ_{1}, …, χ_{c}] ⟶^{ϕ_{X}} Ext∗ ; Â(X, X) of graded rings under which Ext∗ ; Â(X, Y) is a finitely generated graded Â[χ_{1}, …, χ_{c}]-module for any Â-module Y. Using the canonical isomorphism k[χ_{1}, …, χ_{c}] ≃ Â[χ_{1}, …, χ_{c}] ⊗Â k we obtain a homomorphism

k[χ_{1}, …, χ_{c}] ⟶^{ϕ_{X}⊗1} Ext∗ ; Â(X, X)⊗Âk of graded rings under which Ext∗ ; Â(X, Y)⊗Âk is finitely generated over k[χ_{1}, …, χ_{c}]. We denote the polynomial ring k[χ_{1}, …, χ_{c}] by H and the graded H-module Ext∗ ; Â(X, Y) ⊗Â k by E(X, Y). Furthermore, we denote the sequence χ_{1}, …, χ_{c} of Eisenbud operators by χ, so that Â[χ] and k[χ] are short-hand notations for Â[χ_{1}, …, χ_{c}] and k[χ_{1}, …, χ_{c}], respectively.

2000 Mathematics Subject Classification. Primary 13C14, 13C40, 13D07, 14M10; Secondary 20J06.

Key words and phrases. Complete intersections, support varieties.

Let M be an A-module and M̂ = Â ⊗_{A} M its 𝔪-adic completion. The support variety V(M) of M is the algebraic set

V(M) = {α = (α_{1}, …, α_{c}) ∈ k̃^{c} | f(α) = 0 for all f ∈ Ann_{H} E(M̂, M̂)},

where k̃ is the algebraic closure of k. This is equal to the algebraic set defined by the annihilator in H of E(M̂, k).

For an ideal 𝔞 of H we denote by V_{H}(𝔞) the algebraic set in k̃^{c} defined by 𝔞, i.e.

V_{H}(𝔞) = {α = (α_{1}, …, α_{c}) ∈ k̃^{c} | f(α) = 0 for all f ∈ 𝔞}.

Note that the variety V(M) of M is the set V_{H} (Ann_{H} E(M̂, M̂)), and if f is an

element of H then V_{H}(f) is the set of all elements in k̃^{c} on which f vanishes.

2. 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_{R}^{n}(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_{R}^{|η|}(X) → Y, giving the pushout diagram

[figure: 0; Ω_{R}^{|η|}(X); f_{η}; P_{|η|−1}; Ω_{R}^{|η|−1}(X); 0; 0; Y; K_{η}; Ω_{R}^{|η|−1}(X); 0]

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

0 → Ω_{A}^{|η|}(K_{θ}) → K_{ηθ} ⊕ F → K_{η} → 0

of R-modules, where F is free.

Now suppose R is Gorenstein and X is a maximal Cohen-Macaulay (or “MCM” from now on) module. Then there exists a complete resolution

ℙ: ⋯ → P_{2} → P_{1} → P_{0} ⟶^{d} P_{−1} → P_{−2} → ⋯

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 Êxt_{R}^{n}(X, Y) is defined as the n’th homology of the complex Hom_{R}(ℙ, Y). If X and Y are Â-modules and X

is MCM, then Êxt∗ ; Â(X, Y) = ⨁_{i=−∞}^{∞} Êxti ; Â(X, Y) is a module over the ring Â[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 Â[χ] the Â[χ]_{𝔮}-modules Êxt∗ ; Â(X, Y)_{𝔮} and Ext∗ ; Â(X, Y)_{𝔮} are isomorphic.

We are now ready to prove the first result, which shows that when “cutting down” the variety of an MCM Â-module by a homogeneous element, the resulting homogeneous algebraic set is also the variety of an Â-module.

Theorem 2.2. Let η ∈ H^{+} = (χ) be a homogeneous element, and let \overline{η} ∈ Â[χ] be a homogeneous element such that \overline{η} ⊗ 1 corresponds to η under the isomorphism H ≃ Â[χ]⊗Âk. Furthermore, let X be an Â-module, and let θ ∈ Ext∗ ; Â(X, X) be any homogeneous element such that θ ⊗ 1 = ϕ_{X}(\overline{η}) ⊗ 1 in E(X, X) = Ext∗ ; Â(X, X) ⊗Â k. Then there is an inclusion

V(K_{θ}) ⊆ V(X) ∩ V_{H}(η),

and equality holds whenever X is MCM.

Proof. Consider the exact sequence

0 → X → K_{θ} → Ω|η|−1 ; Â(X) → 0

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 V(K_{θ}) ⊆ V_{H}(η).

The exact sequence induces a long exact sequence

[figure: 0; HomÂ(Ω|η|−1 ; Â(X), k); HomÂ(K_{θ}, k); HomÂ(X, k); ∘θ; Ext|η| ; Â(M, k); Ext1 ; Â(K_{θ}, k); Ext1 ; Â(X, k); ∘(−θ); Ext|η|+1 ; Â(M, k); Ext2 ; Â(K_{θ}, k); Ext2 ; Â(X, k); ⋮; ⋮; ⋮; ∘(−1)^{i}θ; Ext|η|+i ; Â(M, k); Exti+1 ; Â(K_{θ}, k); Exti+1 ; Â(X, k); ⋮; ⋮; ⋮]

in cohomology, from which we obtain the short exact sequence

0 → \frac{Ext∗+|η| ; Â(X, k)}{Ext∗ ; Â(X, k) ∘ θ} ⟶^{f} Ext∗+1 ; Â(K_{θ}, k) ⟶^{g} Ker(∘θ)|_{Ext∗+1 ; Â(X,k)} → 0.

Now for any Â-modules W and Z the left and right scalar actions from Â[χ] on Ext∗ ; Â(W, Z), through the ring homomorphisms ϕ_{Z} and ϕ_{W}, respectively, are actually equal (see [AvB, 1.1.2]). Consequently Ext∗ ; Â(X, k) ∘ θ is an Â[χ]-submodule of Ext∗+|η| ; Â(X, k), and the above short exact sequence is a sequence of Â[χ]-modules and maps. Moreover, the end terms are both annihilated by the element \overline{η}. To see this, note that since θ ⊗ 1 = ϕ_{X}(\overline{η}) ⊗ 1 in Ext∗ ; Â(X, X) ⊗Â k, the element ϕ_{X}(\overline{η}) − θ ∈ Ext∗ ; Â(X, X) can be written as a finite sum

ϕ_{X}(\overline{η}) − θ = ∑ m_{i}θ_{i},

where m_{i} ∈ 𝔪̂ and θ_{i} ∈ Ext∗ ; Â(X, X). If G^{∗} = ⊕_{i=0}^{∞}G^{i} is any graded right Ext∗ ; Â(X, X)-module annihilated by θ, and with the property that each graded part G^{i} is finitely generated over Â, then

[G^{i} · ϕ_{X}(\overline{η})] ⊗Â k = [G^{i} · (θ + ∑ m_{i}θ_{i})] ⊗Â k = 0.

This implies that G^{i} · ϕ_{X}(\overline{η}) vanishes itself, hence \overline{η} annihilates G^{∗}. In particular, the element \overline{η} annihilates the end terms in the above short exact sequence.

Now for any i ≥ 0, let w be an element of Exti ; Â(K_{θ}, k), and consider the element \overline{η} · w ∈ Exti+|η| ; Â(K_{θ}, k). Since g(\overline{η} · w) = \overline{η} · g(w) = 0, there must exist an element

z ∈ \frac{Ext∗+|η| ; Â(X,k)}{Ext∗ ; Â(X,k)∘θ} with the property that \overline{η} · w = f(z), giving \overline{η}^{2} · w = f(\overline{η} · z) = 0.

Therefore the element \overline{η}^{2} annihilates Exti ; Â(K_{θ}, k), and so the element η^{2} ∈ H is contained in Ann_{H} E(K_{θ}, k). This gives the inclusion V(K_{θ}) ⊆ V_{H}(η^{2}) = V_{H}(η), 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 \overline{𝔭} ≠ (χ) of Â[χ] corresponding to 𝔭 and containing \overline{η} and the annihilator of Ext∗ ; Â(X, k), and suppose \overline{𝔭} does not contain the annihilator of Ext∗ ; Â(K_{θ}, k). The exact sequence from the beginning of the proof induces a long exact sequence

⋯ → Êxti ; Â(K_{θ}, k) → Êxti ; Â(X, k) ⟶^{∘(−1)^{i}θ} Êxti+|η| ; Â(X, k) → Êxti+1 ; Â(K_{θ}, k) → ⋯

in stable cohomology, which in turn gives the exact sequence

0 → \frac{Êxt∗+|η|−1 ; Â(X, k)}{Êxt∗−1 ; Â(X, k) ∘ θ} → Êxt∗ ; Â(K_{θ}, k),

in which the index ∗ ranges over all the integers. Now let

⋯ → P_{2} ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} P_{−1} ⟶^{d_{−1}} P_{−2} → ⋯

be a complete resolution of X, and consider the group Êxt−n ; Â(X, k) for any nonnegative integer n. Since Êxt−n ; Â(X, k) = Êxt1 ; Â(Ker d_{−(n+2)}, k) = Ext1 ; Â(Ker d_{−(n+2)}, k),

the ring Â[χ] acts on Êxt−n ; Â(X, k) from both sides, and these actions coincide. Therefore Êxt∗ ; Â(X, k) ∘ θ is an Â[χ]-submodule of Êxt∗+|η| ; Â(X, k), and the above short exact sequence is a sequence of Â[χ]-modules.

Now recall from the discussion prior to this theorem that Êxt∗ ; Â(W, Z)\overline{𝔭} ≃ Ext∗ ; Â(W, Z)\overline{𝔭} for any Â-modules W and Z with W MCM. As \overline{𝔭} does not contain the annihilator of Ext∗ ; Â(K_{θ}, k), we see by localizing the above short exact sequence at \overline{𝔭} that Êxt∗ ; Â(X, k)\overline{𝔭} = [Êxt∗ ; Â(X, k) ∘ θ]\overline{𝔭}. Since θ = ϕ_{X}(\overline{η}) − ∑ m_{i}θ_{i} in the ring

Ext∗ ; Â(X, X), and the ideal \overline{𝔭} contains both \overline{η} and the m_{i} (it contains the element \overline{η} by assumption, and contains the ideal 𝔪̂ because it corresponds to the ideal 𝔭 ⊆ H under the isomorphism Â[χ] ⊗Â k ≃ H), the Â(χ)\overline{𝔭}-module [Êxt∗ ; Â(X, k) ∘ θ]\overline{𝔭} must

be contained in \overline{𝔭}Â(χ)\overline{𝔭} · Êxt∗ ; Â(X, k)\overline{𝔭}. Consequently the inclusions

Êxt∗ ; Â(X, k)\overline{𝔭} ⊆ \overline{𝔭}Â(χ)\overline{𝔭} · Êxt∗ ; Â(X, k)\overline{𝔭} ⊆ Êxt∗ ; Â(X, k)\overline{𝔭}

hold, and so Êxt∗ ; Â(X, k)\overline{𝔭} = \overline{𝔭}Â(χ)\overline{𝔭} · Êxt∗ ; Â(X, k)\overline{𝔭}. But Êxt∗ ; Â(X, k)\overline{𝔭}, being isomorphic to Ext∗ ; Â(X, k)\overline{𝔭}, is finitely generated over Â[χ]\overline{𝔭}, hence Nakayama’s Lemma implies Ext∗ ; Â(X, k)\overline{𝔭} = 0. This contradicts the assumption that \overline{𝔭} contains the annihilator of Ext∗ ; Â(X, k), and therefore \overline{𝔭} must contain the annihilator of Ext∗ ; Â(K_{θ}, k). But then Ann_{H} E(K_{θ}, k) ⊆ 𝔭, giving the inclusion

\sqrt{Ann_{H} E(K_{θ}, k)} ⊆ \sqrt{(η, Ann_{H} E(X, k))}

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 M̂. Let η ∈ H^{+} be a homogeneous element, let \overline{η} ∈ Â[χ] be a corresponding element, and consider the element ϕM̂(\overline{η}) ⊗ 1 in Ext∗ ; Â(M̂, M̂) ⊗Â k. Since Ext∗ ; Â(M̂, M̂) is isomorphic to Ext_{A}^{∗}(M, M) ⊗_{A} Â, there is an isomorphism

Ext_{A}^{∗}(M, M) ⊗_{A} k ⟶^{∼} Ext∗ ; Â(M̂, M̂) ⊗Â k

under which the image of an element θ ⊗ 1 ∈ Ext_{A}^{∗}(M, M) ⊗_{A} k is θ̂ ⊗ 1. Hence there exists a homogeneous element θ^{M} ∈ Ext_{A}^{|η|}(M, M) such that θ̂^{M} ⊗ 1 equals the element ϕM̂(\overline{η}) ⊗ 1 in Ext∗ ; Â(M̂, M̂) ⊗Â k. Now if θ^{M} is represented by the exact sequence

0 → M → K_{θ^{M}} → Ω_{A}^{|η|−1}(M) → 0,

then its completion θ̂^{M} is represented by the sequence

0 → M̂ → { \atop K_{θ^{M}}} → Ω|η|−1 ; Â(M̂) → 0,

whose middle term is the completion of an A-module. By Theorem 2.2 the inclusion

V(K_{θ^{M}}) ⊆ V(M) ∩ V_{H}(η)

holds, with equality holding whenever M is MCM, and consequently we obtain the following corollary, showing that every homogeneous algebraic set in k̃^{c} is the variety of an MCM A-module.

Corollary 2.3. Every closed homogeneous variety in k̃^{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_{A}^{dim A}(k). Then V(M) = V(k) = k̃^{c}, and from the theorem and the above discussion we see that there exists a homogeneous element θ_{1} ∈ Ext_{A}^{|η_{1}|}(M, M) with the property that V(K_{θ_{1}}) = V(M)∩V_{H}(η_{1}) = V_{H}(η_{1}). Repeating the process with η_{2}, …, η_{t} we end up with an MCM A-module K such that

V(K) = V_{H}(η_{1}) ∩ ⋯ ∩ V_{H}(η_{t}) = V_{H}(η_{1}, …, η_{t}).
□

Remarks. (i) In an unpublished preprint (as of December 2006), L. Avramov and D. Jorgensen obtain a different proof of Corollary 2.3, based on a result concerning the realization of certain graded modules as cohomology modules. L. Avramov reported it at a meeting at MSRI, Berkeley, in December 2002.

(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]).

3. Decomposition

Before proving the next result, recall that an MCM-approximation of an Amodule X is an exact sequence

0 → Y_{X} → C_{X} ⟶^{f} X → 0

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 = fg 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 → Â is a faithfully flat local homomorphism, an A-module Z has finite projective dimension if and only if the Â-module Ẑ has finite projective dimension, and it follows from [Mat, Theorem 23.3] that Z is MCM if and only if Ẑ 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

0 → Y_{X} → C_{X} ⟶^{f} X → 0

is a minimal MCM-approximation if and only if

0 → Ŷ_{X} → Ĉ_{X} \underrightarrow{f̂} X̂ → 0

is a minimal MCM-approximation.

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.

Theorem 3.1. If for an A-module M we have V(M) = V_{1} ∪ V_{2} where V_{1} and V_{2} are closed homogeneous varieties having trivial intersection, then the completion Ĉ_{M} of the minimal MCM-approximation of M decomposes as Ĉ_{M} = C_{1} ⊕ C_{2} with V(C_{i}) = V_{i}.

Proof. Let

0 → Y → C → M → 0

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(X̂) 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}) in k̃^{c} defined by 𝔞_{i} for i = 1, 2. We then have equalities

{0} = V_{1} ∩ V_{2} = V_{H}(𝔞_{1}) ∩ V_{H}(𝔞_{2}) = V_{H}(𝔞_{1} + 𝔞_{2}),

and so it follows from Hilbert’s Nullstellensatz that for each 1 ≤ i ≤ c we have χ_{i} ∈ \sqrt{𝔞_{1} + 𝔞_{2}}. Therefore \sqrt{𝔞_{1} + 𝔞_{2}} is the graded maximal ideal H^{+} of H, i.e. \sqrt{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

θ_{1} ∈ 𝔞_{1}, dim (V_{2} ∩ V_{H}(θ_{1})) < dim V_{2},
θ_{2} ∈ 𝔞_{2}, dim (V_{1} ∩ V_{H}(θ_{2})) < dim V_{1}.

Now since V_{H}(θ_{1}θ_{2}) = V_{H}(θ_{1}) ∪ V_{H}(θ_{2}) ⊇ V_{1} ∪ V_{2} = V(C), it follows once more from Hilbert’s Nullstellensatz that θ_{1}θ_{2} ∈ \sqrt{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} = \overline{θ}_{i} ⊗ 1, where \overline{θ}_{1} and \overline{θ}_{2} are homogeneous elements of positive degrees in A[χ] with the property that \overline{θ}_{1}\overline{θ}_{2} ∈ Ann_{A[χ]} Ext_{A}^{∗}(C, C). To see the latter, note that 0 = θ_{1}θ_{2} (Ext_{A}^{i}(C, C) ⊗_{A} k) = \overline{θ}_{1}\overline{θ}_{2} Ext_{A}^{i}(C, C) ⊗_{A} k for every i ≥ 0, and since \overline{θ}_{1}\overline{θ}_{2} Ext_{A}^{i}(C, C) is a finitely generated A-module (\overline{θ}_{1}\overline{θ}_{2} commutes with elements in A), the claim follows.

Now consider the images θ_{1}^{C} and θ_{2}^{C} of \overline{θ}_{1} and \overline{θ}_{2} in Ext_{A}^{∗}(C, C). Since θ_{1}^{C}θ_{2}^{C} = 0, the bottom exact sequence in the exact commutative diagram

[figure: 0; Ω_{A}^{|θ_{1}^{C}|+|θ_{2}^{C}|}(C); f_{θ^{C}_{1} θ^{C}_{2}}; Q_{|θ_{1}^{C}|+|θ_{2}^{C}|−1}; Ω_{A}^{|θ_{1}^{C}|+|θ_{2}^{C}|−1}(C); 0; 0; C; K_{θ_{1}^{C}θ_{2}^{C}}; Ω_{A}^{|θ_{1}^{C}|+|θ_{2}^{C}|−1}(C); 0]

splits, where Q_{n} denotes the n’th module in the minimal free resolution of C. Therefore K_{θ_{1}^{C}θ_{2}^{C}} is isomorphic to C ⊕ Ω_{A}^{|θ_{1}^{C}|+|θ_{2}^{C}|−1}(C), and from Lemma 2.1 we see that there exists an exact sequence

(†)  0 → Ω_{A}^{|θ_{1}^{C}|}(K_{θ_{2}^{C}}) → C ⊕ Ω_{A}^{|θ_{1}^{C}|+|θ_{2}^{C}|−1}(C) ⊕ F → K_{θ_{1}^{C}} → 0

for some free module F. From Theorem 2.2 we have V(K_{θ_{i}^{C}}) = V(C) ∩ V_{H}(θ_{i}), hence the equality V(C) = V_{1} ∪V_{2} and the inclusion V_{i} ⊆ V_{H}(θ_{i}) give the equalities

V(K_{θ_{1}^{C}}) = V_{1} ∪ (V_{2} ∩ V_{H}(θ_{1})) ,
V(K_{θ_{2}^{C}}) = V_{2} ∪ (V_{1} ∩ V_{H}(θ_{2})) .

By induction there exist A-modules X_{1}, X_{2}, Y_{1} and Y_{2} such that K_{θ_{1}^{C}} = X_{1} ⊕ X_{2}

and Ω_{A}^{|θ_{1}^{C}|}(K_{θ_{2}^{C}}) = Y_{1} ⊕ Y_{2}, and such that

V(X_{1}) = V_{1},
V(X_{2}) = V_{2} ∩ V_{H}(θ_{1}),
V(Y_{1}) = V_{1} ∩ V_{H}(θ_{2}),
V(Y_{2}) = V_{2}.

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_{A}^{i}(X_{1}, Y_{2}) and Ext_{A}^{i}(X_{2}, Y_{1}) vanish for i ≫ 0. But K_{θ_{1}^{C}} is MCM, implying X_{1} and X_{2} are both MCM, and so it follows from [ArY, Theorem 4.2] that Ext_{A}^{i}(X_{1}, Y_{2}) and Ext_{A}^{i}(X_{2}, Y_{1}) vanish for i ≥ 1. Therefore

Ext_{A}^{1}(K_{θ_{1}^{C}} , Ω_{A}^{|θ_{1}^{C}|}(K_{θ_{2}^{C}})) = Ext_{A}^{1}(X_{1}, Y_{1}) ⊕ Ext_{A}^{1}(X_{2}, Y_{2}),

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_{A}^{|θ_{1}^{C}|+|θ_{2}^{C}|−1}(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.

Acknowledgements

I would like to express my gratitude to Lucho Avramov for numerous comments and improvements. Also, I would like to thank Dave Jorgensen and my supervisor Øyvind Solberg for valuable suggestions and comments on this paper.

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Petter Andreas Bergh Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: bergh@math.ntnu.no