On complexes of finite complete intersection dimension

In [DGI], the authors raised the question whether every nonzero homologically finite complex of finite complete intersection dimension over a local ring is virtually small. In other words, given such a ring A and such a complex M ∈ D(A), is the intersection thick D(A) (M ) ∩ thick D(A) (A) nonzero? We give an affirmative answer to this question, by showing that the thick subcategory generated by M contains a nonzero complex of complexity zero.

In fact, we show that if the complexity of M is c, then thick D(A) (M ) contains a nonzero complex of complexity t for every 0 ≤ t ≤ c. The homologically finite complexes of complexity zero are precisely the complexes of finite projective dimension. Moreover, a homologically finite complex belongs to thick D(A) (A) if and only if its projective dimension is finite. Thus, the results mentioned indeed settle the above question.

Let (A, m, k) be a local (meaning commutative Noetherian local) ring, and denote by D(A) the derived category of (not necessarily finitely generated) A-modules.

for n ≪ 0, and bounded above if M n = 0 for n ≫ 0. The complex is bounded if it is both bounded below and bounded above, and finite if it is bounded and degreewise finitely generated. The homology of M , denoted H(M ), is the complex with H(M ) n = H n (M ), and with trivial differentials.

When H(M ) is finite, then M is said to be homologically finite.

As shown for example in [Rob], when the complex M is homologically finite, then it has a minimal free resolution. Thus, there exists a quasi-isomorphism F ≃ M , where F is a bounded below complex

of finitely generated free modules, and where Im d n ⊆ m F n-1 . This resolution is unique up to isomorphism, and so for each integer n the rank of the free module F n is a well defined invariant of M . This is the nth Betti number β n (M ) of M , and the corresponding generating function

The complexity of a homologically finite complex is not necessarily finite. In fact, by a theorem of Gulliksen (cf. [Gul]), finiteness of the complexities of all homologically finite complexes in D(A) is equivalent to A being a complete intersection ring.

Suppose that our complex M is homologically finite, with a minimal free resolution F . Given a complex N ∈ D(A), the complex Hom A (F, N ) is denoted by R Hom A (M, N ). Up to quasi-isomorphism, this complex is well defined, hence so is the cohomology group

for every integer n. The projective dimension of M , denoted pd A M , is defined as

, which is the same as the supremum of all integers n such that F n is nonzero. Namely, since the complex F is minimal, the differentials in the complex Hom A (F, k) are trivial, and

In particular, the projective dimension of M is finite if and only if its complexity is zero.

The derived category D(A) is triangulated, the suspension functor Σ being the left shift of a complex. Given complexes M and N as above, for each n we may identify the cohomology group Ext

Let A be a local ring. Recall that a quasi deformation of A is a diagram A → R ← Q of local homomorphisms, in which A → R is faithfully flat, and R ← Q is surjective with kernel generated by a regular sequence. A homologically finite complex M ∈ D(A) has finite complete intersection dimension if there exists such a quasi deformation for which pd Q (R ⊗ A M ) is finite. From now on, we write “CI-dimension” instead of “complete intersection dimension”.

The notion of CI-dimension was first introduced for modules in [AGP]. The terminology reflects the fact that a local ring is a complete intersection precisely when all its finitely generated modules have finite CI-dimension. The same holds if we replace “modules” with “homologically finite complexes”.

In order to prove the main result, we need the following lemma. It shows that finite CI-dimension is preserved in thick subcategories.

Lemma 3.1. Let A be a local ring, and let M ∈ D(A) be a homologically finite complex of finite CI-dimension. Then every complex in thick D(A) (M ) has finite CI-dimension.

Proof. Let A → R և Q be a quasi-deformation of A such that pd Q (R ⊗ A M ) is finite. We show by induction that for all n ≥ 1, every complex

Next, suppose the claim holds for all 1, . . . , n, and let X be a complex in thick n

, where Y and Z are complexes in thick n D(A) (M ) and thick 1

the proof is complete.

Next, we prove the main result; the thick subcategory generated by a complex of finite CI-dimension contains complexes of all possible complexities. Theorem 3.2. Let A be a local ring, and let M ∈ D(A) be a nonzero homologically finite complex of finite CI-dimension. Then for every 0 ≤ t ≤ cx M , there exists a nonzero complex in thick D(A) (M ) of complexity t.

Proof. The proof is by induction on the complexity c of M . If c = 0, then there is nothing to prove, so suppose that c is nonzero. We shall construct a nonzero complex in thick

) generated by the central elements in degree two. By [AvS,Corollary 5

where k is the residue field of A. Now

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