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Canonical modules of complexes

We define the notion of the canonical module of a complex. We then consider Serre's conditions for a complex and study their relationship to the local cohomology of the canonical module and its ring of endomorphisms.


Introduction
The notion of the canonical module of a ring is an important tool in commutative algebra. P. Schenzel defined in [12] the notion of the canonical module of a module. The canonical module always satisfies Serre's condition (S 2 ). Schenzel related higher Serre's conditions to the vanishing of certain local cohomology modules of the canonical module. The purpose of this article is to extend these results to complexes. We utilize the powerful tools of hyperhomological algebra.
Let (R, m) be a Noetherian local ring admitting a dualizing complex. We work within the derived category D f b (R) of bounded complexes of R-modules with finitely generated homology. Generalizing the work of Schenzel, we define for any complex M ∈ D f b (R) and any i ∈ Z the ith module of deficiency K i M by setting K i M = H i (RHom R (M, D R ), where D R denotes the dualizing complex of R normalized so that H dim R (D R ) = 0 and H i (D R ) = 0 for i > dim R. The canonical module of M is then K M = K dim M M . Note that by local duality the local cohomology module H i m (M) is the Matlis dual of K i M . In particular, modules of deficiency measure how far the complex is from being Cohen-Macaulay.
Given k ∈ N, we say that a complex M satisfies Serre's condition for all prime ideals p ∈ Supp M. It is convenient to consider complexes satisfying the condition dim R M = dim Rp M p + dim R/p for every p ∈ Supp R M. Here Supp M means the homological support of M. It then follows from our Theorem 4.12 that (S k ) is equivalent to the natural homomorphism being bijective for all i ≥ −k + 2, and injective for i = −k + 1. Note that K M ⊗ L R K M = Hom R (K M , K M ). It makes also sense, for any l ∈ Z, to look at the condition (S k,l ) saying that for all prime ideals p ∈ Supp M. Observe that (S k ) always implies (S k,sup M ). It now turns out in Corollary 4.14 that (S k,l ) is equivalent to the natural homomorphism H i (M) → K i+t K M being bijective for i ≥ l − k + 2, and injective for i = l − k + 1. In the case M is a module and l = 0, this reduces to the result of Schenzel mentioned in the beginning.
Finally, we look at the complex M † := RHom R (M, D R where D(M) and D(H s (M)) denote the dimension filtrations of M and H s (M) respectively.

Preliminaries
The purpose of this section is to fix notation and recall some definitions and results of hyperhomological algebra relevant to this article. As a general reference, we mention [2] and references therein. For more details, see also [3], [4], [8] and [9].
In the following R is always a commutative Noetherian ring. If R is local, then m denotes the maximal ideal and k the residue field of R.
Throughout this article we work within the derived category D(R) of R-modules. We use homological grading so that the objects of D(R) are complexes of R-modules of the form M : . . .
The derived category is triangulated, the suspension functor Σ being defined by the formulas (ΣM) n = M n−1 and d ΣM n = −d n−1 . The symbol "≃" is reserved for isomorphisms in D(R). We use the subscripts "b", "+" and "−" to denote the homological boundness, the homological boundness from below and the homological boundness from above, respectively. The superscript "f " denotes the homological finiteness. So the full subcategory of D(R) consisting of complexes with finitely generated homology modules is denoted by D f (R). As usual, we identify the category of R-modules as the full subcategory of D(R) of complexes M satisfying H i (M) = 0 for i = 0. For a complex M ∈ D(R), by sup M and inf M, we mean its homological supremum and infimum. Let M and N be complexes of R-modules. We use the standard notations M ⊗ L R N and RHom R (M, N) for the left derived tensor product complex and the right derived homomomorphism complex, respectively. Moreover, we set Also note the formula The set of all anchor primes for M is denoted by Anc R M. The anchor primes play the role of minimal primes for complexes. Note Definition 2.1. Let (R, m) be a ring, and let M ∈ D + (R). We say that M is equidimensional, if Proposition 2.2. Let (R, m) be a catenary ring, and let M ∈ D + (R). Then the following conditions are equivalent: Here the first equality holds true, since R/q is a catenary integral do- If (R, m) is a local ring, the derived local cohomology functor with respect to m is denoted by RΓ m . As usual, we set [8, 2.4]). If R admits a dualizing complex, we denote by D R the dualizing complex normalized with sup We obtain a contravariant functor (−) † : where E R (k) denotes the injective envelope of k. We will frequently use the formulas [5,Proposition 7.2.7]). By biduality, we then have sup M = dim R M † and inf M = depth R M † . Also observe that Here the dagger dual on the left-hand side is taken with respect to the normalized dualizing complex of the localization R p .
Proposition 2.3. Let (R, m) be a local ring admitting a dualizing complex, and let M ∈ D f b (R). Then the following statements are equivalent: Proof. We know by Proposition 2.2 and formula (7) that M † being equidimensional is equivalent to On the other hand, by using formula (9) we get where the last equality comes from formula (7). So a) and b) are equivalent.

Modules of deficiency of a complex
We extend the definition given by P. Schenzel in [12, p. 62] to the case of complexes: Remark 3.2. Obviously, the modules of deficiency are finitely generated. Using formulas (7) and (8), we get Example 3.3. Any finitely generated module is a canonical module of a complex. Indeed, if K is a finitely generated R-module and t ∈ Z, set M = −t K † . Since dim R M = t by formula (7), it now follows by biduality that Lemma 3.4. Let (R, m) be a local ring admitting a dualizing complex, Proof. a) Using formula (9) we get Our next aim is to investigate the associated primes of modules of deficiency. From now on we set Lemma 3.5. Let (R, m) be a local ring admitting a dualizing complex, and let M ∈ D f b (R). Then the following statements hold for all i ∈ Z: Proof. a) Using formula (2) we have On the other hand, we have (5). It now follows that We can now identify the set of the associated primes and the support of the canonical module of a complex.
Proposition 3.6. Let (R, m) be a local ring admitting a dualizing complex, and let M ∈ D f b (R). Then a) Proof. a) Let p ∈ Supp R M. We apply formula (3) to M † . Because sup M † = dim R M by formula (7), it thus follows that p ∈ Ass R K M if and only if depth Rp (M † ) p = − dim R M. Using formulas (9) and (7) we get where the first inequality is clear by the definition of Krull dimension and the subsequent equality holds true, since R/q is a catenary integral domain. Because by a), this shows that Taking into account inequality (1), we get Suppose then that the above equality holds for p ∈ Supp R M. By Lemma 3.4 b) we have (K M ) p ∼ = K Mp = 0. Thus p ∈ Supp R K M , and we are done.
Remark 3.7. We observe that by Proposition 3.6 a) and Lemma 3.5 c)

Main results
We want to generalize Serre's conditions to complexes. It is convenient to begin with the following very general definition: The following proposition is now immediate:    Given an integer n, recall that the soft truncation of a complex M above at n is the complex In D(R) we now have an exact triangle Remark 4.8. Clearly (11) H otherwise.
In particular, when M ≃ 0, we have C M ≃ 0 if and only if M is Cohen-Macaulay. If this is not the case, then inf C M = depth M.
Remark 4.9. Because sup M † = dim M by formula (7), we obtain an exact triangle An application of the functor (−) † to (12) yields an exact triangle 10. Let (R, m) be a local ring admitting a dualizing complex, and let Indeed, by using (9) and Proposition 2.2, we get Mp so that by (9) (C † M ) p ≃ C †p Mp as wanted. Lemma 4.11. Let (R, m) be a local ring admitting a dualizing complex, and let M, N ∈ D f b (R). Suppose that M is an equidimensional complex. Then the following conditions are equivalent: c) ⇔ d) : Using adjointness and formula (7) we get The claim follows, because by [5, Proposition 6.3.9 b), (E.6.3.1)] and Remark 4.8 we have Proof. By applying the functor RHom R (N, −) on (13), we get the exact triangle N ) is by Lemma 4.11 equivalent to sup RHom R (N, C † M ) ≤ −k, a look at the corresponding long exact sequence of homology implies the claim.
In particular, by taking N = M, this immediately applies to Serre's condition (S k ). For this case, we observe the following Proof. By adjointness . Since Hom R (K M , K M ) = 0, it follows from formula (7) and [2, The desired isomorphism now follows from the corresponding long exact sequence of homology, because sup RHom R (K M , C M ) ≤ sup C M by [2, Proposition A.4.6].
Let us then consider the conditions (S k,l ).
Corollary 4.14. Let (R, m) be a local ring admitting a dualizing complex. Let k, l ∈ Z and M ∈ D f b (R). Set t = dim R M. If M is equidimensional, then the following conditions are equivalent: ) is bijective for i ≥ l − k + 2, and surjective for i = l − k + 1.
Proof. The equivalence of a) and b) follows immediately from Theorem 4.12 by taking N = l R whereas the homomorphism of c) is by local duality the Matlis-dual of that of b).
If R is a ring and N is an R-module, we use the notation We now turn to look at the dagger dual: Proof. Note first that by using formula (9) together with formulas (7) and (8)  We also have inf(M † ) p = dim R/p + inf M †p p = dim R/p + depth Rp M p . The claim then follows from Proposition 4.3.
In order to apply Theorem 4.12 in this case, we need   is bijective for all i ≥ −k + 2, and injective for i = −k + 1.
Proof. Note that dim R M † = s by formula (7). By dagger duality K M † = H s (M). By adjointness and biduality we then get . The claim is then a direct consequence of Lemma 4.18 and Theorem 4.12.
In a similar way, Corollary 4.14 yields    Let R be a ring. Recall that a filtration of Spec R is a descending sequence F : . . . ⊇ F i−1 ⊇ F i ⊇ F i+1 ⊇ . . . of subsets of Spec R such that i F i = ∅, F i = Spec R for some i ∈ Z and each p ∈ F i \ F i+1 is a minimal element of F i with respect to inclusion. Let E F (M) denote the Cousin complex corresponding to a complex M ∈ Observe that we here grade the Cousin complex homologically in contrary to the general tradition. For more details about Cousin complexes we refer to [11,Chapter IV,§3]. Note that if M is an R-module, then the Cousin complex studied by Sharp (see [15], for example) is a com-