The Cohen-Macaulay property in derived commutative algebra
aa r X i v : . [ m a t h . A C ] J a n THE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVEALGEBRA
LIRAN SHAULA
BSTRACT . By extending some basic results of Grothendieck and Foxby about local co-homology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementingresults of Jørgensen and resolving a recent conjecture of Minamoto. When these inequali-ties are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make adetailed study of this notion, showing that much of the classical theory of Cohen-Macaulayrings and modules can be generalized to the derived setting, and that there are many natu-ral examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-ringsare local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allow-ing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but wealso discuss extensions of it to non-negative DG-rings, which could lead to the concept ofCohen-Macaulayness in topology. C ONTENTS
0. Introduction 11. Preliminaries 52. Local cohomology Krull dimension over commutative local DG-rings 123. Depth and local cohomology over commutative local DG-rings 154. Local-Cohen-Macaulay commutative DG-rings 195. Regular sequences and the derived Bass conjecture 226. Local-Cohen-Macaulay DG-modules 307. Trivial extension DG-rings and the local-Cohen-Macaulay property 348. Finite maps, localizations and global Cohen-Macaulay DG-rings 369. Some remarks on non-negatively graded commutative DG-rings 39References 400. I
NTRODUCTION
In classical commutative algebra, the classes of Gorenstein and Cohen-Macaulay ringsare among the most important classes of local rings. In particular, the theory of Cohen-Macaulay rings and modules is among the most deep and influential parts of commutativealgebra, with numerous applications in commutative algebra, algebraic geometry and com-binatorics.The Gorenstein condition has been introduced long ago to higher algebra and relatedfields. Its first incarnation was probably in the work [11] of Félix, Halperin and Thomasabout Gorenstein spaces in topology. Some other occurrences of it are in the works of
Mathematics Subject Classification
Avramov and Foxby [1] and Frankild, Iyengar and Jørgensen [13, 14] about GorensteinDG-rings, of Dwyer, Greenlees and Iyengar [10] about Gorenstein S -algebras (where S isthe sphere spectrum), in the work of Lurie about Gorenstein spectral algebraic spaces [21,Chapter 6.6.5] and many more.Despite the great success of the Gorenstein condition in higher algebra, and of theCohen-Macaulay condition in classical commutative algebra, until now it was completelymissing from higher algebra. The aim of this paper is to extend the theory of Cohen-Macaulay rings and Cohen-Macaulay modules to the setting of commutative noetheriandifferential graded rings.We work with commutative non-positive DG-rings A = L n = −∞ A n with a differentialof degree +1 . These include (and in characteristic zero are equivalent to) the normaliza-tions of the simplicial commutative rings, so they include affine derived schemes.Given a commutative DG-ring (or a ring) A , we denote by D ( A ) the unbounded derivedcategory of A -modules. For M ∈ D ( A ) , its amplitude is the number (or + ∞ ) amp( M ) = sup { i | H i ( M ) = 0 } − inf { i | H i ( M ) = 0 } . For a ring A , we denote by dim( A ) the Krull dimension of A , and similarly for an A -module M , dim( M ) is the Krull dimension of M . To describe the main results of thispaper, let us first summarize some important facts from the classical theory of Cohen-Macaulay rings which we are going to generalize: Classical Theorem A.
Let ( A, m ) be a noetherian local ring. Then the following areequivalent:(1) The ring A is Cohen-Macaulay.(2) There is an A -regular sequence x , . . . , x d ∈ m ⊆ A of length d = dim( A ) which is a system of parameters of A , and moreover, for each ≤ i ≤ d , the ring A i = A/ ( x , . . . , x i ) is a Cohen-Macaulay ring and dim( A i ) = dim( A ) − i .(3) One has amp (RΓ m ( A )) = 0 , i.e, the local cohomology of A is concentrated in asingle degree.(4) The m -adic completion ( b A, b m ) is a Cohen-Macaulay ring.(5) The Bass conjecture holds: there exists a finitely generated A -module M = 0 offinite injective dimension; that is, = M ∈ D bf ( A ) such that amp( M ) = 0 and inj dim A ( M ) < ∞ .If moreover A has a dualizing complex R then this is also equivalent to:(6) One has amp ( R ) = 0 , i.e, A has a dualizing module.Furthermore, the following rings satisfy these equivalent statements:(a) Gorenstein rings.(b) Local rings A with dim( A ) = 0 . A few remarks are in order. First, that item (5) holds is the proof of the Bass conjecture,which was introduced in [3], due to Peskine and Szpiro (see [24]). A noetherian localring has a dualizing complex if and only if it a quotient of a Gorenstein ring. This is atheorem of Kawasaki (see [20]), proving a conjecture of Sharp. The fact that a local ringwith a dualizing complex is Cohen-Macaulay if and only if it has a dualizing module is aconsequence of Grothendieck’s local duality theorem.Next, let us recall some basics of the theory of Cohen-Macaulay modules over Cohen-Macaulay rings which we will generalize:
HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 3
Classical Theorem B.
Let ( A, m ) be a noetherian local Cohen-Macaulay ring with a du-alizing module R . Denote by CM ( A ) the category of Cohen-Macaulay A -modules, and by MCM ( A ) its full subcategory of maximal Cohen-Macaulay A -modules. Then the followinghold: • The functor D ( − ) := R Hom A ( − , R ) induces a duality on CM ( A ) . More pre-cisely, if M ∈ CM ( A ) , then D ( M ) is a shift of an object of CM ( A ) , and thenatural map M → D ( D ( M )) is an isomorphism. • The above duality restricts to a duality on
MCM ( A ) , so that if M ∈ MCM ( A ) ,then its dual D ( M ) is a shift of an object in MCM ( A ) .Moreover, we have that A, R ∈ MCM ( A ) . We wish to generalize these results to derived commutative algebra. We say that a DG-ring A is noetherian if the ring H ( A ) is a noetherian ring, and for each i < the H ( A ) -module H i ( A ) is finitely generated. If A is noetherian and (H ( A ) , ¯ m , k ) is a local ring,we say that ( A, ¯ m ) (or ( A, ¯ m , k ) ) is a noetherian local DG-ring. We will recall in Section1.5 the notion of local cohomology of a DG-ring A with respect to a finitely generatedideal in H ( A ) . In particular, if ( A, ¯ m ) is a noetherian local DG-ring, attached to it is thelocal cohomology functor RΓ ¯ m : D ( A ) → D ( A ) . The analogue of the notion of a dualizing complex over a DG-ring is called a dualizing DG-module, and is recalled in Section 1.4. The notion of a regular sequence in the DG-settingis recalled in Definition 5.2.As a first step to generalize Classical Theorem A, we prove the following new inequal-ities about the amplitude of local cohomology and of dualizing DG-modules, and on thelength of regular sequences over noetherian local DG-rings:
Theorem 1.
The following inequalities hold:(1) If ( A, ¯ m ) is a noetherian local DG-ring with bounded cohomology then amp( A ) ≤ amp (RΓ ¯ m ( A )) . (2) If ( A, ¯ m ) is a noetherian local DG-ring with bounded cohomology, and seq . depth( A ) denotes the maximal length of an A -regular sequence contained in ¯ m , then seq . depth( A ) ≤ dim(H ( A )) . (3) If A is a noetherian DG-ring, and R is a dualizing DG-module over A then amp( A ) ≤ amp( R ) . This result is contained in Theorem 4.1 and Corollary 5.5 below. It is worth noting thatitem (2) above is non-trivial: Given a noetherian local DG-ring ( A, ¯ m ) , and given ¯ x ∈ ¯ m ,the noetherian local DG-ring A// ¯ x is given by a Koszul-complex type construction whichis recalled in the beginning of Section 5. Unlike rings, the situation for DG-rings is thateven if ¯ x ∈ ¯ m is A -regular, it could happen that dim(H ( A// ¯ x )) = dim(H ( A )) (seeExample 7.2).Given a commutative DG-ring A , and a finitely generated ideal ¯ a ⊆ H ( A ) , the derived ¯ a -adic completion of A , denoted by LΛ( A, ¯ a ) , is a commutative DG-ring, defined in [27],and recalled in Section 1.6 below.In view of the inequalities in Theorem 1, it is natural to study DG-rings for which theseare equalities. Let us say that a noetherian local DG-ring ( A, ¯ m ) with bounded cohomol-ogy is local-Cohen-Macaulay if there is an equality seq . depth( A ) = dim(H ( A )) . We LIRAN SHAUL characterize local-Cohen-Macaulay DG-rings in the next result which is a precise derivedanalogue of Classical Theorem A.
Theorem 2.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology. Thenthe following are equivalent:(1) The DG-ring A is local-Cohen-Macaulay, i.e, seq . depth( A ) = dim(H ( A )) .(2) There exists an A -regular sequence ¯ x , . . . , ¯ x d ∈ ¯ m ⊆ H ( A ) of length d =dim(H ( A )) which is a system of parameters of H ( A ) , and moreover, for each ≤ i ≤ d , the DG-ring A i = A// (¯ x , . . . , ¯ x i ) is local-Cohen-Macaulay, andthere is an equality dim(H ( A i )) = dim(H ( A )) − i .(3) There is an equality amp (RΓ ¯ m ( A )) = amp( A ) .(4) The derived ¯ m -adic completion LΛ( A, ¯ m ) is local-Cohen-Macaulay.(5) The analogue of the Bass conjecture holds: there exists = M ∈ D bf ( A ) suchthat amp( M ) = amp( A ) , and inj dim A ( M ) < ∞ . If moreover A has a dualizing DG-module R then this is also equivalent to:(6) There is an equality amp ( R ) = amp( A ) .Furthermore, the following DG-rings are local-Cohen-Macaulay:(a) Local Gorenstein DG-rings.(b) Local DG-rings ( A, ¯ m ) with dim(H ( A )) = 0 . The proof of this result takes the majority of Sections 4 and 5 below. The reason for theterminology local-Cohen-Macaulay is that, unlike the case of rings, this property neednot be preserved under localization. See Section 8 below for a discussion and for a globalvariant of this property.Next, we study local-Cohen-Macaulay DG-modules over a local-Cohen-Macaulay DG-ring, and prove an analogue of Classical Theorem B. We give in Section 6 below definitionsof local-Cohen-Macaulay DG-modules and maximal local-Cohen-Macaulay DG-modules,and show in Section 6 that:
Theorem 3.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, and let R be adualizing DG-module over A . Denote by CM ( A ) the category of local-Cohen-MacaulayDG-modules over A . Let MCM ( A ) be the full subcategory of CM ( A ) which consists ofmaximal local-Cohen-Macaulay DG-modules over A . Then the following hold: • The functor
R Hom A ( − , R ) induces a duality on CM ( A ) . • The above duality restricts to a duality on
MCM ( A ) , so that if M ∈ MCM ( A ) ,then its dual R Hom A ( M, R ) is an object in MCM ( A ) .Moreover, we have that A, R ∈ MCM ( A ) . Let us now describe the rest of the contents of this paper. In Section 1 we gather variouspreliminaries about DG-rings that will be used throughout this paper. In sections 2 and 3 wemake a detailed study of local cohomology in the DG setting. The main result is Theorem2.15 which is a DG version of Grothendieck’s vanishing and non-vanishing theorems forlocal cohomology. We introduce the notion of a local-Cohen-Macaulay DG-ring in Section4, give examples, and study some its basic properties. Then, in Section 5 we study regularsequences, associated primes and other related notions in the DG-setting, following worksof Christensen and Minamoto. Using these ideas and our results about local cohomology,we show in Corollary 5.20 that: To be precise, we show that the other conditions imply this condition, and that the converse holds under theadditional assumption that A has a noetherian model, see Remark 5.25 for details why this extra assumption isneeded. HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 5
Theorem 4.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology. Thenthere exists a maximal A -regular sequence ¯ x , . . . , ¯ x n ∈ ¯ m such that ¯ x , . . . , ¯ x n can becompleted to system of parameters of H ( A ) . Using this result, we then prove in Theorem 5.22 a DG-version of the Bass conjecture,and a bit more generally:
Theorem 5.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology.(1) If A is local-Cohen-Macaulay, there exists ≇ M ∈ D bf ( A ) such that inj dim A ( M ) < ∞ , and such that amp( M ) = amp( A ) .(2) Assume further that A has a noetherian model. For any ≇ M ∈ D bf ( A ) such that inj dim A ( M ) < ∞ , we have that amp( M ) ≥ amp( A ) . If there exists such M with amp( M ) = amp( A ) , then A is local-Cohen-Macaulay. This amplitude inequality, which was mentioned in the abstract, solves a recent conjec-ture of Minamoto.Section 6 introduces local-Cohen-Macaulay and maximal local-Cohen-Macaulay DG-modules over a local DG-ring. Among its results, we prove the following general resultabout the structure of dualizing DG-modules over local DG-rings.
Theorem 6.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology. Setting n = amp( A ) and d = dim(H ( A )) , let R be a dualizing DG-module over A , normalizedso that inf( R ) = − d . Then the following hold:(1) For every ≤ i ≤ d there is an inequality dim (cid:0) H − i + n ( R ) (cid:1) ≤ i (2) We have that A is a local-Cohen-Macaulay DG-ring if and only if dim (cid:16) H sup( R ) ( R ) (cid:17) = d. This result is contained in Theorem 6.7.In Section 7 we consider the problem of determining when are DG-rings that arise fromtrivial extensions of local rings by cochain complexes are Cohen-Macaulay. We show inparticular that any Cohen-Macaulay module over a local ring give rise to a local-Cohen-Macaulay DG-ring.Section 8 discusses two points where the DG theory diverges from the classical theory:independence of the Cohen-Macaulay property from the base, and localization. We explainthe reason for this divergence, and construct an example of a local-Cohen-Macaulay DG-ring which has a localization who is not local-Cohen-Macaulay. We then give a globaldefinition of the notion of a Cohen-Macaulay DG-ring, and prove in Corollary 8.7 that anylocal-Cohen-Macaulay DG-ring which has a dualizing DG-module, and whose spectrumis irreducible, is a Cohen-Macaulay DG-ring in the global sense.In the final Section 9 we briefly discuss the problem of defining Cohen-Macaulay DG-rings in the case where DG-rings are non-negatively graded. Such DG-rings arise in topol-ogy. We explain that our amplitude inequalities described above do not hold in the non-negative case, and suggest a possible way to overcome this problem.1. P
RELIMINARIES
In this section we will gather various preliminaries about commutative DG-rings thatwill be used throughout this paper. A complete reference about derived categories of dif-ferential graded rings is the book [32], and a good summary is in [31, Section 1]. However,
LIRAN SHAUL our terminology will sometimes diverge from the terminology of [32], and we will explic-itly indicate such changes in terminology.1.1.
Basics about commutative DG-rings, noetherian conditions.
A differential gradedring (abbreviated DG-ring) is a graded ring A = ∞ M n = −∞ A n equipped with a Z -linear differential d : A → A of degree +1 , such that the Leibniz rule(1.1) d ( a · b ) = d ( a ) · b + ( − i · a · d ( b ) is satisfied for any a ∈ A i , b ∈ A j and any i, j ∈ Z . We will further say that A iscommutative (called strongly commutative in [32]) if b · a = ( − i · j · a · b , and moreover,if i is odd, then a = 0 . All DG-rings in this paper will be assumed to be commutative .A DG-ring A is called non-positive if A i = 0 for all i > . From now on, in the rest ofthis paper except Section 9, we will assume that all DG-rings are non-positive .Taking cohomology, note that H ( A ) has the structure of a commutative ring. We willoften denote it by ¯ A := H ( A ) . It is called the cohomological reduction of A . Note thatthere is a natural map of DG-rings π A : A → ¯ A . The set A of degree zero elements of A is also a commutative ring, and H ( A ) is a quotient of it.A differential graded-module M over A is a graded A -module M equipped with adifferential d : M → M of degree +1 which satisfies a Leibniz rule similar to (1.1).The DG-modules over A form an abelian category, denoted by DGMod( A ) , in whichthe morphisms are given by degree A -linear homomorphisms which respect the differ-ential. Inverting quasi-isomorphisms in DGMod( A ) , we obtain the derived category ofDG-modules over A , denoted by D ( A ) . It is a triangulated category. For any M ∈ D ( A ) ,and any n ∈ Z , we have that H n ( M ) is an H ( A ) -module.For any n ∈ Z there are smart truncation functors smt >n , smt ≤ n : D ( A ) → D ( A ) such that for all M ∈ D ( A ) , there are equalities H i (cid:0) smt >n ( M ) (cid:1) = (cid:26) H i ( M ) , if i > n , , if i ≤ n ,and H i (cid:0) smt ≤ n ( M ) (cid:1) = (cid:26) H i ( M ) , if i ≤ n , , if i > n .Moreover, there is a distinguished triangle smt ≤ n ( M ) → M → smt >n ( M ) → smt ≤ n ( M )[1] in D ( A ) .Given M ∈ D ( A ) , the infimum and supremum of M are the numbers (or ±∞ ) inf( M ) = inf { n ∈ Z | H n ( M ) = 0 } , sup( M ) = sup { n ∈ Z | H n ( M ) = 0 } . In the book [32], what we denote here by inf( M ) (resp. sup( M ) ) is denoted by inf(H( M )) (resp., sup(H( M )) ). We prefer the shorter notation used here, as this pa-per is entirely cohomological in nature, and we will never need to consider the non-cohomological infimum and supremum.The full subcategory of D ( A ) consisting of DG-modules M with inf( M ) > −∞ isdenoted by D + ( A ) , and the full subcategory of D ( A ) consisting of DG-modules M with HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 7 sup( M ) < ∞ is denoted by D − ( A ) . These are triangulated subcategories of D ( A ) . Theirintersection is also a triangulated subcategory of D ( A ) , the category of bounded DG-modules (called cohomologically bounded DG-modules in [32]), which will be denotedby D b ( A ) .Given M ∈ D b ( A ) , its amplitude is the number amp( M ) = sup( M ) − inf( M ) ∈ N . If M / ∈ D b ( A ) , we set amp( M ) = + ∞ . Again, in [32], this is called the cohomologicalamplitude, and is denoted by amp(H( M )) . We will say that a DG-ring A has bounded co-homology if amp( A ) < ∞ . Under the assumption that A is non-positive, this is equivalentto assuming that inf( A ) > −∞ .A DG-ring A is called noetherian (the terminology in [32] is cohomologically pseudo-noetherian) if the commutative ring H ( A ) is a noetherian ring, and for all i < , the H ( A ) -module H i ( A ) is finitely generated. See [28, Theorem 6.6] for a justification ofthis definition.If A is a noetherian DG-ring, we say that M ∈ D ( A ) has finitely generated cohomol-ogy if for all n ∈ Z , the H ( A ) -modules H n ( M ) are finitely generated. We denote by D f ( A ) the full triangulated subcategory of D ( A ) consisting of DG-modules with finitelygenerated cohomology. We also set D − f ( A ) = D f ( A ) ∩ D − ( A ) . Similarly we will consider D +f ( A ) , D bf ( A ) . All these are full triangulated subcategories of D ( A ) .If A is a noetherian DG-ring, and if the noetherian ring H ( A ) is a local ring withmaximal ideal ¯ m , we will say that ( A, ¯ m ) is a noetherian local DG-ring.We say that a noetherian DG-ring A has a noetherian model if there exist commutativeDG-rings B, C , . . . , C n , A , . . . , A n , and quasi-isomorphisms of DG-rings A ← C → A ← C → A ← C → · · · → A n ← B such that B is a noetherian ring, and for each i < , the B -module B i is finitely gen-erated. As shown in the proof of [30, Lemma 7.8], a sufficient condition for A to have anoetherian model is that there exists a noetherian ring k and a map of DG-rings k → A ,such that the induced map k → H ( A ) is essentially of finite type. The above zig-zag ofquasi-isomorphisms implies that the triangulated categories D ( A ) and D ( B ) are equiva-lent, and this equivalence respects standard derived functors (see [32, Theorem 12.7.2]), sowith regards to statements about their derived categories, we will be able to freely replace A with B .We do not know if any noetherian DG-ring has a noetherian model, though all noether-ian DG-rings that arise in nature do have a noetherian model. Nevertheless, with the excep-tion of one point (see Remark 5.25), we will avoid using the noetherian model assumptionin this paper, and will almost always make only the weaker noetherian assumption, withoutassuming the existence of a noetherian model.1.2. Reduction functors.
Given a commutative DG-ring A , the natural map of DG-rings π A : A → H ( A ) induces two functors R Hom A (H ( A ) , − ) : D + ( A ) → D + ( A ) , and − ⊗ L A H ( A ) : D − ( A ) → D − ( A ) which are sometimes called the reduction functors, and are extremely useful in studying D ( A ) . One reason for their usefulness is the following property: for any M ∈ D + ( A ) , by LIRAN SHAUL [28, Proposition 3.3] we have that(1.2) inf( M ) = inf (cid:0) R Hom A (H ( A ) , M ) (cid:1) , H inf( M ) ( M ) ∼ = H inf( M ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) Dually, for any M ∈ D − ( A ) , it follows from the proof of [30, Proposition 3.1] that(1.3) sup( M ) = sup (cid:0) M ⊗ L A H ( A ) (cid:1) , H sup( M ) ( M ) ∼ = H sup( M ) (cid:0) M ⊗ L A H ( A ) (cid:1) Injective DG-modules.
This section follows [28]. Given a commutative DG-ring A , and given M ∈ D + ( A ) , the injective dimension of M , denoted by inj dim A ( M ) , isdefined in [28, Section 2], similarly to the definition of injective dimension over rings.An injective DG-module is a DG-module M ∈ D + ( A ) such that either M ∼ = 0 , or theinjective dimension of M is , and moreover inf( M ) = 0 . The category of injectives over A is denoted by Inj( A ) . The functor H is an equivalence of categories H : Inj( A ) → Inj(H ( A )) . In particular, if ( A, ¯ m ) is a commutative noetherian local DG-ring, there is, unique up toisomorphism, DG-module E ∈ Inj( A ) such that H ( E ) is the injective hull of the residuefield of the local ring (H ( A ) , ¯ m ) . Following [28, Section 7], we will denote this DG-module by E ( A, ¯ m ) .1.4. Dualizing DG-modules.
We now recall the notion of a dualizing DG-module over anoetherian DG-ring. These generalize Grothendieck’s notion of a dualizing complex overnoetherian rings (see [16, Chapter V]). References for all facts in this section are [14, 30].Let A be a commutative noetherian DG-ring. We say that a DG-module R ∈ D +f ( A ) is adualizing DG-module over A if R has finite injective dimension over A , and the canonicalmap A → R Hom A ( R, R ) is an isomorphism in D ( A ) . It follows that if R is a dualizing DG-module over A , then forany M ∈ D f ( A ) , the natural map M → R Hom A (R Hom A ( M, R ) , R ) is an isomorphism in D ( A ) .Similarly to the (non-)uniqueness theorem for dualizing complexes over rings, a similarresult is true over DG-rings. In particular, if ( A, ¯ m ) is a noetherian local DG-ring, andif R , R are dualizing DG-modules over A , then there exists n ∈ Z such that R ∼ = R [ n ] . We say that a dualizing DG-module R over a noetherian local DG-ring ( A, ¯ m ) isnormalized if inf( R ) = − dim(H ( A )) .1.5. Local cohomology over commutative DG-rings.
Following [5, 27], let us recall thenotion of local cohomology over commutative DG-rings. Let A be a commutative DG-ring, and let ¯ a ⊆ H ( A ) be a finitely generated ideal. Recall that an H ( A ) -module ¯ M iscalled ¯ a -torsion if for any ¯ m ∈ ¯ M there exists n ∈ N such that ¯ a n · ¯ m = 0 , equivalently, if ¯ M = lim −→ Hom H ( A ) (H ( A ) / ¯ a n , ¯ M ) . The category of all ¯ a -torsion modules is a thick abelian subcategory of Mod(H ( A )) . Thisimplies that the category D ¯ a − tor ( A ) consisting of DG-modules M such that for all n ∈ Z ,the H ( A ) -module H n ( M ) is ¯ a -torsion, is a triangulated subcategory of D ( A ) . One canshow (see [5, 27] for details) that the inclusion functor D ¯ a − tor ( A ) ֒ → D ( A ) HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 9 has a right adjoint D ( A ) → D ¯ a − tor ( A ) , and composing this right adjoint with the inclusion, one obtains a triangulated functor RΓ ¯ a : D ( A ) → D ( A ) , which we call the derived torsion or local cohomology functor of A with respect to ¯ a .In case A = H ( A ) is a commutative noetherian ring, this coincide with Grothendieck’slocal cohomology functor, the (total) right derived functor of the ¯ a -torsion functor Γ ¯ a ( − ) := lim −→ Hom A ( A/ ¯ a n , − ) : Mod( A ) → Mod( A ) . Warning: If A = H ( A ) is a ring which is not noetherian, the functor RΓ ¯ a might bedifferent in general from the right derived functor of the ¯ a -torsion functor. In this paper,when we write RΓ ¯ a we will always mean the functor introduced above, which may bedifferent from the right derived functor of the ¯ a -torsion functor (The condition which guar-antees these functors coincide is that the ideal ¯ a is weakly proregular, which is always thecase if H ( A ) is noetherian). Remark 1.4.
Given a commutative DG-ring A and a finitely generated ideal ¯ a ⊆ H ( A ) ,we will sometimes need to work both local cohomology functor of A with respect to ¯ a ,and with the local cohomology functor of H ( A ) with respect to ¯ a . The former is a functor D ( A ) → D ( A ) , while the latter is a functor D (H ( A )) → D (H ( A )) . The notation weintroduced above doesn’t allow one to distinguish between the two. To fix this issue, whenwe work with both of these functors, we will denote the former by RΓ A ¯ a : D ( A ) → D ( A ) , and the latter by RΓ ¯ A ¯ a : D (H ( A )) → D (H ( A )) . To actually compute the functor RΓ ¯ a ( − ) , we recall the telescope complex, following[15, 25]. Given a commutative ring A , and given a ∈ A , the telescope complex associatedto A and a is the complex → ∞ M n =0 A → ∞ M n =0 A → in degrees , , with the differential being defined by d ( e i ) = (cid:26) e , if i = 0 , e i − − a · e i , if i ≥ . where we have let e , e , . . . denote the standard basis of the countably generated free A -module ⊕ ∞ n =0 A . We denote this complex by Tel( A ; a ) . Given a finite sequence a = a , . . . , a n ∈ A , we set Tel( A ; a , . . . , a n ) = Tel( A ; a ) ⊗ A Tel( A ; a ) ⊗ A · · · ⊗ A Tel( A ; a n ) . The complex
Tel( A ; a ) is a bounded complex of free A -modules, called the telescopecomplex associated to A and a .One important property of the telescope complex is its behavior with respect to basechange. Let A be a commutative ring, let a be a finite sequence of elements of A , let B beanother commutative ring, and let f : A → B be a ring homomorphism. Denoting by b the image of a by f , there is an isomorphism of complexes of B -modules: Tel( A ; a ) ⊗ A B ∼ = Tel( B ; b ) . As is well known (see for instance [25, Proposition 4.8]), if A is a noetherian ring, a ⊆ A , and a is a finite sequence of elements of A that generates a , there is a naturalisomorphism RΓ a ( M ) ∼ = Tel( A ; a ) ⊗ A M for every M ∈ D ( A ) .More generally, as shown in [27, Corollary 2.13], if A is a commutative DG-ring and ¯ a ⊆ H ( A ) is a finitely generated ideal, and if a = ( a , . . . , a n ) is a finite sequence ofelements of A , whose image in H ( A ) generates ¯ a , then there is a natural isomorphism(1.5) RΓ ¯ a ( M ) ∼ = Tel( A ; a ) ⊗ A A ⊗ A M for any M ∈ D ( A ) .If ¯ a , ¯ b ⊆ H ( A ) are two finitely generated ideals such that √ ¯ a = √ ¯ b , then by [27,Corollary 2.15], there is a natural isomorphism(1.6) RΓ ¯ a ( − ) ∼ = RΓ ¯ b ( − ) . Derived completion of DG-modules and derived completion of DG-rings.
Let A be a commutative DG-ring, and let ¯ a ⊆ H ( A ) be a finitely generated ideal. As explainedin [27], the local cohomology functor RΓ ¯ a : D ( A ) → D ( A ) has a left adjoint, which wedenote by LΛ ¯ a : D ( A ) → D ( A ) , and call the derived ¯ a -adic completion functor. Thereason for this name is that if A = H ( A ) is a noetherian ring, it coincides with the leftderived functor of the ¯ a -adic completion functor Λ ¯ a ( − ) := lim ←− − ⊗ A A/ ¯ a n which wasintroduced in [15]. As for derived torsion, in case A is a ring, the condition under whichthese two operations coincide is that ¯ a is a weakly proregular ideal. In case A is a ring, theleft adjoint we discuss here is studied in [29, Section 091N].Similarly to the previous section, if A is a commutative DG-ring and ¯ a ⊆ H ( A ) is afinitely generated ideal, and if a = ( a , . . . , a n ) is a finite sequence of elements of A ,whose image in H ( A ) generates ¯ a , then there is a natural isomorphism LΛ ¯ a ( M ) ∼ = Hom A (Tel( A ; a ) ⊗ A A, M ) for any M ∈ D ( A ) .Given a commutative DG-ring A and a finitely generated ideal ¯ a ⊆ H ( A ) , it is ex-plained [27, Section 4] that the DG-module LΛ ¯ a ( A ) has the structure of a commutativenon-positive DG-ring, called the derived ¯ a -adic completion of A . We denote this DG-ringby LΛ( A, ¯ a ) . If A is noetherian then its derived completion is also noetherian, and if ( A, ¯ m ) is a noetherian local DG-ring, then LΛ( A, ¯ m ) is a noetherian local DG-ring. Furthermore,we also have the following result, complementing [27]: Proposition 1.7.
Let ( A, ¯ m ) be a commutative noetherian local DG-ring, and let B =LΛ( A, ¯ m ) be the derived ¯ m -adic completion of A . Then for all i ≤ , we have that H i ( B ) ∼ = Λ ¯ m (H i ( A )) . In particular amp( B ) = amp( A ) .Proof. Let E := E ( A, ¯ m ) be the injective DG-module corresponding to the maximal ideal ¯ m . By [28, Theorem 7.22], we have that B ∼ = R Hom A ( E, E ) . Hence, it follows from [28, Theorem 4.10, Corollary 4.12] that H i ( B ) = H i (R Hom A ( E, E )) ∼ = Hom H ( A ) (H − i ( E ) , H ( E )) ∼ =Hom H ( A ) (cid:0) Hom H ( A ) (H i ( A ) , H ( E )) , H ( E ) (cid:1) . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 11
Since H ( E ) is exactly the injective hull of the residue field of the local ring H ( A ) ,the result follows from Matlis duality over the noetherian local ring (H ( A ) , ¯ m ) . Finally,the equality amp( B ) = amp( A ) follows from faithfulness of adic completion on finitelygenerated modules. (cid:3) Localization and support.
Given a commutative DG-ring A , we recall, following[30, Section 4], that one may localize it at at prime ideals of H ( A ) . Given a prime ¯ p ∈ Spec(H ( A )) , the localization A ¯ p is defined as follows: let π A : A → H ( A ) be thecanonical surjection, and π A : A → H ( A ) its degree component. Let p = ( π A ) − (¯ p ) .Then p ∈ Spec( A ) , and one sets A ¯ p := A ⊗ A A p . More generally, given M ∈ D ( A ) , we define M ¯ p := M ⊗ A A ¯ p ∼ = M ⊗ A A p ∈ D ( A ¯ p ) . Since A p is flat over A , it follows that for all n ∈ Z , we have that(1.8) H n ( A ¯ p ) = H n ( A ) ¯ p , H n ( M ¯ p ) = H n ( M ) ¯ p . It follows that if A is a noetherian DG-ring, then ( A ¯ p , ¯ p · H ( A ) ¯ p ) is a noetherian localDG-ring, and that amp( A ¯ p ) ≤ amp( A ) . Lemma 1.9.
Let A be a commutative DG-ring, and let M ∈ D ( A ) . For any ¯ p ∈ Spec(H ( A )) , there is a natural isomorphism M ¯ p ⊗ L A H ( A ) ∼ = M ⊗ L A H ( A ) ¯ p in D (H ( A )) .Proof. This follows from associativity of the derived tensor product, and the fact that A ¯ p ⊗ L A H ( A ) ∼ = H ( A ) ¯ p . (cid:3) Definition 1.10.
Let A be a commutative DG-ring, and let M ∈ D ( A ) . We define thesupport of M over A to be the set Supp A ( M ) := { ¯ p ∈ Spec(H ( A )) | M ¯ p ≇ } . It follows from the definition and from (1.8) that
Supp A ( M ) = [ n ∈ Z Supp H ( A ) (H n ( M )) . Proposition 1.11.
Let A be a commutative DG-ring, and let M ∈ D − ( A ) . Then there isan equality Supp A ( M ) = Supp H ( A ) (cid:0) M ⊗ L A H ( A ) (cid:1) . Proof.
Given ¯ p ∈ Spec(H ( A )) , we have that ¯ p ∈ Supp A ( M ) if and only if M ¯ p ≇ .Note that M ¯ p ∈ D − ( A ) . By (1.3), we have that M ¯ p ≇ if and only if M ¯ p ⊗ L A H ( A ) ≇ .By Lemma 1.9, this is equivalent to M ⊗ L A H ( A ) ¯ p ∼ = (cid:0) M ⊗ L A H ( A ) (cid:1) ¯ p ≇ if and only if ¯ p ∈ Supp H ( A ) (cid:0) M ⊗ L A H ( A ) (cid:1) , as claimed. (cid:3)
2. L
OCAL COHOMOLOGY K RULL DIMENSION OVER COMMUTATIVE LOCAL
DG-
RINGS
The next definition follows [12, Section 3]:
Definition 2.1.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let M ∈ D − ( A ) be abounded above DG-module. We define the local cohomology Krull dimension of M tobe lc . dim( M ) := sup ℓ ∈ Z (cid:8) dim(H ℓ ( M )) + ℓ (cid:9) where dim(H ℓ ( M )) is the usual Krull dimension of the H ( A ) -module H ℓ ( M ) ; that is,the Krull dimension of the noetherian ring H ( A ) / ann(H ℓ ( M )) . Remark 2.2.
The name local cohomology Krull dimension is justified by Theorem 2.15below.
Remark 2.3.
Since any H ( A ) -module ¯ M has ≤ dim( ¯ M ) ≤ dim(H ( A )) , we necessarily have(2.4) sup( M ) ≤ lc . dim( M ) ≤ sup( M ) + dim(H ( A )) for any M ∈ D − ( A ) , with lc . dim( M ) = sup( M ) + dim(H ( A )) if and only if(2.5) dim(H sup( M ) ( M )) = dim(H ( A )) . In particular, it follows, since A is non-positive, that(2.6) lc . dim( A ) = dim(H ( A )) . Remark 2.7.
The paper [4] discusses another notion of Krull dimension for differentialgraded algebras, and shows that in some nice cases, it coincides with dim(H ( A )) , as inthe above definition.We now discuss two results we will need about bounds of local cohomology over com-mutative local rings (and not DG-rings, as in the rest of this paper). These results areprobably well known, and we simply wish to emphasize that they also hold for unboundedcomplexes, as we will need to apply them in an unbounded situation. Proposition 2.8.
Let ( A, m ) be a noetherian local ring, and let M ∈ D − f ( A ) . Then lc . dim( M ) = sup (RΓ m ( M )) . Proof. If M has bounded cohomology, then this is precisely [12, Proposition 3.14(d)]. Inthe general case, let d = dim( A ) . Then amp(RΓ m ( A )) ≤ d . Let n = sup( M ) − ( d + 1) ,and set M ′ := smt ≤ n ( M ) , M ′′ := smt >n ( M ) . Then there is a distinguished triangle M ′ → M → M ′′ → M ′ [1] in D ( A ) . Applying the triangulated functor RΓ m ( − ) , we obtain a distinguished triangle: RΓ m ( M ′ ) → RΓ m ( M ) → RΓ m ( M ′′ ) → RΓ m ( M ′ )[1] If i < sup( M ) − d , we must have i + dim(H i ( M )) < sup( M ) HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 13 so it follows that lc . dim( M ) = lc . dim( M ′′ ) . Since RΓ m ( A ) ∈ [0 , d ] , and RΓ m ( M ′ ) ∼ =RΓ m ( A ) ⊗ L A M ′ , we have that H i (RΓ m ( M ′ )) = 0 , for i ≥ sup( M ) Hence, the above distinguished triangle and the fact that M ′′ is a bounded complex implythere are equalities sup (RΓ m ( M )) = sup (RΓ m ( M ′′ )) = lc . dim( M ′′ ) = lc . dim( M ) . (cid:3) Similarly, the following unbounded version of [12, Proposition 3.7] holds. We omit thesimilar proof.
Proposition 2.9.
Let ( A, m ) be a noetherian local ring, and let M ∈ D − ( A ) . Then sup (RΓ m ( M )) ≤ lc . dim( M ) . Proposition 2.10.
Let
A, B be commutative DG-rings, let ¯ a ⊆ H ( A ) be a finitely gener-ated ideal, let f : A → B be a map of DG-rings, and let ¯ b := H ( f )(¯ a ) · H ( B ) be the ideal in H ( B ) generated by the image of ¯ a . Given M ∈ D ( A ) , there is a naturalisomorphism RΓ ¯ a ( M ) ⊗ L A B ∼ = RΓ ¯ b ( M ⊗ L A B ) in D ( B ) .Proof. Let ¯ x , . . . , ¯ x n be a finite sequence of elements in H ( A ) that generates ¯ a , and let x , . . . , x n be lifts of these elements to A . Then by [27, Corollary 2.13], we have that RΓ ¯ a ( M ) ∼ = Tel( A ; x , . . . , x n ) ⊗ A A ⊗ A M. Hence, by the base change property of the telescope complex RΓ ¯ a ( M ) ⊗ L A B ∼ = Tel( B ; f ( x ) , . . . , f ( x n )) ⊗ B (cid:0) M ⊗ L A B (cid:1) . Since the images of f ( x ) , . . . , f ( x n ) in H ( B ) generate ¯ b , we deduce from [27, Corollary2.13] that the latter is naturally isomorphic to RΓ ¯ b ( M ⊗ L A B ) , as claimed. (cid:3) In the next results we will use the notation RΓ ¯ A and RΓ A introduced in Remark 1.4.Similarly, we will discuss both the local cohomology Krull dimension of DG-modules overa DG-ring A and of complexes over the ring H ( A ) . To distinguish between them, we willdenote the former by lc . dim A ( − ) and the latter by lc . dim ¯ A ( − ) . Proposition 2.11.
Let A be a noetherian DG-ring, let ¯ a ⊆ H ( A ) be a an ideal, and let M ∈ D ( A ) . Then there is a natural isomorphism RΓ A ¯ a ( M ) ⊗ L A H ( A ) ∼ = RΓ ¯ A ¯ a (cid:0) M ⊗ L A H ( A ) (cid:1) in D (H ( A )) .Proof. This follows from applying Proposition 2.10 to the map A → H ( A ) . The as-sumption that H ( A ) is noetherian is required, in order for the local cohomology func-tor discussed in the DG context to coincide with the classical local cohomology functor RΓ ¯ A ¯ a . (cid:3) Lemma 2.12.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let M ∈ D − ( A ) . Then thereis an equality sup (cid:0) RΓ A ¯ m ( M ) (cid:1) = sup (cid:16) RΓ ¯ A ¯ m ( M ⊗ L A H ( A )) (cid:17) . If moreover M ∈ D − f ( A ) , then there is also an equality sup (cid:0) RΓ A ¯ m ( M ) (cid:1) = lc . dim ¯ A ( M ⊗ L A H ( A )) . Proof.
Letting x , . . . , x n ∈ A be a finite sequence of elements of A such that theirimage in H ( A ) generates ¯ m , it follows from (1.5) that RΓ ¯ m ( M ) ∼ = Tel( A ; x , . . . , x n ) ⊗ A A ⊗ A M Since the DG-module
Tel( A ; x , . . . , x n ) ⊗ A A is bounded-above, we deduce that theDG-module RΓ ¯ m ( M ) is also bounded above. Hence, by (1.3) we have that sup (cid:0) RΓ A ¯ m ( M ) (cid:1) = sup (cid:0) RΓ A ¯ m ( M ) ⊗ L A H ( A ) (cid:1) = sup (cid:16) RΓ ¯ A ¯ m ( M ⊗ L A H ( A )) (cid:17) where the second equality follows from Proposition 2.11. This proves the first claim.Finally, if in addition M ∈ D − f ( A ) , then we have that M ⊗ L A H ( A ) ∈ D − f (H ( A )) , so it follows from Proposition 2.8 that sup (cid:16) RΓ ¯ A ¯ m ( M ⊗ L A H ( A )) (cid:17) = lc . dim ¯ A ( M ⊗ L A H ( A )) . proving the second claim. (cid:3) Proposition 2.13.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let M ∈ D − ( A ) . Thenthere is an equality lc . dim A ( M ) = lc . dim ¯ A ( M ⊗ L A H ( A )) . Proof.
By our definition, lc . dim A ( M ) = sup ℓ ∈ Z (cid:8) dim(H ℓ ( M )) + ℓ (cid:9) . First, exactly as in the proof of [12, Proposition 3.5], note that this number satisfies(2.14) lc . dim( M ) = sup ¯ p ∈ Spec(H ( A )) (cid:8) dim(H ( A ) / ¯ p ) + sup( M ¯ p ) (cid:9) . Now, given ¯ p ∈ Spec(H ( A )) , by (1.3) we have that sup( M ¯ p ) = sup (cid:0) M ¯ p ⊗ L A H ( A ) (cid:1) . By Lemma 1.9, we have that M ¯ p ⊗ L A H ( A ) ∼ = (cid:0) M ⊗ L A H ( A ) (cid:1) ¯ p . Hence, using [12, Proposition 3.5] we see that lc . dim A ( M ) is equal to sup ¯ p ∈ Spec(H ( A ) n dim(H ( A ) / ¯ p ) + sup (cid:0) M ⊗ L A H ( A ) (cid:1) ¯ p o = lc . dim ¯ A ( M ⊗ L A H ( A )) as claimed. (cid:3) The next result is a DG version of Grothendieck’s vanishing and non-vanishing theo-rems for local cohomology (see [7, Theorems 6.1.2 and 6.1.4]).
HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 15
Theorem 2.15.
Let ( A, ¯ m ) be a noetherian local DG-ring, and M ∈ D − ( A ) . Then sup (RΓ ¯ m ( M )) ≤ lc . dim( M ) . If moreover M ∈ D − f ( A ) then sup (RΓ ¯ m ( M )) = lc . dim( M ) . In particular, for M ∈ D − f ( A ) we have that H lc . dim( M )¯ m ( M ) := H lc . dim( M ) (RΓ ¯ m ( M )) = 0 . Proof.
Take M ∈ D − ( A ) . By Lemma 2.12 we have that sup (cid:0) RΓ A ¯ m ( M ) (cid:1) = sup (cid:16) RΓ ¯ A ¯ m ( M ⊗ L A H ( A )) (cid:17) , and by Proposition 2.9, since M ⊗ L A H ( A ) ∈ D − (H ( A )) , we have that sup (cid:16) RΓ ¯ A ¯ m ( M ⊗ L A H ( A )) (cid:17) ≤ lc . dim ¯ A ( M ⊗ L A H ( A )) so the first claim follows from Proposition 2.13. Now, assume further that M ∈ D − f ( A ) .Then by Lemma 2.12 sup (cid:0) RΓ A ¯ m ( M ) (cid:1) = lc . dim ¯ A ( M ⊗ L A H ( A )) , and by Proposition 2.13 the latter is equal to lc . dim( M ) , as claimed. (cid:3) Corollary 2.16.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let d = dim(H ( A )) .Then H d (RΓ ¯ m ( A )) = 0 . and H i (RΓ ¯ m ( A )) = 0 for all i > d .Proof. This follows from (2.6) and Theorem 2.15. (cid:3)
Proposition 2.17.
Let ( A, ¯ m ) be a commutative noetherian local DG-ring, and let B =LΛ( A, ¯ m ) be the derived ¯ m -adic completion of A . Then lc . dim( B ) = lc . dim( A ) .Proof. According to [27, Proposition 4.16], we have that H ( B ) = Λ ¯ m (H ( A )) . Hence, since B is non-positive, we obtain lc . dim( B ) = dim (cid:0) H ( B ) (cid:1) = dim (cid:0) Λ ¯ m (H ( A )) (cid:1) = dim (cid:0) H ( A ) (cid:1) = lc . dim( A ) . (cid:3)
3. D
EPTH AND LOCAL COHOMOLOGY OVER COMMUTATIVE LOCAL
DG-
RINGS
Our definition of depth is identical to the usual homological definition of depth overlocal rings:
Definition 3.1.
Let ( A, ¯ m , k ) be a noetherian local DG-ring, and let M ∈ D + ( A ) . Wedefine the depth of M to be the number depth A ( M ) := inf (R Hom A ( k , M )) . Note that this definition is not invariant under translations, but is very useful for workingwith local cohomology. In Definition 5.2 below we will give a modified definition of depththat is invariant under translations. It follows from the definition that depth A (0) = −∞ .Dually to Proposition 2.13, we have the following reduction formula for the depth, and,as in the case of rings, a connection to local cohomology: Proposition 3.2.
Let ( A, ¯ m , k ) be a noetherian local DG-ring. Then for any M ∈ D + ( A ) ,there are equalities inf (RΓ ¯ m ( M )) = depth A ( M ) = depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) . Proof.
Since M ∈ D + ( A ) , and since the functor RΓ ¯ m ( − ) has finite cohomological di-mension, RΓ ¯ m ( M ) ∈ D + ( A ) . Hence, by (1.2), inf (RΓ ¯ m ( M )) = inf (cid:0) R Hom A (H ( A ) , RΓ ¯ m ( M )) (cid:1) . According to [28, Proposition 7.23], there is an isomorphism
R Hom A (H ( A ) , RΓ A ¯ m ( M )) ∼ = RΓ ¯ A ¯ m (cid:0) R Hom A (H ( A ) , M ) (cid:1) , in D (H ( A )) , so it is enough to compute the infimum of the latter. Since R Hom A (H ( A ) , M ) ∈ D + (H ( A )) , it follows from [12, Proposition 3.8] that inf (cid:16) RΓ ¯ A ¯ m (cid:0) R Hom A (H ( A ) , M ) (cid:1)(cid:17) = inf (cid:0) R Hom H ( A ) ( k , R Hom A (H ( A ) , M )) (cid:1) . The latter is by definition depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) . Furthermore, the adjunction isomorphism shows that
R Hom H ( A ) ( k , R Hom A (H ( A ) , M )) ∼ = R Hom A ( k , M ) . proving the claim. (cid:3) Over a local ring, it follows immediately from the definition of local cohomology thatthe depth of any complex is greater or equal to its infimum, and, as explained, in [12, (3.3)],they are equal if and only if the maximal ideal is an associated prime of the bottommostcohomology of the complex. Similarly, we have:
Proposition 3.3.
Let ( A, ¯ m , k ) be a noetherian local DG-ring. Then for any = M ∈ D + ( A ) , there are inequalities inf (RΓ ¯ m ( M )) = depth A ( M ) ≥ inf( M ) . Moreover, there is an equality depth A ( M ) = inf( M ) if and only if ¯ m is an associated prime of H inf( M ) ( M ) .Proof. By Proposition 3.2, depth A ( M ) = depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) As remarked above, depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) ≥ inf (cid:0) R Hom A (H ( A ) , M ) (cid:1) , HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 17 with equality if and only if ¯ m is an associated prime of H inf ( R Hom A (H ( A ) ,M ) ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) . Hence, the result follows from the equalities inf (cid:0)
R Hom A (H ( A ) , M ) (cid:1) = inf( M ) and H inf( M ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) = H inf( M ) ( M ) of (1.2). (cid:3) We shall need the following upper bound satisfied by depth over rings:
Proposition 3.4.
Let ( A, m ) be a noetherian local ring, and let M ∈ D +f ( A ) . Then thereis an inequality: depth A ( M ) ≤ dim(H inf( M ) ( M )) + inf( M ) . Proof. If M ∈ D bf ( A ) , then this statement is exactly [12, Proposition 3.17]. In the generalcase, let n := dim(H inf( M ) ( M )) + inf( M ) , and consider the following distinguishedtriangle in D ( A ) : M ′ → M → M ′′ → M ′ [1] where M ′ := smt ≤ n ( M ) , M ′′ := smt >n ( M ) . Applying the triangulated functor RΓ m , and passing to cohomology, we have for each i ∈ Z the following exact sequence of local cohomology modules: H i − m ( M ′′ ) → H i m ( M ′ ) → H i m ( M ) → H i m ( M ′′ ) From the definition of local cohomology, and since inf( M ′′ ) > n , it follows that H i m ( M ′′ ) = 0 for all i ≤ n . Hence, H i m ( M ′ ) ∼ = H i m ( M ) for all i ≤ n . Since M ′ ∈ D bf ( A ) , according to [12, Proposition 3.17] there exists i ≤ dim(H inf( M ′ ) ( M ′ )) + inf( M ′ ) such that H i m ( M ′ ) = 0 . Since dim(H inf( M ′ ) ( M ′ )) + inf( M ′ ) = dim(H inf( M ) ( M )) + inf( M ) = n, we deduce that there exists some i ≤ dim(H inf( M ) ( M )) + inf( M ) such that H i m ( M ) = 0 , as claimed. (cid:3) The next result is a DG-version of Proposition 3.4.
Proposition 3.5.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let M ∈ D +f ( A ) . Thenthere is an inequality: depth A ( M ) ≤ dim(H inf( M ) ( M )) + inf( M ) . Proof.
By Proposition 3.2, we have that depth A ( M ) = depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) . Since M ∈ D +f ( A ) , we have that R Hom A (H ( A ) , M ) ∈ D +f (H ( A )) . Hence, by Propo-sition 3.4, we obtain an inequality depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) ≤ dim (cid:0) H i (R Hom A (H ( A ) , M ) + i (cid:1) where i := inf (cid:0) R Hom A (H ( A ) , M ) (cid:1) . According to (1.2), there are equalities inf( M ) = inf (cid:0) R Hom A (H ( A ) , M ) (cid:1) , H i ( M ) = H i (cid:0) R Hom A (H ( A ) , M ) (cid:1) . Combining all of the above, we see that depth A ( M ) = depth H ( A ) (cid:0) R Hom A (H ( A ) , M ) (cid:1) ≤ dim (H inf( M ) ( M )) + inf( M ) , as claimed. (cid:3) Corollary 3.6.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology, andlet d = dim(H ( A )) . Then there is an inequality inf (RΓ ¯ m ( A )) ≤ inf( A ) + d. If there is an equality inf (RΓ ¯ m ( A )) = inf( A ) + d then (3.7) dim (cid:16) H inf( A ) ( A ) (cid:17) = dim(H ( A )) . Proof.
By Proposition 3.2 and Proposition 3.5 we have that(3.8) inf (RΓ ¯ m ( A )) = depth A ( A ) ≤ dim(H inf( A ) ( A )) + inf( A ) . Since H inf( A ) ( A ) is an H ( A ) -module, we have that dim(H inf( A ) ( A ) ≤ d , proving theclaim. If inf (RΓ ¯ m ( A )) = inf( A ) + d , it follow from the (3.8) that d ≤ dim(H inf( A ) ( A )) ,which implies that dim(H inf( A ) ( A )) = d . (cid:3) Proposition 3.9.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology, andlet B = LΛ( A, ¯ m ) be the derived ¯ m -adic completion of A . Then depth( B ) = depth( A ) .Proof. In this proof we will use the terminology of [27]. Let ( C, c ) be a weakly proregularresolution of ( A, ¯ m ) (in the sense of [27, Definition 2.1]). Since C → A is a quasi-isomorphism, we have that depth( C ) = depth( A ) . Let c be the ideal in C generated by c , and let ¯ c := c · H ( A ) . The isomorphism H ( C ) → H ( A ) sends ¯ c to ¯ m . According to[27, Theorem 4.8], there is an isomorphism(3.10) B = LΛ( A, ¯ m ) ∼ = Λ c ( C ) = lim ←− n C ⊗ C C / c n . Let us denote the latter by b C . The ideal of definition of the local DG-ring b C is given by b ¯ c := ¯ c · H ( b C ) . The isomorphism (3.10) implies that depth( B ) = depth( b C ) . By Proposition 2.10, thereis an isomorphism RΓ ¯ c ( C ) ⊗ L C b C ∼ = RΓ b ¯ c ( b C ) . Hence, we have that depth( b C ) = inf (cid:16) RΓ ¯ c ( C ) ⊗ L C b C (cid:17) . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 19
To compute the infimum of the latter, we may apply the forgetful functor D ( b C ) → D ( C ) ,and consider the DG-module RΓ ¯ c ( C ) ⊗ L C b C ∈ D ( C ) . It follows from [27, Corollary 2.13] that there is an isomorphism b C ∼ = LΛ ¯ c ( C ) in D ( C ) , so there is an isomorphism RΓ ¯ c ( C ) ⊗ L C b C ∼ = RΓ ¯ c (LΛ ¯ c ( C )) . By the MGM equivalence (a result dual to [27, Proposition 2.7] proven similarly to it), wehave that RΓ ¯ c (LΛ ¯ c ( C )) ∼ = RΓ ¯ c ( C ) , so that inf (cid:16) RΓ ¯ c ( C ) ⊗ L C b C (cid:17) = depth( C ) , which implies that depth( A ) = depth( C ) = depth( b C ) = depth( B ) , as claimed. (cid:3)
4. L
OCAL -C OHEN -M ACAULAY COMMUTATIVE
DG-
RINGS
We are now ready to prove items (1) and (3) of Theorem 1 from the introduction. Theyare contained in the following result:
Theorem 4.1.
The following inequalities hold:(1) If ( A, ¯ m ) is a noetherian local DG-ring with amp( A ) < ∞ and d = dim(H ( A )) then amp( A ) ≤ amp (RΓ ¯ m ( A )) ≤ amp( A ) + d. (2) If A is a noetherian DG-ring, and R is a dualizing DG-module over A then amp( A ) ≤ amp( R ) . If moreover A has is local and has bounded cohomology, and d = dim(H ( A )) then d < ∞ and amp( R ) ≤ amp( A ) + d. Proof. (1) Let ( A, ¯ m ) be a noetherian local DG-ring. Let n = amp( A ) , d = dim(H ( A )) ,and suppose that n < ∞ . By Corollary 2.16, we have that sup (RΓ ¯ m ( A )) = d. By Corollary 3.6, there is an inequality inf (RΓ ¯ m ( A )) ≤ inf( A ) + d = d − n. Combining these two facts we obtain: amp (RΓ ¯ m ( A )) = sup (RΓ ¯ m ( A )) − inf (RΓ ¯ m ( A )) ≥ d + ( n − d ) = n = amp( A ) . On the other hand, by Proposition 3.3, inf (RΓ ¯ m ( A )) ≥ inf( A ) = − n, so that amp (RΓ ¯ m ( A )) ≤ d + n = amp( A ) + d. (2) Let A be a noetherian DG-ring, and R a dualizing DG-module over A . If amp( A ) = ∞ then by [30, Corollary 7.3] we have that amp( R ) = ∞ . We may thus assume that amp( A ) = n < ∞ . Let ¯ p ∈ Supp(H − n ( A )) . Then A ¯ p is a noetherian local DG-ring,and since H − n ( A ) ¯ p = 0 , there is an equality amp( A ) = amp( A ¯ p ) . Since localizationof DG-rings is cohomologically essentially smooth (in the sense of [26, Definition 6.4]),according to [26, Corollary 6.11] the DG-module R ¯ p is a dualizing DG-module over A ¯ p .By local duality for local DG-rings ([28, Corollary 7.29]), there is an equality amp (RΓ ¯ p ( A ¯ p )) = amp ( R ¯ p ) . By (1) of this theorem, we have that amp (RΓ ¯ p ( A ¯ p )) ≥ amp( A ¯ p ) = amp( A ) and since amp( R ) ≥ amp ( R ¯ p ) , we deduce that amp( R ) ≥ amp( A ) . Next, observethat by [30, Proposition 7.5], the complex R Hom A (H ( A ) , R ) is a dualizing complexover H ( A ) , so by [16, Corollary V.7.2] we have that d = dim(H ( A )) < ∞ . Finally,assuming that A is local, the inequality amp( R ) ≤ amp( A ) + d. follows from local duality for local DG-rings and from (1). (cid:3) In view of Theorem 4.1, it makes sense to define:
Definition 4.2.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ . We say that A is local-Cohen-Macaulay if amp(RΓ ¯ m ( A )) = amp( A ) . Example 4.3.
Let ( A, m ) be a noetherian local ring. Then A is Cohen-Macaulay in theclassical sense if and only if A is local-Cohen-Macaulay in the sense of Definition 4.2. Proposition 4.4.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ , and let R be a dualizing DG-module over A . Then A is local-Cohen-Macaulay if and only if amp( A ) = amp( R ) .Proof. This follows immediately from the equality amp(RΓ ¯ m ( A )) = amp( R ) establishedin [28, Corollary 7.29]. (cid:3) Following [13, 14], recall that a noetherian local DG-ring ( A, ¯ m ) is called Gorenstein if amp( A ) < ∞ and inj dim A ( A ) < ∞ . In this case, note that A is a dualizing DG-moduleover A . Hence, by Proposition 4.4 we have: Proposition 4.5.
Let ( A, ¯ m ) be a noetherian local Gorenstein DG-ring. Then A is local-Cohen-Macaulay. (cid:3) Just like noetherian local rings, noetherian local DG-rings need not to have dualizingDG-modules. However, passing to their derived completion, we showed in [28, Proposi-tion 7.21] that the derived completion has a dualizing DG-module. It is thus convenientto know that the local-Cohen-Macaulay property is preserved by the derived completionoperation. In Example 7.8 below we construct a local-Cohen-Macaulay DG-ring A whichis not equivalent to a ring, such that A does not have a dualizing DG-module. Proposition 4.6.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ , and let B = LΛ( A, ¯ m ) be its derived ¯ m -adic completion. Then A is local-Cohen-Macaulay if andonly if B is local-Cohen-Macaulay. HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 21
Proof.
This follows from the equalities amp( A ) = amp( B ) , lc . dim( A ) = lc . dim( B ) , depth( A ) = depth( B ) shown in Propositions 1.7, 2.17 and 3.9. (cid:3) Next we wish to show that zero-dimensional DG-rings are local-Cohen-Macaulay. Firstwe need the following lemma about local cohomology with respect to nilpotent ideals overDG-rings:
Lemma 4.7.
Let A be a noetherian DG-ring, let ¯ a ⊆ H ( A ) be an ideal, and supposethat there is some n ∈ N such that ¯ a n = 0 . Then the functor RΓ ¯ a : D ( A ) → D ( A ) isisomorphic to the identity functor.Proof. Let M ∈ D ( A ) . In general (even without the nilpotent assumption), it follows from[27, Corollary 2.13] that RΓ ¯ a ( M ) ∼ = RΓ ¯ a ( A ) ⊗ L A M, so it is enough to show that RΓ ¯ a ( A ) ∼ = A . By Proposition 2.11, we have that RΓ A ¯ a ( A ) ⊗ L A H ( A ) ∼ = RΓ ¯ A ¯ a (cid:0) H ( A ) (cid:1) ∼ = H ( A ) where the last isomorphism follows from the fact that since ¯ a is nilpotent, the additivefunctor Γ ¯ a : Mod (cid:0) H ( A ) (cid:1) → Mod (cid:0) H ( A ) (cid:1) is equal to the identity functor, so its right derived functor is also the identity functor.Hence, by [30, Proposition 3.3(1)], it follows that RΓ ¯ a ( A ) ∼ = A , as claimed. (cid:3) Proposition 4.8.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ , andsuppose that dim(H ( A )) = 0 . Then A is local-Cohen-Macaulay.Proof. Since (H ( A ) , ¯ m ) is a zero-dimensional local ring, its maximal ideal ¯ m must benilpotent. Hence, by Lemma 4.7 we have amp (RΓ ¯ m ( A )) = amp( A ) , so that A is local-Cohen-Macaulay. (cid:3) Example 4.9.
Let ϕ : ( A, m ) → ( B, n ) be a local homomorphism between noetherianlocal rings. Assume that ϕ is a finite ring map, and that it makes B an A -module of finiteflat dimension. Let D = A/ m ⊗ L A B be the derived fiber of ϕ . This is a commutative localDG-ring with H ( D ) = B/ m B . The assumption that ϕ is finite implies that B/ m B is azero dimensional ring, and the finite flat dimension assumption implies that amp( D ) < ∞ .Hence, by Proposition 4.8, D is a local-Cohen-Macaulay DG-ring.Given a local homomorphism ( A, m ) → ( B, n ) of finite flat dimension, it follows from[1, Theorem 4.4] and [1, Theorem 7.1] that if A is Gorenstein then B is Gorenstein if andonly if the local DG-ring D = A/ m ⊗ L A B is Gorenstein. Unfortunately, it turns out not tobe the case for the local-Cohen-Macaulay property, as the next example shows: Example 4.10.
Let ( A, m ) be a regular local ring, and let B = A/I be a quotient of A which is not Cohen-Macaulay. For a specific example, on can take A = k [[ x, y, z ]] , B = k [[ x, y, z ]] / ( xy, xz ) . Since A is regular, B has finite flat dimension over A , so byExample 4.9, the derived fiber D of the map ϕ : A → B is a local-Cohen-Macaulay DG-ring. Since B is not a Cohen-Macaulay ring, it follows from [2, (8.9)] that the map ϕ isnot a Cohen-Macaulay homomorphism in the sense of [2, (8.1)]. Note also that since B isnot Gorenstein, it follows from [1, Theorem 7.1] that D is not a Gorenstein DG-ring. The minimal non-vanishing cohomology of a local-Cohen-Macaulay DG-ring satisfiesthe following property:
Proposition 4.11.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, and let n = amp( A ) . Then the H ( A ) -module H − n ( A ) satisfies dim (cid:0) H − n ( A ) (cid:1) = dim (cid:0) H ( A ) (cid:1) . Proof.
Let d = dim(H ( A )) . By the proof of Theorem 4.1, if A is local-Cohen-Macaulay,we must have that inf (RΓ ¯ m ( A )) = inf( A ) + d. Hence, by equation (3.8) in Corollary 3.6, we deduce that dim (cid:0) H − n ( A ) (cid:1) = dim (cid:0) H ( A ) (cid:1) . (cid:3)
5. R
EGULAR SEQUENCES AND THE DERIVED B ASS CONJECTURE
In this section we study regular sequences and system of parameters over noetherianlocal DG-rings. Much of this section is inspired, and based on, the work of Christensenon regular sequences and system of parameters acting on chain complexes over local rings([8, 9]). We first recall the notions of a regular sequence in the DG-setting, and the notionof a quotient DG-module, essentially following Minamoto ([22, Section 3.2]).Given a commutative DG-ring A and ¯ x ∈ H ( A ) , the identification H ( A ) = H (R Hom A ( A, A )) = Hom D ( A ) ( A, A ) implies that ¯ x induces a map A → A in D ( A ) , which we also denote by ¯ x . We denote themapping cone of the map ¯ x by A// ¯ x , so there is a distinguished triangle(5.1) A ¯ x −→ A → A// ¯ x → A [1] in D ( A ) . It is shown in [22, Section 3.2] that A// ¯ x has the structure of a commutativeDG-ring. If A is noetherian, then A// ¯ x is also noetherian, and if amp( A ) < ∞ then amp( A// ¯ x ) < ∞ . It follows from (5.1) and the fact that A is non-positive that H ( A// ¯ x ) ∼ = H ( A ) / ¯ x. In particular, if A is a noetherian local DG-ring, then A// ¯ x is also a noetherian local DG-ring.Given M ∈ D ( A ) , applying the triangulated functor M ⊗ L A − to the triangle (5.1), weobtain another distinguished triangle in D ( A ) : M ¯ x ⊗ L A M −−−−→ M → M// ¯ xM → M [1] where we have set M// ¯ xM := A// ¯ x ⊗ L A M. In particular we see that
M// ¯ xM ∈ D ( A// ¯ x ) .Given a finite set of elements ¯ x , . . . , ¯ x n ∈ H ( A ) , setting B = A// ¯ x , we defineinductively A// (¯ x , . . . , ¯ x n ) = B// (¯ x , . . . , ¯ x n ) , where we identified ¯ x , . . . , ¯ x n with their images in H ( A ) / ¯ x . Similarly one defines M// (¯ x , . . . , ¯ x n ) M = M ⊗ L A A// (¯ x , . . . , ¯ x n ) ∈ D ( A// (¯ x , . . . , ¯ x n )) . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 23
Definition 5.2.
Let A be a commutative DG-ring, and let M ∈ D + ( A ) .(1) An element ¯ x ∈ H ( A ) is called M -regular if it is H inf( M ) ( M ) -regular; that is, ifthe multiplication map ¯ x × − : H inf( M ) ( M ) → H inf( M ) ( M ) is injective.(2) Inductively, a sequence ¯ x , . . . ¯ x n ∈ H ( A ) is called M -regular if ¯ x is M -regular,and the sequence ¯ x , . . . ¯ x n is M// ¯ x M -regular.(3) Assuming ( A, ¯ m ) is a noetherian local DG-ring, the sequential depth of M , de-noted by seq . depth A ( M ) is the length of an M -regular sequence contained in ¯ m of maximal length.It is shown in [22, Proposition 3.15] that if ( A, ¯ m ) is a noetherian local DG-ring, and if = M ∈ D +f ( A ) then(5.3) seq . depth A ( M ) = depth A ( M ) − inf( M ) . To be precise, Minamoto calls the right hand side of this equality the cohomological depthof M , and shows in that proposition that it coincides with what we called here the sequen-tial depth of M . In view of this formula, we make the following definition which is avariation on the local cohomology Krull dimension of a DG-module which is not affectedby shifts. Definition 5.4.
Given a noetherian local DG-ring ( A, ¯ m ) , and given M ∈ D − ( A ) , wedefine the derived Krull dimension of M to be the number der . dim A ( M ) = lc . dim A ( M ) − sup( M ) . Corollary 5.5.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ . Then thereis an inequality seq . depth A ( A ) ≤ dim(H ( A )) = der . dim A ( A ) , with equality if and only if A is local-Cohen-Macaulay.Proof. It follows from Theorem 2.15 and Proposition 3.2 that amp (RΓ ¯ m ( A )) = lc . dim( A ) − depth A ( A ) . By Theorem 4.1(1), since sup( A ) = 0 , we have that lc . dim( A ) − depth A ( A ) ≥ amp( A ) = − inf( A ) which implies by (5.3) that seq . depth A ( A ) ≤ lc . dim( A ) = der . dim A ( A ) . Moreover, we see that this is an equality if and only if amp (RΓ ¯ m ( A )) = lc . dim( A ) − depth A ( A ) = amp( A ) if and only if A is local-Cohen-Macaulay. (cid:3) Assuming ¯ x ∈ ¯ m , it is shown in [22, Lemma 3.13] that inf( M ) − ≤ inf( M// ¯ xM ) ≤ inf( M ) , and that inf( M// ¯ xM ) = inf( M ) if and only if ¯ x is M -regular. We further note that byNakayama’s lemma, it is always the case that sup( M// ¯ xM ) = sup( M ) . It follows that if M ∈ D bf ( A ) , and if ¯ x ∈ ¯ m is M -regular, then(5.6) amp( M// ¯ xM ) = amp( M ) . Remark 5.7.
Minamoto’s discussion of the above in [22] uses a slightly different termi-nology, as instead of working with elements of H ( A ) , he works with lifting of them tothe ring A . Given M ∈ D ( A ) and n ∈ Z , since the A action on H n ( M ) factors through H ( A ) , it follows that our definitions are equivalent to his.The next definitions are DG-versions of [8, Definition 2.3] and [9, Definition 2.1]. Wenote that [8, 9] uses homological notation, making the definitions look slightly different.The support of a DG-module was defined in Definition 1.10. Definition 5.8.
Let ( A, ¯ m ) be a commutative noetherian local DG-ring.(1) Given M ∈ D + ( A ) , the set of associated primes of M is given by Ass A ( M ) := { ¯ p ∈ Supp A ( M ) | depth A ¯ p ( M ¯ p ) = inf( M ¯ p ) } . (2) Given M ∈ D − ( A ) , we set W A ( M ) := { ¯ p ∈ Supp A ( M ) | lc . dim A ( M ) ≤ sup( M ¯ p ) + dim(H ( A ) / ¯ p ) } . We now show that these sets of prime ideals are often finite.
Proposition 5.9.
Let ( A, ¯ m ) be a commutative noetherian local DG-ring, and let M ∈ D bf ( A ) . Then the set Ass A ( M ) of associated primes of M is a finite set.Proof. Let ¯ p ∈ Ass A ( M ) , so that depth A ¯ p ( M ¯ p ) = inf( M ¯ p ) . It follows from Proposition3.3 that ¯ p · H ( A ) ¯ p is an associated prime of the H ( A ) ¯ p -module H inf( M ¯ p ) ( M ¯ p ) ∼ = (cid:16) H inf( M ¯ p ) ( M ) (cid:17) ¯ p . This implies (for instance, by [29, Tag 0310]) that ¯ p is an associated prime of the H ( A ) -module H inf( M ¯ p ) ( M ) . It follows that Ass A ( M ) ⊆ sup( M ) [ n =inf( M ) Ass H ( A ) (H n ( M )) . Since H n ( M ) is a finitely generated H ( A ) -module, it has only finitely many associatedprimes, so the fact that H n ( M ) = 0 only for finitely many n ∈ Z implies the result. (cid:3) Lemma 5.10.
Let ( A, ¯ m ) be a commutative noetherian local DG-ring, let M ∈ D bf ( A ) ,and let ¯ p ∈ Supp A ( M ) . Then there is an inequality lc . dim A ( M ) ≥ lc . dim A ¯ p ( M ¯ p ) + dim(H ( A ) / ¯ p ) . Proof.
By the definition of the local cohomology Krull dimension, there is some n ∈ Z such that H n ( M ¯ p ) = 0 and lc . dim A ¯ p ( M ¯ p ) = n + dim ¯ A ¯ p (H n ( M ¯ p )) . By basic properties of Krull dimension, the finitely generated H ( A ) -module H n ( M ) sat-isfies dim (H n ( M ) ¯ p ) + dim (cid:0) H ( A ) / ¯ p (cid:1) ≤ dim (H n ( M )) . Hence, we obtain lc . dim A ¯ p ( M ¯ p ) + dim(H ( A ) / ¯ p ) = n + dim ¯ A ¯ p (H n ( M ¯ p )) + dim(H ( A ) / ¯ p ) ≤ n + dim (H n ( M )) ≤ lc . dim A ( M ) as claimed. (cid:3) HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 25
Proposition 5.11.
Let ( A, ¯ m ) be a commutative noetherian local DG-ring, and let M ∈ D bf ( A ) . Then the set W A ( M ) is a finite set.Proof. Let ¯ p ∈ W A ( M ) . By the definition of W A ( M ) and Lemma 5.10 there are inequal-ities lc . dim A ¯ p ( M ¯ p ) + dim(H ( A ) / ¯ p ) ≤ lc . dim A ( M ) ≤ sup( M ¯ p ) + dim(H ( A ) / ¯ p ) . This implies that lc . dim A ¯ p ( M ¯ p ) ≤ sup( M ¯ p ) . so by (2.4) there is an equality(5.12) lc . dim A ¯ p ( M ¯ p ) = sup( M ¯ p ) . Since by definition of the local cohomology Krull dimension, we have that lc . dim A ¯ p ( M ¯ p ) ≥ sup( M ¯ p ) + dim ¯ A ¯ p (cid:16) H sup( M ¯ p ) ( M ¯ p ) (cid:17) , it follows from (5.12) that dim ¯ A ¯ p (cid:16) H sup( M ¯ p ) ( M ¯ p ) (cid:17) = 0 . Since there is an isomorphism H sup( M ¯ p ) ( M ¯ p ) ∼ = (cid:16) H sup( M ¯ p ) ( M ) (cid:17) ¯ p , we deduce that ¯ p is a minimal prime ideal of the finitely generated H ( A ) -module H sup( M ¯ p ) ( M ) . It follows that W A ( M ) ⊆ sup( M ) [ n =inf( M ) Min H ( A ) (H n ( M )) . Since H n ( M ) is a finitely generated H ( A ) -module, it has only finitely many minimalprimes, so the fact that H n ( M ) = 0 only for finitely many n ∈ Z implies the result. (cid:3) Proposition 5.13.
Let ( A, ¯ m ) be a noetherian local DG-ring, let M ∈ D + ( A ) , let ¯ x ∈ ¯ m ⊆ H ( A ) , and assume that ¯ x / ∈ [ ¯ p ∈ Ass A ( M ) ¯ p . Then ¯ x is M -regular.Proof. Suppose ¯ x is not M -regular. Then the map ¯ x : H inf( M ) ( M ) → H inf( M ) ( M ) is not injective, so that ¯ x ∈ H ( A ) is a zero-divisor for H inf( M ) ( M ) . Hence, by [29, Tag00LD], there exists an associated prime ¯ p of the H ( A ) -module H inf( M ) ( M ) such that ¯ x ∈ ¯ p . By [29, Tag 0310], we deduce that ¯ p · H ( A ) ¯ p is an associated prime of H inf( M ) ( M ) ¯ p .Moreover, according to [29, Tag 0586], we have that ¯ p ∈ Supp H ( A ) (H inf( M ) ( M )) . Since H inf( M ) ( M ¯ p ) ∼ = H inf( M ) ( M ) ¯ p , we deduce that ¯ p ∈ Supp A ( M ) , and that inf( M ¯ p ) = inf( M ) . It follows from Proposition 3.3 that ¯ p ∈ Ass A ( M ) , which gives a contradiction. Hence, ¯ x is M -regular. (cid:3) Given a commutative ring A , a complex of A -modules M , and x ∈ A , we denote by K A ( x ; M ) the Koszul complex of M with respect to x . Explicitly, we have that K A ( x ; A ) = 0 → A · x −→ A → , K A ( x ; M ) = K A ( x ; A ) ⊗ A M where the complex K A ( x ; A ) is concentrated in degrees − , . Lemma 5.14.
Let A be a commutative DG-ring, let M ∈ D ( A ) , and let ¯ x ∈ H ( A ) . Thenthere is an isomorphism ( M// ¯ xM ) ⊗ L A H ( A ) ∼ = K ¯ A (cid:0) ¯ x ; M ⊗ L A H ( A ) (cid:1) in D (H ( A )) .Proof. To compute the left hand side, we apply the functor − ⊗ L A M ⊗ L A H ( A ) to the distinguished triangle A · ¯ x −→ A → A// ¯ x → A [1] in D ( A ) . We obtain a distinguished triangle M ⊗ L A H ( A ) → M ⊗ L A H ( A ) → ( M// ¯ xM ) ⊗ L A H ( A ) → (cid:0) M ⊗ L A H ( A ) (cid:1) [1] in D (H ( A )) . Since the cone of the leftmost map in this triangle is exactly the Koszulcomplex K ¯ A (cid:0) ¯ x ; M ⊗ L A H ( A ) (cid:1) the result follows. (cid:3) Lemma 5.15.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let M ∈ D − ( A ) . Then thereis an equality of sets of prime ideals: W A ( M ) = W H ( A )0 (cid:0) M ⊗ L A H ( A ) (cid:1) . Proof.
By Proposition 1.11 there is an equality
Supp A ( M ) = Supp H ( A ) (cid:0) M ⊗ L A H ( A ) (cid:1) . By Proposition 2.13, there is an equality lc . dim A ( M ) = lc . dim ¯ A (cid:0) M ⊗ L A H ( A ) (cid:1) . By (1.3) and Lemma 1.9, for any ¯ p ∈ Spec(H ( A )) , there is an equality sup( M ¯ p ) = sup (cid:0) ( M ⊗ L A H ( A )) ¯ p (cid:1) . These three equalities and the definition of W imply the result. (cid:3) Proposition 5.16.
Let ( A, ¯ m ) be a noetherian local DG-ring, let M ∈ D − f ( A ) , let ¯ x ∈ ¯ m ⊆ H ( A ) , and assume that ¯ x / ∈ [ ¯ p ∈ W A ( M ) ¯ p . Then lc . dim( M// ¯ xM ) = lc . dim( M ) − . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 27
Proof.
By Proposition 2.13 we have that lc . dim A ( M// ¯ xM ) = lc . dim ¯ A (cid:0) M// ¯ xM ⊗ L A H ( A ) (cid:1) . By Lemma 5.14 this is equal to lc . dim ¯ A (cid:0) K H ( A ) (cid:0) ¯ x ; M ⊗ L A H ( A ) (cid:1)(cid:1) . It follows from Lemma 5.15 that ¯ x / ∈ [ ¯ p ∈ W H0( A )0 ( M ⊗ L A H ( A ) ) ¯ p . Hence, by [9, Proposition 2.8], there is an equality lc . dim ¯ A (cid:0) K H ( A ) (cid:0) ¯ x ; M ⊗ L A H ( A ) (cid:1)(cid:1) = lc . dim ¯ A (cid:0) M ⊗ L A H ( A ) (cid:1) − and by Proposition 2.13, the latter is equal to lc . dim A ( M ) − , as claimed. (cid:3) Lemma 5.17.
Let ( A, ¯ m ) be a noetherian local DG-ring, let M ∈ D bf ( A ) , and supposethat:(1) seq . depth A ( M ) > ; that is depth A ( M ) > inf( M ) (2) der . dim A ( M ) > ; that is lc . dim A ( M ) > sup( M ) .Then there exists ¯ x ∈ ¯ m such that ¯ x is M -regular, and moreover lc . dim( M// ¯ xM ) =lc . dim( M ) − .Proof. It follows from assumption (1) that ¯ m / ∈ Ass A ( M ) , and from assumption (2) that ¯ m / ∈ W A ( M ) . By Proposition 5.9 and Proposition 5.11, the two sets of prime ideals Ass A ( M ) , W A ( M ) are finite sets, and since they do not include ¯ m , by the prime avoidancelemma, there exists ¯ x ∈ ¯ m such that ¯ x / ∈ [ ¯ p ∈ Ass A ( M ) ¯ p , and moreover ¯ x / ∈ [ ¯ p ∈ W A ( M ) ¯ p . It follows from Proposition 5.13 that ¯ x is M -regular, and from Proposition 5.16 that lc . dim( M// ¯ xM ) = lc . dim( M ) − , as claimed. (cid:3) The following theorem is a DG version of [9, Theorem 3.6].
Theorem 5.18.
Let ( A, ¯ m ) be a noetherian local DG-ring, and let M ∈ D bf ( A ) . Assumethat amp (RΓ ¯ m ( M )) ≥ amp( M ) . Then there exists a maximal M -regular sequence ¯ x , . . . , ¯ x n ∈ ¯ m such that for each ≤ i ≤ n , there is an equality (5.19) lc . dim ( M// (¯ x , . . . , ¯ x i ) M ) = lc . dim( M ) − i. Proof.
The proof is by induction on seq . depth A ( M ) . If seq . depth A ( M ) = 0 , thereis nothing to prove, as in this case, a maximal M -regular sequence is empty. Suppose seq . depth A ( M ) > ; that is, suppose that depth A ( M ) > inf( M ) . By Theorem 2.15 andProposition 3.2, we have that amp (RΓ ¯ m ( M )) = lc . dim( M ) − depth( M ) . By assumption, this number is greater or equal to amp( M ) = sup( M ) − inf( M ) , whichimplies that der . dim( M ) = lc . dim( M ) − sup( M ) ≥ depth( M ) − inf( M ) > . It follows that the conditions of Lemma 5.17 are satisfied, so there exists ¯ x ∈ ¯ m suchthat ¯ x is M -regular, and moreover lc . dim( M// ¯ xM ) = lc . dim( M ) − . Since ¯ x is M -regular, we have that depth( M// ¯ xM ) = depth( M ) − . Also, by (5.6) we have that amp( M// ¯ xM ) = amp( M ) , and inf( M// ¯ xM ) = inf( M ) , so that seq . depth A ( M// ¯ xM ) = seq . depth A ( M ) − . Applying Theorem 2.15 and Proposition 3.2 again, we obtain amp (RΓ ¯ m ( M// ¯ xM )) = lc . dim( M// ¯ xM ) − depth( M// ¯ xM ) == (lc . dim( M ) − − (depth( M ) −
1) == amp (RΓ ¯ m ( M )) ≥ amp( M ) = amp( M// ¯ xM ) . By the induction hypothesis, there is a maximal
M// ¯ xM -regular sequence ¯ x , . . . ¯ x n ∈ ¯ m such that lc . dim (( M// ¯ xM ) // (¯ x , . . . ¯ x i )) = lc . dim( M// ¯ xM ) − ( i −
1) = lc . dim( M ) − i, which implies that ¯ x , . . . ¯ x n ∈ ¯ m is a maximal M -regular sequence which satisfies (5.19). (cid:3) Corollary 5.20.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology.Then there exists a maximal A -regular sequence ¯ x , . . . , ¯ x n ∈ ¯ m such that ¯ x , . . . , ¯ x n canbe completed to system of parameters of H ( A ) .Proof. It follows from Theorem 4.1(1) that A satisfies the assumption of Theorem 5.18, sothere exists a maximal A -regular sequence ¯ x , . . . , ¯ x n ∈ ¯ m which satisfies lc . dim ( A// (¯ x , . . . , ¯ x i )) = lc . dim( A ) − i. The result now follows from the fact that lc . dim ( A// (¯ x , . . . , ¯ x i )) = dim (cid:0) H ( A ) / (¯ x , . . . , ¯ x i ) (cid:1) , and that lc . dim( A ) = dim (cid:0) H ( A ) (cid:1) . (cid:3) Corollary 5.21.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring. Then thereexists a maximal A -regular sequence ¯ x , . . . , ¯ x n ∈ ¯ m which is a system of parameters of H ( A ) . Moreover, for each ≤ i ≤ n , the DG-ring A// (¯ x , . . . , ¯ x i ) is local-Cohen-Macaulay.Proof. The first claim follows immediately from Corollary 5.20 since the fact that A islocal-Cohen-Macaulay implies that seq . depth( A ) = dim(H ( A )) . To prove the secondclaim, note that seq . depth( A// ¯ x ) = seq . depth( A ) − , and that dim(H ( A// ¯ x )) =dim(H ( A )) − , so by Corollary 5.5, the DG-ring A// ¯ x is local-Cohen-Macaulay, andthe general result follows by induction on seq . depth( A ) = dim(H ( A )) . (cid:3) We now prove a DG-version of the Bass conjecture about local-Cohen-Macaulay rings.
Theorem 5.22.
Let ( A, ¯ m ) be a noetherian local DG-ring with bounded cohomology.(1) If A is local-Cohen-Macaulay, there exists ≇ M ∈ D bf ( A ) such that inj dim A ( M ) < ∞ , and such that amp( M ) = amp( A ) . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 29 (2) Assume further that A has a noetherian model. For any ≇ M ∈ D bf ( A ) such that inj dim A ( M ) < ∞ , we have that amp( M ) ≥ amp( A ) . If there exists such M with amp( M ) = amp( A ) , then A is local-Cohen-Macaulay.Proof. (1) Let d = dim(H ( A )) . By Corollary 5.21, there exists a system of parameters ¯ x , . . . , ¯ x d ∈ ¯ m of H ( A ) which is a maximal A -regular sequence. Consider the DG-module N = A// (¯ x , . . . , ¯ x d ) ∈ D bf ( A ) . It follows from (5.6) that amp( N ) = amp( A ) .Moreover, note that by our construction of N , we have that flat dim A ( N ) < ∞ . Let E = E ( A, ¯ m ) ∈ D ( A ) be the injective DG-module corresponding to the injective hull ¯ E of the residue field H ( A ) / ¯ m . Finally, let M = R Hom A ( N, E ) . Note that by [28,Theorem 4.10], we have for all n ∈ Z an isomorphism(5.23) H n ( M ) ∼ = Hom H ( A ) (H − n ( N ) , ¯ E ) , so since ¯ E is a cogenerator of Mod(H ( A )) , we have that H n ( M ) = 0 if and only if H − n ( N ) = 0 . We deduce that amp( M ) = amp( N ) . Since flat dim A ( N ) < ∞ and inj dim A ( E ) < ∞ , it follows from the adjunction isomorphism R Hom A ( − , M ) = R Hom A ( − , R Hom A ( N, E )) ∼ = R Hom A ( − ⊗ L A N, E ) that inj dim A ( M ) < ∞ . It remains to show that the cohomologies of M are finitelygenerated over H ( A ) . To see this, note first that the fact that ¯ x , . . . , ¯ x d is a system ofparameters of H ( A ) implies that H ( N ) = H ( A// (¯ x , . . . , ¯ x d )) = H ( A ) / (¯ x , . . . , ¯ x d ) is a zero-dimensional local ring. For any n ∈ Z , we have that H − n ( N ) is a finitelygenerated H ( N ) -module, and since the H ( A ) -action on H − n ( N ) factors through thering H ( N ) , we deduce that the H ( A ) -module H − n ( N ) is artinian, and hence of finitelength. It follows from Matlis duality that its Matlis dual, which by (5.23) is H n ( M ) isfinitely generated, proving the claim.(2) Since A has a noetherian model, M ∈ D bf ( A ) and inj dim A ( M ) < ∞ , it follows from[18, Theorem B] that amp( M ) ≥ amp (RΓ ¯ m ( A )) . By Theorem 4.1(1), we have that amp (RΓ ¯ m ( A )) ≥ amp( A ) , which implies that amp( M ) ≥ amp( A ) . These inequalities also imply that if amp( M ) =amp( A ) , then we must have amp (RΓ ¯ m ( A )) = amp( A ) , so that A is local-Cohen-Macaulay. (cid:3) Remark 5.24.
Item (2) above solves a recent conjecture of Minamoto under the mildnoetherian model assumption. See [22, Conjecture 3.36].
Remark 5.25.
Unlike the rest of this paper, in item (2) above we had to impose the noether-ian model assumption, in addition to our standing assumption that DG-rings are noetherian.The reason for this is our use of the results of [17, 18] which made this assumption. Weconjecture that this assumption is redundant, both here and in general, in the main theoremsof [17, 18].
6. L
OCAL -C OHEN -M ACAULAY
DG-
MODULES
In this section we will define and study local-Cohen-Macaulay DG-modules and max-imal local-Cohen-Macaulay DG-modules over local-Cohen-Macaulay DG-rings. Over anoetherian local ring, one can define both Cohen-Macaulay modules and, more generally,Cohen-Macaulay complexes. Our notion of a local-Cohen-Macaulay DG-module gener-alizes Cohen-Macaulay modules and not Cohen-Macaulay complexes. As Yekutieli andZhang observed in [33, Theorem 6.2], the category of Cohen-Macaulay complexes overa ring is an abelian category, so this makes finding a definition which generalizes Cohen-Macaulay complexes particularly appealing, but it is currently not clear to us how to dothat.For Cohen-Macaulay modules, however, we do have the following generalization:
Definition 6.1.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ . We say that M ∈ D bf ( A ) is a local-Cohen-Macaulay DG-module if there are equalities amp( M ) = amp( A ) = amp (RΓ ¯ m ( M )) . We denote by
CM( A ) the full subcategory of D bf ( A ) consisting of local-Cohen-MacaulayDG-module.It is clear from this definition and Definition 4.2 that A is local-Cohen-Macaulay as aDG-module if and only if it is local-Cohen-Macaulay as a DG-ring. Proposition 6.2.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, and let R be a dualizing DG-module over A . Let M ∈ D bf ( A ) be such that amp( M ) = amp( A ) .Then M ∈ CM( A ) if and only if amp (R Hom A ( M, R )) = amp( A ) . In this case we also have that
R Hom A ( M, R ) ∈ CM( A ) . Proof.
Let ¯ E be the H ( A ) -module which is the injective hull of H ( A ) / ¯ m . By the DGlocal duality theorem ([28, Theorem 7.26]), we have that H n (RΓ ¯ m ( M )) = Hom H ( A ) (cid:0) Ext − nA ( M, R ) , ¯ E (cid:1) . Since ¯ E is an injective cogenerator, we deduce that amp (RΓ ¯ m ( M )) = amp (R Hom A ( M, R )) , proving the first claim. The second claim follows from the duality isomorphism M ∼ = R Hom A (R Hom A ( M, R ) , R ) that holds because R is a dualizing DG-module. (cid:3) Proposition 6.3.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, and let R be a dualizing DG-module over A . Then R ∈ CM( A ) .Proof. By Proposition 4.4 we have that amp( R ) = amp( A ) , and since R is dualizing, R Hom A ( R, R ) = A, so the result follows from Proposition 6.2. (cid:3) HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 31
In view of the local duality theorem, the above duality of local-Cohen-Macaulay DG-modules might seem tautological. More interesting is its restriction to maximal local-Cohen-Macaulay DG-modules which we now discuss.Recall that over a noetherian local ring A , a Cohen-Macaulay A -module M is calledmaximal Cohen-Macaulay if dim( M ) = dim( A ) . To generalize this to local DG-rings,we recall from (2.4) that if A is a ( A, ¯ m ) be a noetherian local DG-ring, and M ∈ D bf ( A ) that lc . dim( M ) ≤ sup( M ) + dim(H ( A )) . It is thus make sense to define: Definition 6.4.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ , and let M ∈ CM( A ) . We say that M is a maximal local-Cohen-Macaulay DG-module if lc . dim( M ) =sup( M ) + dim(H ( A )) . We denote by MCM( A ) the full subcategory of CM( A ) consist-ing of maximal local-Cohen-Macaulay DG-modules over A .It follows from this definition and from (2.6) that if A is a local-Cohen-Macaulay DG-ring, then A ∈ MCM( A ) .We will soon prove that dualizing DG-modules over a local-Cohen-Macaulay DG-ringare also maximal Cohen-Macaulay DG-modules. Before proving this, let us discuss thesituation over rings. Let A be a local ring, and let a R be a dualizing complex over A ,normalized so that inf( R ) = − dim( A ) . According to [29, tag 0A7U], for any ≤ i ≤ dim( A ) we have that dim (cid:0) H − i ( R ) (cid:1) ≤ i and H i ( R ) = 0 for i / ∈ [ − d, . Moreover, by [29, tag 0AWE], there is an equality dim (cid:0) H − d ( R ) (cid:1) = d. It follows from these facts and the definition of local cohomology Krull dimension that(6.5) lc . dim( R ) = 0 = inf( R ) + dim( A ) . Hence, it happens that lc . dim( R ) = sup( R ) + dim( A ) if and only if amp( R ) = 0 , if andonly if A is a Cohen-Macaulay ring. Let us record this fact, as we are not aware of anyliterature containing it. Proposition 6.6.
Let ( A, m ) be a noetherian local ring of dimension d , and let R be adualizing complex over A . Then A is Cohen-Macaulay if and only if lc . dim( R ) = sup( R ) + d. (cid:3) The next result generalizes this discussion to DG-rings. In the DG-setting, unlike thecase of rings, there can be more than one cohomology module of R of dimension d (andthere are at least two such cohomologies if A is local-Cohen-Macaulay and amp( A ) > ). Theorem 6.7.
Let ( A, ¯ m ) be a noetherian local DG-ring, let d = dim(H ( A )) , and let R be a dualizing DG-module over A . Assume that R is normalized so that inf( R ) = − d .Then the following holds:(1) There is an equality dim ¯ A (cid:16) H inf( R ) ( R ) (cid:17) = dim ¯ A (cid:0) H ( A ) (cid:1) . (2) If moreover amp( A ) = n < ∞ , then for every ≤ i ≤ dim ¯ A (H ( A )) there is aninequality dim ¯ A (cid:0) H − i + n ( R ) (cid:1) ≤ i (3) Assuming amp( A ) = n < ∞ , we have that A is local-Cohen-Macaulay if andonly if there is an equality dim ¯ A (cid:16) H sup( R ) ( R ) (cid:17) = dim ¯ A (cid:0) H ( A ) (cid:1) , if and only if lc . dim A ( R ) = sup( R ) + dim(H ( A )) , if and only if R ∈ MCM( A ) .Proof. (1) By assumption inf( R ) = − d . According to (1.2), we have that inf (cid:0) R Hom A (H ( A ) , R ) (cid:1) = − d, H − d ( R ) = H − d (cid:0) R Hom A (H ( A ) , R ) (cid:1) . Hence, it is enough to show that dim ¯ A (cid:0) H − d (cid:0) R Hom A (H ( A ) , R ) (cid:1)(cid:1) = d, and this follows from [29, tag 0AWE], because by [30, Proposition 7.5], the complex R Hom A (H ( A ) , R ) is a dualizing complex over H ( A ) .(2) According to [28, Proposition 7.25], we have that RΓ ¯ m ( R ) = E ( A, ¯ m ) , so in particular sup (RΓ ¯ m ( R )) = n. Hence, by Theorem 2.15, we deduce that lc . dim( R ) = n . Given ≤ i ≤ dim ¯ A (H ( A )) ,by the definition of Krull dimension of DG-modules, there are inequalities n = lc . dim( R ) = sup ℓ ∈ Z (cid:8) dim(H ℓ ( R )) + ℓ (cid:9) ≥ dim ¯ A (cid:0) H − i + n ( R ) (cid:1) + ( − i + n ) which implies that dim ¯ A (cid:0) H − i + n ( R ) (cid:1) ≤ i. (3) The fact that dim ¯ A (cid:16) H sup( R ) ( R ) (cid:17) = dim ¯ A (cid:0) H ( A ) (cid:1) , if and only if lc . dim A ( R ) = sup( R ) + dim(H ( A )) , follows from the definition of dimension and from (2.5). To see that this is equivalent to A being local-Cohen-Macaulay, note that, as we have seen in the proof of (2), lc . dim A ( R ) = n . It follows that lc . dim A ( R ) = sup( R ) + dim(H ( A )) if and only if sup( R ) = n − d , if and only if amp( R ) = sup( R ) − inf( R ) = ( n − d ) − ( − d ) = n, and by Proposition 4.4, this is equivalent to A being local-Cohen-Macaulay. Finally, if A is local-Cohen-Macaulay, we have seen that R ∈ CM( A ) , so the above implies that R ∈ MCM( A ) , while if R ∈ MCM( A ) , in particular R ∈ CM( A ) , so that A is local-Cohen-Macaulay. (cid:3) Remark 6.8.
In view of the above result, it is natural to wonder if for a dualizing DG-module R over a local-Cohen-Macaulay DG-ring A , one has that dim(H i ( R )) = d for all − d ≤ i ≤ n − d . This is not the case. In fact, it could happen that H i ( R ) = 0 for all − d < i < n − d . Such an example will be given in Example 7.7 below. HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 33
As we have seen above, the topmost and bottommost cohomologies of a dualizingDG-module over a noetherian local-Cohen-Macaulay DG-ring have maximal dimension.The next result shows that this is the case for every maximal local-Cohen-Macaulay DG-module.
Proposition 6.9.
Let ( A, ¯ m ) be a noetherian local DG-ring with amp( A ) < ∞ , and let M ∈ MCM( A ) . Then dim ¯ A (cid:16) H inf( M ) ( M ) (cid:17) = dim ¯ A (cid:16) H sup( M ) ( M ) (cid:17) = dim(H ( A )) . Proof.
The second equality follows from the definition and from (2.5). Since M is maxi-mal local-Cohen-Macaulay, we have that depth A ( M ) = lc . dim( M ) − sup( M ) + inf( M ) = dim (cid:0) H ( A ) (cid:1) + inf( M ) , so by Proposition 3.5 we deduce that dim (cid:0) H ( A ) (cid:1) + inf( M ) ≤ dim ¯ A (cid:16) H inf( M ) ( M ) (cid:17) + inf( M ) , which implies that dim (cid:0) H ( A ) (cid:1) ≤ dim ¯ A (cid:16) H inf( M ) ( M ) (cid:17) , so these numbers must be equal. (cid:3) We now discuss characterizations of local-Cohen-Macaulay and maximal local-Cohen-Macaulay DG-modules using the notion of a regular sequence from the previous section.
Proposition 6.10.
Let ( A, ¯ m ) be a noetherian local DG-ring, let M ∈ D bf ( A ) , and supposethat amp( M ) = amp( A ) . Then M ∈ CM( A ) if and only if the length of a maximal M -regular sequence contained in ¯ m is equal to lc . dim( M ) − sup( M ) ; that is if and only if seq . depth A ( M ) = der . dim A ( M ) . If M ∈ CM( A ) , then M ∈ MCM( A ) if and only if seq . depth A ( M ) = der . dim( A ) .Proof. It follows from the definitions and from from Theorem 2.15 and Proposition 3.2that seq . depth A ( M ) = der . dim A ( M ) if and only if amp(RΓ ¯ m ( M )) = amp( M ) if andonly if M ∈ CM( A ) . In that case, it also follows from the definitions that M ∈ MCM( A ) if and only if seq . depth A ( M ) = der . dim( A ) . (cid:3) Next we show that like Cohen-Macaulay DG-modules, over local-Cohen-MacaulayDG-rings maximal local-Cohen-Macaulay DG-modules are also self-dual:
Proposition 6.11.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, let R be adualizing DG-module over A , and let M ∈ MCM( A ) . Then R Hom A ( M, R ) ∈ MCM( A ) . Proof.
It is clear that
R Hom A ( M, R ) ∈ CM( A ) , so we only need to show that lc . dim (R Hom A ( M, R )) − sup (R Hom A ( M, R )) = dim(H ( A )) . According to Theorem 2.15 we have that lc . dim (R Hom A ( M, R )) = sup (RΓ ¯ m (R Hom A ( M, R ))) , and by the DG local duality theorem ([28, Theorem 7.26]), the latter is equal to − inf( M ) .The local duality theorem also implies that sup (R Hom A ( M, R )) = − inf (RΓ ¯ m ( M )) , and by Proposition 3.2 the latter is equal to − depth( M ) . Hence, lc . dim (R Hom A ( M, R )) − sup (R Hom A ( M, R )) = − inf( M ) + depth( M ) = seq . depth( M ) and since M is maximal local-Cohen-Macaulay, the latter is equal to dim(H ( A )) , asclaimed. (cid:3)
7. T
RIVIAL EXTENSION
DG-
RINGS AND THE LOCAL -C OHEN -M ACAULAY PROPERTY
In this section we study trivial extension DG-rings. This will allow us to construct manyexamples of local-Cohen-Macaulay DG-rings. We first recall the construction, following[19, Section 1] (but note that we use cohomological notation, unlike the homological nota-tion used there).Let A be a commutative DG-ring, and let M ∈ D ( A ) . Suppose that sup( M ) < .Donating by d A the differential of A and by d M the differential of M , we give the gradedabelian group A ⊕ M the structure of a commutative DG-ring by letting the differential be d A ⊕ M (cid:18) am (cid:19) = (cid:18) d A ( a ) d M ( m ) (cid:19) , and defining multiplication by the rule (cid:18) a m (cid:19) · (cid:18) a m (cid:19) = (cid:18) a · a a · m + m · a (cid:19) . This gives A ⊕ M the structure of a commutative (non-positive) DG-ring, which we willdenote by A ⋉ M . We note that there are natural maps of DG-rings A → A ⋉ M → A ,such that their composition is equal to A .The next result follows from the definition: Proposition 7.1.
Let A be a commutative noetherian DG-ring, let M ∈ D − f ( A ) , andassume that sup( M ) < . Then A ⋉ M is a commutative noetherian DG-ring with H ( A ⋉ M ) = H ( A ) . In particular, if ( A, ¯ m ) is a local DG-ring, then ( A ⋉ M, ¯ m ) is a local DG-ring. (cid:3) This construction will allow us to construct many interesting examples and counterex-amples. For instance, let us show that the notion of a regular sequence behaves different inthe DG-setting:
Example 7.2.
Let k be a field, let R = k [[ x, y ]] / ( x · y ) , and let M be the R -module R/ ( x ) ∼ = k [[ y ]] . Consider the DG-ring A = R ⋉ M [1] , and consider y ∈ H ( A ) = R .Since y is M -regular, and M = H inf( A ) ( A ) , we see that y is A -regular. However, since dim( R ) = dim( R/y ) , we have that dim(H ( A )) = dim(H ( A//y )) .Next, we provide a sufficient condition for the trivial extension to be local-Cohen-Macaulay: Theorem 7.3.
Let ( A, m ) be a noetherian local ring, and M ∈ D bf ( A ) . Assume that:(1) sup( M ) < .(2) lc . dim( M ) = inf( M ) + dim( A ) .(3) lc . dim( M ) ≤ depth( A ) .Then the trivial extension A ⋉ M is a noetherian local-Cohen-Macaulay DG-ring if andonly if M is a Cohen-Macaulay complex over A . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 35
Proof.
By Proposition 7.1 and the definition, the DG-ring A ⋉ M is a noetherian localDG-ring with bounded cohomology, and it holds that amp( A ⋉ M ) = − inf( M ) . Tocompute amp (cid:0) RΓ A ⋉ M m ( A ⋉ M ) (cid:1) , we may apply the forgetful functor D ( A ⋉ M ) → D ( A ) , and since local cohomologycommutes with it, and the ideals of definition of A and A ⋉ M coincide, we may computeinstead the number amp (cid:0) RΓ A m ( A ⋉ M ) (cid:1) , where we now compute local cohomology over the local ring A . Since in the category D ( A ) we have that A ⋉ M = A ⊕ M , so that RΓ A m ( A ⋉ M ) ∼ = RΓ A m ( A ) ⊕ RΓ A m ( M ) . Since lc . dim( M ) ≤ depth( A ) , we must have depth( M ) ≤ depth( A ) . Hence, inf (cid:0) RΓ A m ( A ⋉ M ) (cid:1) = inf (cid:0) RΓ A m ( M ) (cid:1) = depth( M ) . Similarly, since lc . dim( M ) = inf( M ) + dim( A ) ≤ sup( M ) + dim( A ) < dim( A ) , wehave that sup (cid:0) RΓ A m ( A ⋉ M ) (cid:1) = sup (cid:0) RΓ A m ( A ) (cid:1) = dim( A ) = lc . dim( M ) − inf( M ) . Hence, amp (cid:0) RΓ A m ( A ⋉ M ) (cid:1) = lc . dim( M ) − inf( M ) − depth( M ) =(lc . dim( M ) − depth( M )) + ( − inf( M )) =(lc . dim( M ) − depth( M )) + amp( A ⋉ M ) We see that A ⋉ M is local-Cohen-Macaulay if and only if lc . dim( M ) − depth( M ) = 0 if and only if M is a Cohen-Macaulay complex over A . (cid:3) Example 7.4.
Let ( A, m ) be a noetherian local ring, let M be a finitely generated A -module, and suppose that dim( M ) = dim( A ) . Then M [dim( A )] satisfies the conditionsof Theorem 7.3, so A ⋉ M [dim( A )] is a local-Cohen-Macaulay DG-ring if and only if M is a Cohen-Macaulay module if and only if M is a maximal Cohen-Macaulay module. Example 7.5.
A bit more generally, if ( A, m ) is a noetherian local ring, and M is anarbitrary finitely generated A -module, we can replace A with A/ ann( M ) , and considerthe DG-ring B = A/ ann( M ) ⋉ M [dim( M )] . Then the conditions of Theorem 7.3 hold, and we see that B is a local-Cohen-MacaulayDG-ring if and only if M is a Cohen-Macaulay module over A . Example 7.6.
Let ( A, m ) be a noetherian local ring, let R be a dualizing complex over A , and suppose that sup( R ) < . Jørgensen proved in [19] that in this case A ⋉ R is aGorenstein DG-ring. In particular, by Proposition 4.5, A ⋉ R is a local-Cohen-MacaulayDG-ring. Here is an alternative proof of the latter fact: By (6.5), condition (2) of Theorem7.3 is satisfied. Moreover, Grothendieck’s local duality implies that: > sup( R ) = inf( R )+amp( R ) = inf( R )+dim( A ) − depth( A ) = lc . dim( R ) − depth( A ) , so that lc . dim( R ) < depth( A ) . Hence, all the conditions of Theorem 7.3 are satisfied,and we deduce that A ⋉ R is a local-Cohen-Macaulay DG-ring. Example 7.7.
Following Remark 6.8, let ( A, m ) be a Cohen-Macaulay local ring with adualizing module M . Then by the above mentioned Theorem of Jørgensen from [19], wehave that B = A ⋉ M [dim( A )] is a Gorenstein DG-ring. In particular, B is a local-Cohen-Macaulay DG-ring, and a dualizing DG-module over itself. Moreover, for all inf( B )
Let ( A, m ) be a noetherian local ring of Krull dimension d > which doesnot have a dualizing complex, but does have a maximal Cohen-Macaulay module M . Togive a concrete example of such A, M , in [23, Section 6, Example 1] there is an exampleof Cohen-Macaulay ring which does not have a canonical module. Then one can take thisring to be A , and M = A . By Theorem 7.3, the DG-ring B = A ⋉ M [ d ] is a local-Cohen-Macaulay DG-ring. Since H ( B ) = A does not have a dualizing complex, it follows from[30, Proposition 7.5] that B does not have a dualizing DG-module.8. F INITE MAPS , LOCALIZATIONS AND GLOBAL C OHEN -M ACAULAY
DG-
RINGS
In this section we discuss analogues of the following two basic properties of Cohen-Macaulay rings and modules:(1) (
Independence of base ring ): Assume f : ( A, m ) → ( B, n ) is a local map betweentwo noetherian local rings which is a finite ring map, and that M is a finitely generated B -module. Then M is Cohen-Macaulay over B if and only if M is Cohen-Macaulay over A .(2) ( Localization ): Let ( A, m ) be a noetherian local Cohen-Macaulay ring, and let p ∈ Spec( A ) . Then ( A p , p · A p ) is also Cohen-Macaulay.Unfortunately, both of these properties fail in general in the DG-case, and for the samereason: change of amplitude. Any commutative ring A has amp( A ) = 0 , but different DG-rings have different amplitude. Regarding item (1) above, if one assumes that amp( A ) =amp( B ) , then its DG-generalization is true. Regarding item (2), it could happen that fora DG-ring A , and ¯ p ∈ Spec(H ( A )) there is a strict inequality amp( A ¯ p ) < amp( A ) ,and when that happens, it could happen that A is local-Cohen-Macaulay but A ¯ p is not.Again, when amp( A ¯ p ) = amp( A ) , it holds that if A is local-Cohen-Macaulay then A ¯ p is. On the positive side, we will see below that often for local-Cohen-Macaulay DG-rings, there is always an equality amp( A ¯ p ) = amp( A ) , so that under the assumption thatthe topological space Spec(H ( A )) is irreducible, the local-Cohen-Macaulay property isstable under localization. Proposition 8.1. (Independence of base DG-ring): Let ( A, ¯ m ) and ( B, ¯ n ) be two noether-ian local DG-rings, such that amp( A ) = amp( B ) < ∞ . Let f : A → B be a map ofDG-rings such that the induced map H ( f ) : H ( A ) → H ( B ) is a finite ring map whichis local. Given M ∈ D bf ( B ) , we have that M is a local-Cohen-Macaulay DG-module over B if and only if M is a local-Cohen-Macaulay DG-module over A . Before proving this result, we shall need the following lemma.
Lemma 8.2.
Let ( A, ¯ m ) and ( B, ¯ n ) be two noetherian local DG-rings, let f : A → B be amap of DG-rings, and suppose that H ( f )( ¯ m ) · H ( B ) contains some power of ¯ n . Letting Q : D ( B ) → D ( A ) be the forgetful functor, there is an isomorphism RΓ ¯ m ◦ Q ( − ) ∼ = Q ◦ RΓ ¯ n ( − ) of functors D ( B ) → D ( A ) . HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 37
Proof.
Let a be a finite sequence of elements of A whose image in H ( A ) generates ¯ m ,let b be a finite sequence of elements of B whose image in H ( B ) generates ¯ n , and let N ∈ D ( N ) . By (1.5) there are natural isomorphisms: RΓ ¯ m ( Q ( N )) ∼ = RΓ ¯ m ( B ⊗ L B N ) ∼ = Tel( A ; a ) ⊗ A A ⊗ L A B ⊗ L B N By associativity of the derived tensor product, this is naturally isomorphic to
Tel( A ; a ) ⊗ A B ⊗ L B B ⊗ L B N. Our assumption on H ( f ) and (1.6) imply that Tel( A ; a ) ⊗ A B ⊗ L B B ∼ = Tel( B ; b ) ⊗ L B B, so the result follows from (1.5). (cid:3) Proof of Proposition 8.1.
It follows from Lemma 8.2 that there is an equality amp (RΓ ¯ m ( M )) = amp (RΓ ¯ m ( N )) . If M is local-Cohen-Macaulay over A then amp( M ) = amp( A ) = amp (RΓ ¯ m ( M )) , and the assumption that amp( A ) = amp( B ) then implies that M is local-Cohen-Macaulayover B . The claim that M ∈ CM( B ) = ⇒ M ∈ CM( A ) follows similarly. (cid:3) Next we discuss the problem of localization. We begin with a counterexample.
Example 8.3.
Let ( A, m ) be a noetherian local ring. Assume that there is a finitely gen-erated A -module M which is maximal Cohen-Macaulay, such that for some p ∈ Spec( A ) we have that p / ∈ Supp( M ) , and the ring A p is not Cohen-Macaulay. As a concrete ex-ample, let k be a field, take A to be the localization of k [ x, y, z ] / ( y z, xyz ) at the origin, M = A/zA and p = ( x, y ) . It follows from Theorem 7.3 that B = A ⋉ M [dim( A )] is alocal-Cohen-Macaulay DG-ring. Localizing at p ∈ H ( B ) , the fact that M p = 0 impliesthat B p = ( A ⋉ M [dim( A )]) p ∼ = A p which is not local-Cohen-Macaulay.Two features of this counterexample that caused this unfortunate phenomena are that amp( B p ) < amp( B ) , and that Spec(H ( B )) is reducible. Theorem 8.4.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, and let ¯ p ∈ Spec(H ( A )) . Suppose that A has a dualizing DG-module. Assume that amp( A ¯ p ) =amp( A ) ; equivalently, that ¯ p ∈ Supp(H inf( A ) ( A )) . Then (cid:0) A ¯ p , ¯ p · H ( A ¯ p ) (cid:1) is a local-Cohen-Macaulay DG-ring.Proof. Assume R is a dualizing DG-module over A . It follows from [26, Corollary 6.11]that R ¯ p is a dualizing DG-module over A ¯ p . It follows from the definition of localizationthat amp( R ¯ p ) ≤ amp( R ) , while by Theorem 4.1 we have that amp( R ¯ p ) ≥ amp( A ¯ p ) = amp( A ) . Since A is local-Cohen-Macaulay, by Proposition 4.4 we have that amp( A ) = amp( R ) ,so the above inequalities imply that amp( R ¯ p ) = amp( A ¯ p ) . Proposition 4.4 now impliesthat A ¯ p is local-Cohen-Macaulay. (cid:3) Proposition 8.5.
Let ( A, m ) be a noetherian local ring such that Spec( A ) is irreducible;equivalently, such that A has a unique minimal prime ideal. Let M be a finitely generated A -module such that dim( M ) = dim( A ) . Then Supp( M ) = Spec( A ) .Proof. Since M is finitely generated by [29, tag 00L2], we have that(8.6) Supp( M ) = { p ∈ Spec( A ) | ann( M ) ⊆ p } . Let q be the unique minimal prime ideal of A . Since dim( M ) = dim( A ) , so that Supp( M ) contains a chain of primes of length dim( A ) , we must have q ∈ Supp( M ) . It follows from(8.6) that ann( M ) ⊆ q , and since for all p ∈ Spec( A ) , we have that q ⊆ p , we deduce that Supp( M ) = Spec( A ) . (cid:3) Corollary 8.7.
Given a noetherian local-Cohen-Macaulay DG-ring ( A, ¯ m ) , assume that A has a dualizing DG-module. If Spec(H ( A )) is irreducible, then for all ¯ p ∈ Spec(H ( A )) ,the DG-ring (cid:0) A ¯ p , ¯ p · H ( A ¯ p ) (cid:1) is local-Cohen-Macaulay.Proof. By Proposition 4.11, we have that dim (cid:16) H inf( A ) ( A ) (cid:17) = dim(H ( A )) , so by Proposition 8.5, we deduce that Supp (cid:16) H inf( A ) ( A ) (cid:17) = Spec(H ( A )) . Hence, for all ¯ p ∈ Spec(H ( A )) , we have that amp( A ¯ p ) = amp( A ) , so the result followsfrom Theorem 8.4. (cid:3) We now make a global definition of Cohen-Macaulay DG-rings:
Definition 8.8.
Let A be a noetherian DG-ring with bounded cohomology. We say that A is a Cohen-Macaulay DG-ring if for all ¯ p ∈ Spec(H ( A )) , the DG-ring A ¯ p is a local-Cohen-Macaulay DG-ring. Proposition 8.9.
Let A be a noetherian DG-ring which is Gorenstein. Then A is Cohen-Macaulay.Proof. For any ¯ p ∈ Spec(H ( A )) , it follows from [26, Corollary 6.11] that A ¯ p is a localGorenstein DG-ring, so by Proposition 4.5, A ¯ p is local-Cohen-Macaulay. (cid:3) Corollary 8.10.
Let A be a noetherian DG-ring with bounded cohomology, and assume A has a dualizing DG-module. Suppose that Spec(H ( A )) is locally irreducible; equiva-lently, any maximal ideal of Spec(H ( A )) contains a unique minimal prime ideal. Then A is a Cohen-Macaulay DG-ring if and only if for any maximal ideal ¯ m ∈ Spec(H ( A )) , theDG-ring A ¯ m is local-Cohen-Macaulay.Proof. The assumption that
Spec(H ( A )) is locally irreducible implies that for any max-imal ideal ¯ m ∈ Spec(H ( A )) , the ring H ( A ¯ m ) has an irreducible spectrum, and by as-sumption the DG-ring A ¯ m is local-Cohen-Macaulay. Since the property of having a singleminimal prime ideal is preserved by localization, the result now follows from Corollary8.7. (cid:3) Corollary 8.11.
Let A be a noetherian DG-ring with bounded cohomology, and assume A has a dualizing DG-module. Let n = amp( A ) , and suppose that the H ( A ) -module H − n ( A ) has full support. Then A is a Cohen-Macaulay DG-ring if and only if for anymaximal ideal ¯ m ∈ Spec(H ( A )) , the DG-ring A ¯ m is local-Cohen-Macaulay. HE COHEN-MACAULAY PROPERTY IN DERIVED COMMUTATIVE ALGEBRA 39
Proof.
This follows from Theorem 8.4. (cid:3)
We finish this section with the following observation about the cohomology of a du-alizing DG-module over a local-Cohen-Macaulay DG-ring, as it follows from applyinglocalization to it.
Proposition 8.12.
Let ( A, ¯ m ) be a noetherian local-Cohen-Macaulay DG-ring, and let R be a dualizing DG-module over A . Then Supp (cid:16) H inf( A ) ( A ) (cid:17) ⊆ Supp (cid:16) H inf( R ) ( R ) (cid:17) , and Supp (cid:16) H inf( A ) ( A ) (cid:17) ⊆ Supp (cid:16) H sup( R ) ( R ) (cid:17) . Proof.
Given ¯ p ∈ Supp (cid:0) H inf( A ) ( A ) (cid:1) , we have that amp( A ¯ p ) = amp( A ) , so by the proofof Theorem 8.4, we have that amp( R ¯ p ) = amp( R ) . This in turn implies that ¯ p ∈ Supp (cid:16) H inf( R ) ( R ) (cid:17) , and that ¯ p ∈ Supp (cid:16) H sup( R ) ( R ) (cid:17) . (cid:3)
9. S
OME REMARKS ON NON - NEGATIVELY GRADED COMMUTATIVE
DG-
RINGS
In this final section we will work with non-negatively graded commutative DG-rings A = ∞ M n =0 A n with a differential of degree +1 . We wish to discuss here the question: when is such aDG-ring Cohen-Macaulay? We briefly discuss a possible answer to this question. Wewill continue to work with the assumption that A is noetherian local and has boundedcohomology; that is, we assume that (H ( A ) , ¯ m ) is a noetherian local ring, for each i > ,the H ( A ) -module H i ( A ) is finitely generated, and for i >> we have that H i ( A ) = 0 .Similarly to Section 1.5, the results of [5] still apply in the non-negative setting, andthere is a local cohomology functor RΓ A ¯ m : D ( A ) → D ( A ) . Note that in the non-negative setting, the map H ( A ) → A goes in the other direction,so there is a forgetful functor Q : D ( A ) → D (H ( A )) . It follows from [6, Proposition 2.7]that local cohomology commutes with Q , so that there is an isomorphism(9.1) Q ◦ RΓ A ¯ m ( − ) ∼ = RΓ ¯ A ¯ m ◦ Q ( − ) of functors D ( A ) → D (H ( A )) .This implies that working with local cohomology in the non-negative setting is actuallyeasier, as the computation immediately reduce to the local ring H ( A ) . This allows us toconstruct the following counterexample to Theorem 4.1 in the non-negative setting: Example 9.2.
Let ( B, m ) be a noetherian local Cohen-Macaulay ring of dimension , andlet k = B/ m . Consider the non-negative noetherian local DG-ring A = B ⋉k [ − . Thenit follows from (9.1) that, as a complex of B -modules RΓ m ( A ) ∼ = RΓ m ( B ) ⊕ k [ − is concentrated in degree +1 , so in particular amp (RΓ m ( A )) = 0 < amp( A ) = 1 . The reason for the existence of this counterexample is that(9.3) dim (cid:16) H sup( A ) (cid:17) < dim (cid:0) H ( A ) (cid:1) . This of course cannot happen in the non-positive setting.Now, for non-positive local-Cohen-Macaulay DG-rings, not only that (9.3) is an equal-ity, but also, according to Proposition 4.11, there is a dual equality: dim (cid:16) H inf( A ) (cid:17) = dim (cid:0) H ( A ) (cid:1) . If one assumes that the inequality in (9.3) is an equality, then it follows that the analogueof Theorem 4.1 does hold in the non-negative setting.In view of this discussion, we propose that a noetherian local non-negative DG-ring ( A, ¯ m ) with amp( A ) < ∞ will be called local-Cohen-Macaulay if it satisfies the followingtwo conditions:(1) There is an equality dim (cid:16) H sup( A ) ( A ) (cid:17) = dim (cid:0) H ( A ) (cid:1) . (2) There is an equality amp (RΓ ¯ m ( A )) = amp( A ) . To give further evidence that this is a good definition, consider a non-negative Goren-stein DG-rings ( A, ¯ m ) . It is often the case that if R is a dualizing complex over H ( A ) ,then R Hom H ( A ) ( A, R ) is a dualizing DG-module over A . Assuming the uniqueness the-orem for dualizing DG-modules holds in this setting, the assumption that A is Gorensteinimplies that A is isomorphic to a shift of R Hom H ( A ) ( A, R ) . Then Grothendieck’s localduality implies that amp (RΓ ¯ m ( A )) = amp( A ) . Moreover, a calculation shows that depth( A ) = dim(H ( A )) , and from this one candeduce that dim (cid:16) H sup( A ) ( A ) (cid:17) = dim (cid:0) H ( A ) (cid:1) , so that A is local-Cohen-Macaulay in the above sense. Acknowledgments.
The author is grateful for the anonymous referee for many comments that helped im-proving the manuscript. The author thanks Amnon Yekutieli for many useful sugges-tions. The author was partially supported by the Israel Science Foundation (grant no.1346/15). This work has been supported by Charles University Research Centre programNo.UNCE/SCI/022. R
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