Cohen-Macaulay Homological Dimensions

REND.SEM.MAT.UNIV.PADOVA, Vol. 117 2007)

Cohen-Macaulay Homological Dimensions.

HENRIK HOLM *) - PETER JéRGENSEN **)

ABSTRACT - We define three new homological dimensions: Cohen-Macaulay injective, projective, and flat dimension, and show that they inhabit a theory similar to that of classical injective, projective, and flat dimension. In particular, we show that finiteness of the new dimensions characterizes Cohen-Macaulay rings with dualizing modules.

1. Introduction.

The classical theory of homological dimensions is very important to commutative algebra. In particular, it is useful that there are a number of finiteness conditions on these dimensions which characterize regular rings. For instance, if the projective dimension of each finitely generated A- module is finite, then A is a regular ring. Several attempts have been made to mimic this success by defining homological dimensions whose finiteness would characterize other rings than the regular ones. These efforts have given us complete intersection dimension [2], Gorenstein dimension [1], and Cohen-Macaulay dimension [8]. The normal practice has not been to mimic the full classical theory, which comprises both injective, projective, and flat dimension for arbitrary modules, but rather to focus on projective dimension for finitely generated modules. Hence complete intersection dimension and Cohen-Macaulay dimension only exist in this restricted sense, and the same used to be the case for Gorenstein dimension.

*) Indirizzo dell'A.: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, 8000 AÊrhus C, Denmark. E-mail: [email protected] **) Indirizzo dell'A.: School of Mathematics and Statistics, Merz-Court New- castle University, Newcastle-Upon-Tyne NE1 7RU. E-mail: [email protected] 2000 Mathematics Subject Classification. 13D05, 13D25. 88 Henrik Holm - Peter Jùrgensen

However, recent years have seen much work on the Gorenstein theory which now contains both Gorenstein injective, projective, and flat dimension. These dimensions inhabit a nice theory similar to classical homological algebra. A summary is given in [3], although this reference is al- ready a bit dated. The purpose of this paper is to develop a similar theory in the Cohen- Macaulay case. So we will define Cohen-Macaulay injective, projective, and flat dimension, denoted CMid, CMpd, and CMfd, and show that they inhabit a theory with a number of desirable features. The main result is the following, whereby finiteness of the Cohen-Macaulay dimensions characterizes Cohen-Macaulay rings with dualizing modules.

THEOREM. Let A be a local commutative noetherian ring. Then the following conditions are equivalent. i) A is a Cohen-Macaulay ring with a dualizing module.

ii) CMidA M < 1 for each A-module M. iii) CMpdA M < 1 for each A-module M. _ iv) CMfdAM < 1 for each A-module M. Another main result is a bound for the Cohen-Macaulay dimensions in terms of the Krull dimension dim A of the ring.

THEOREM. Let A be a local commutative noetherian ring and let M be an A-module. Then

CMidA M < 1,CMidA M dim A;

CMpdA M < 1,CMpdA M dim A;

CMfdA M < 1,CMfdA M dim A: These results are proved as theorems 5.1 and 5.4; in fact, theorem 5.1 is somewhat more general than the first of the above theorems. There are also some other results: Theorem 5.6 compares our Cohen- Macaulay projective dimension to Gerko's Cohen-Macaulay dimension from [8], and to the Gorenstein projective dimension, see [3, chp. 4]. And propositions 5.7 and 5.8 show Auslander-Buchsbaum and Bass formulae for the Cohen-Macaulay dimensions. As tools to define the Cohen-Macaulay dimensions, we use change of rings. If A is a ring with a semi-dualizing module C as defined in [4]), then we can consider the trivial extension ring AÂj C, and if M is a complex of A- modules, then we can consider M as a complex of AÂj C)-modules and take Cohen-Macaulay Homological Dimensions 89

Gorenstein homological dimensions of M over AÂj C. The infima of these over all semi-dualizing modules C define the Cohen-Macaulay dimensions of M. Our use of trivial extension rings was in part inspired by [8], and there are also some similarities to [7]. It is worth noting that while finiteness of our Cohen-Macaulay dimensions characterises Cohen-Macaulay rings with dualizing modules, there are Cohen-Macaulay rings without dualizing modules, see [6]. It would be of interest to develop similar dimensions whose finiteness char- acterised all Cohen-Macaulay rings, but we do not yet know how to do that. The paper is organized as follows. Section 2 defines the Cohen-Ma- caulay dimensions and gives an example of their computation. Sections 3 and 4 establish a number of technical tools by studying the trivial extension ring AÂj C when C is a semi-dualizing module for A, and giving some bounds on the injective dimension of C. And finally, section 5 proves all the main results as described. Let us end the introduction with a few blanket items. Throughout, A is a commutative noetherian ring, and complexes of A-modules have the form

ÁÁÁ!Mi1 ! Mi ! MiÀ1 !ÁÁÁ: Isomorphisms in categories are denoted by and equivalences of functors by '.

2. Cohen-Macaulay dimensions.

This section introduces Cohen-Macaulay injective, projective, and flat dimension, and gives an example of their computation.

DEFINITION 2.1. Let C be an A-module. The direct sum A È C can be equipped with the product

a; c) Á a1; c1) aa1; ac1 ca1): This turns A È C into a ring which is called the trivial extension of A by C and denoted AÂj C.

There are ring homomorphisms

A ! AÂj C ! A; a 7À! a; 0); a; c) 7À! a 90 Henrik Holm - Peter Jùrgensen whose composition is the identity on A. These homomorphisms allow us to view any A-module as an AÂj C)-module and any AÂj C)-module as an A- module, and we shall do so freely. In particular, if M is a complex of A-modules with suitably bounded homology, then we can consider the Gorenstein homological dimensions over AÂj C, Gid M; Gpd M; and Gfd M; AÂj C AÂj C AÂj C where Gid, Gpd, and Gfd denote the Gorenstein injective, projective, and flat dimensions as described e.g. in [3]. Let us briefly recall the definitions of the Gorenstein dimensions. A complex N with HiN 0 for i 0 has GpdN n if it is quasi-isomorphic to a complex G which has the form

ÁÁÁ!0 ! Gn ! GnÀ1 !ÁÁÁ and consists of Gorenstein projective modules. The Gorenstein projective modules are the kernels of differentials in complete projective resolutions, that is, exact complexes P of projective modules for which HomP; Q)is also exact for each projective module Q. Dualizing this gives the definition of Gorenstein injective dimension. Finally, Gorenstein flat dimension is defined as follows. A complex N with

HiN 0 for i 0 has GfdN n if it is quasi-isomorphic to a complex H which has the form

ÁÁÁ!0 ! Hn ! HnÀ1 !ÁÁÁ and consists of Gorenstein flat modules. The Gorenstein flat modules are the kernels of differentials in exact complexes of flat modules which remain exact when tensored with an injective module.

DEFINITION 2.2. A semi-dualizing module C for A is a finitely generated module for which the canonical map A ! HomAC; C) is an isomorphism, i while ExtAC; C) 0 for each i51. A semi-dualizing module is called dualizing if it has finite injective dimension. Note that equivalently, a finitely generated module C is semi-dualizing if the canonical morphism A ! RHomAC; C) is an isomorphism in the derived category DA). An example of a semi-dualizing module is A itself, so A always has at least one semi-dualizing module. On the other hand, if A has a dualizing module, then it is necessarily Cohen-Macaulay, as follows for instance by [3, A.8.5.3)]. Cohen-Macaulay Homological Dimensions 91

The theory of semi-dualizing modules and complexes) is developed in [4].

DEFINITION 2.3. Let M and N be complexes of A-modules such that

Hi M 0 for i 0 and Hi N 0 for i 0. The Cohen-Macaulay injective, projective, and flat dimensions of M and N over A are

CMidA M inf f GidAÂj C M j C is a semi-dualizing module g; CMpd N inf f Gpd N j C is a semi-dualizing module g; A AÂj C

CMfdA N inf f GfdAÂj C N j C is a semi-dualizing module g:

REMARK 2.4. From the definition of the Cohen-Macaulay homological dimensions, the connection to Cohen-Macaulayness is less than obvious, but as described in the introduction, we shall prove in section 5 that the Cohen-Macaulay dimensions have the same relation to Cohen-Macaulay rings with dualizing modules as classical injective, projective, and flat dimension have to regular rings.

EXAMPLE 2.5. Let the ring A be local and artinian. Then it is easy directly to compute the Cohen-Macaulay dimensions of any module. Namely, for a local artinian ring, the injective hull Ek) of the residue class field is a dualizing module. By [7, thm. 5.6], the trivial extension

AÂj Ek) is Gorenstein, and it is clearly also local and artinian. But over a local artinian Gorenstein ring, each module is both Goren- stein injective, projective, and flat. So if M is an A-module and we view it as an AÂj Ek)-module, then Gid M 0; Gpd M 0; and Gfd M 0; AÂj Ek) AÂj Ek) AÂj Ek) and hence

CMidA M 0; CMpdA M 0; and CMfdA M 0:

REMARK 2.6. The result in example 2.5 is a special case of the theorems in the introduction, since the local artinian ring A is Cohen-Macaulay and has Krull dimension zero.

3. Lemmas on the trivial extension

This section studies the homological properties of the trivial extension

AÂj C. This is essential for subsequent developments since our Cohen-

Macaulay dimensions are defined in terms of AÂj C. 92 Henrik Holm - Peter Jùrgensen

Some of the material in this section is related to [7]. See in particular [7, prop. 4.35 and thm. 5.2].

LEMMA 3.1. Let C be an A-module and F a faithfully flat AÂj C)- module.

i) If I is an injective A-module, then HomAF; I), with the AÂj C)- structure coming from the first variable, is an injective AÂj C)-module. If

I is faithfully injective, then so is HomAF; I). ii) Each injective AÂj C)-module is a direct summand in a module HomAAÂj C; I), with the AÂj C)-structure coming from the first variable, where I is an injective A-module.

PROOF. i) This holds because adjunction gives

1 HomAÂj CÀ; HomAF; I)) ' HomAF AÂj C À; I) on AÂj C)-modules.

ii) Note first the handy special case of equation 1) with F AÂj C,

j 2 HomAÂj CÀ; HomAAÂ C; I)) ' HomAÀ; I):

To see that an injective AÂj C)-module J is a direct summand in a module of the form HomAAÂj C; I), it is enough to embed it into such a module. For this, first view J as an A-module and choose an embedding J,!I into an injective A-module I. Then use equation 2) to convert the morphism of A- modules J,!I to a morphism of AÂj C)-modules J ! HomAAÂj C; I). It is not hard to check that this last morphism is in fact injective. p

The following lemma is closely related to [8, sec. 3], although that paper was not phrased in terms of derived categories. The lemma and its proof use the right derived Hom functor, RHom, and the left derived tensor functor, L , which are defined on derived categories.

LEMMA 3.2. Let C be a semi-dualizing module for A. i) There is an isomorphism

AÂj C RHomAAÂj C; C) in the derived category DAÂj C), where the AÂj C)-structure of the RHom comes from the first variable. ii) There is a natural equivalence

j RHomAÂj CÀ; AÂ C) ' RHomAÀ; C) of functors from DA) to DA). Cohen-Macaulay Homological Dimensions 93

iii) If M is in DA) then the biduality morphisms

M ! RHomARHomAM; C); C);

j j M !RHomAÂj CRHomAÂj CM; AÂ C); AÂ C) are identified under the equivalence from part ii). iv) There is an isomorphism

j RHomAÂj CA; AÂ C) C in DAÂj C).

PROOF. i) Since C is semi-dualizing, there is a canonical isomorphism in DA),

3 A È C ! RHomAC È A; C):

Let us write out more carefully how this arises. Let C ! I be an injective resolution of C. Then the canonical map A ! HomC; I) is a quasi- isomorphism, and there is clearly a quasi-isomorphism C ! HomAA; I). Combining these maps gives a quasi-isomorphism

A È C ! HomAC; I) È HomAA; I); that is, a quasi-isomorphism

4 A È C ! HomAC È A; I): The right-hand side here is

HomAC È A; I) RHomAC È A; C); so the quasi-isomorphism 4) represents the isomorphism 3) in DA). But it is easy to check that in fact, the morphism 4) respects the action of AÂj C on both sides, so

AÂj C RHomAAÂj C; C) in DAÂj C). ii) This is a computation,

a) j j RHomAÂj CÀ; AÂ C) ' RHomAÂj CÀ; RHomAAÂ C; C))

b) L ' RHom AÂj C) À; C) A AÂj C

' RHomAÀ; C); where a) is by part i) and b) is by adjunction. 94 Henrik Holm - Peter Jùrgensen

iii) and iv) These are easy to obtain from ii). p

LEMMA 3.3. Let C be a semi-dualizing module for A and let I be an injective A-module.

i) A and C are Gorenstein projective over AÂj C.

ii) HomAA; I) I and HomAC; I) are Gorenstein injective over AÂj C.

PROOF. i) To see that A is Gorenstein projective over AÂj C, by [3, prop. 2.2.2)] we need to see that the homology of

j RHomAÂj CA; AÂ C) is concentrated in degree zero and that the biduality morphism

j j A ! RHomAÂj CRHomAÂj CA; AÂ C); AÂ C) is an isomorphism. The first of these conditions follows from lemma 3.2ii). The second condition holds because the biduality morphism can be identified with

A ! RHomARHomAA; C); C) by lemma 3.2iii), and this in turn can be identified with the canonical morphism

A ! RHomAC; C) which is an isomorphism.

To see that C is Gorenstein projective over AÂj C can be done by the same method.

ii) We will prove that HomAC; I) is Gorenstein injective over AÂj C, the case of HomAA; I) I being similar. Since C is Gorenstein projective over AÂj C, it has a complete projective resolution P; see [3, def. 4.2.1)]. In particular, P is exact and the zeroth cycle module Z0P)isisomorphictoC.SinceC is finitely generated, we can assume that P consists of finitely generated AÂj C)- modules by [3, thms. 4.1.4) and 4.2.6)].

Lemma 3.1i) shows that J HomAP; I) is a complex of injective AÂj C)- modules. It is clear that J is exact and that Z0J) HomAZ0P); I)

HomAC; I), so if we can prove that HomAÂj CK; J) is exact for each injective AÂj C)-module K, it will follow that J is a complete injective re- Cohen-Macaulay Homological Dimensions 95

solution of HomAC; I)overAÂj C whence HomAC; I) is Gorenstein injective over AÂj C; see [3, def. 6.1.1)]. But

HomAÂj CK; J) HomAÂj CK; HomAP; I)) HomAK AÂj C P; I);

so it is enough to see that K AÂj C P is exact. And by lemma 3.1ii), the module K is a direct summand in HomAAÂj C; L) for some injective A- j module L, and hence it is enough to see that HomAAÂ C; L) AÂj C P is exact. But P consists of finitely generated projective AÂj C)-modules which by [3, A.2.11)] implies

j j HomAAÂ C; L) AÂj C P HomAHomAÂj CP; AÂ C); L); and this is exact because L is an injective A-module while the complex

j HomAÂj CP; AÂ C) is exact because P is a complete projective resolution.

LEMMA 3.4. Let C be a semi-dualizing module for A and let I be an injective A-module. Then there is an equivalence of functors on DAÂj C),

j RHomAÂj CHomAAÂ C; I); À) ' RHomAHomAC; I); À); where the AÂj C)-structure of HomAAÂj C; I) comes from the first variable.

PROOF. First note that

HomAAÂj C; I) ' RHomAAÂj C; I)

a) ' RHomARHomAAÂj C; C); I)

b) L ' AÂj C) A RHomAC; I) where a) is by lemma 3.2i) and b) is by [3, A.4.24)]. Hence

j RHomAÂj CHomAAÂ C; I); À)

L ' RHom AÂj C) RHom C; I); À) AÂj C A A c) ' RHomARHomAC; I); À);

' RHomAHomAC; I); À) where c) is by adjunction. p 96 Henrik Holm - Peter Jùrgensen

4. Bounds on the injective dimension of C

This section studies the injective dimension of a semi-dualizing module C. If this dimension is finite, then C is a dualizing module, and this forces the ring A to be Cohen-Macaulay. The main result is proposition 4.5 which will be used in section 5 to show that finiteness of Cohen-Macaulay dimensions characterizes Cohen-Ma- caulay rings with a dualizing module.

LEMMA 4.1. Let C be a semi-dualizing module for A and let M be an A- module which is Gorenstein injective over AÂj C. Then there exists a short exact sequence of A-modules,

0 0 ! M ! HomAC; I) ! M ! 0; such that i) The A-module I is injective. 0 ii) The A-module M is Gorenstein injective over AÂj C. iii) For each injective A-module J, the sequence stays exact if one applies to it the functor HomAHomAC; J); À).

PROOF. Since M is Gorenstein injective over AÂj C, it has a complete injective resolution; see [3, def. 6.1.1)]. From this can be extracted a short exact sequence of AÂj C)-modules, 0 ! N ! K ! M ! 0; where K is injective and N Gorenstein injective over AÂj C, which stays

j exact if one applies to it the functor HomAÂj CL; À) for any injective AÂ C)- module L. In particular, the sequence stays exact if one applies to it the functor

j HomAÂj CHomAAÂ C; J); À) for any injective A-module J, because HomAAÂj C; J) is an injective AÂj C)-module by lemma 3.1i). By lemma 3.1ii), the injective AÂj C)-module K is a direct summand 0 in HomAAÂj C; I) for some injective A-module I.IfK È K 0 HomAAÂj C; I), then by adding K to both the first and the second module in the short exact sequence, we may assume that the sequence has the form

h 0 ! N ! HomAAÂj C; I) ! M ! 0:

The module N is still Gorenstein injective over AÂj C,andthesequence Cohen-Macaulay Homological Dimensions 97 still stays exact if one applies to it the functor

j HomAÂj CHomAAÂ C; J); À) for any injective A-module J. Now let us consider in detail the homomorphism h. Elements of the a g source HomAAÂj C; I) have the form a; g) where A ! I and C ! I are homomorphisms of A-modules. The AÂj C)-module structure of

HomAAÂj C; I) comes from the first variable, and one checks that it takes the form

a; c) Á a; g) aa xgc); ag); where xgc) is the homomorphism A ! I given by a7!agc). The target of h is M which is an A-module. When viewed as an AÂj C)- module, M is annihilated by the ideal 0Âj C,so

5 0 0; c) Á ha; g) h0; c) Á a; g)) hxgc); 0); where the last follows from the previous equation. In fact, this implies 6 ha; 0) 0 a for each A ! I. To see so, note that there is a surjection F ! HomAC; I) with F free, and hence a surjection C A F ! C A HomAC; I). The target here is isomorphic to I by [4, prop. 4.4) and obs. 4.10)], so there is a surjection C A F ! I.AsC A F is a direct sum of copies of C, this means that, given an element i in I, it is possible to find homomorphisms g1; ...; gt : C ! I and elements c1; ...; ct in C with i g1c1) ÁÁÁgtct). Hence the homomorphism A !a I given by a 7! ai satisfies a x x ÁÁÁx ; g1c1)ÁÁÁgtct) g1c1) gtct) and so equation 5) implies equation 6). To make use of this, observe that the canonical exact sequence of

AÂj C)-modules

0 ! C ! AÂj C ! A ! 0; 7 c 7À! 0; c); a; c) 7À! a induces an exact sequence

0 ! HomAA; I) ! HomAAÂj C; I) ! HomAC; I) ! 0 because I is injective. Equation 6) means that the composition of 98 Henrik Holm - Peter Jùrgensen

h HomAA; I) ! HomAAÂj C; I)andHomAAÂj C; I) ! M is zero, so h HomAAÂj C; I) ! M factors through the surjection HomAAÂj C; I) ! ! HomAC; I). This means that we can construct a commutative diagram of AÂj C)-modules with exact rows,

We will show that if we view the lower row as a sequence of A-modules, then it is a short exact sequence with the properties claimed in the lemma. i) The A-module I is injective by construction. ii) Applying the Snake Lemma to the above diagram embeds the vertical arrows into exact sequences. The leftmost of these gives the short exact sequence

0 0 ! HomAA; I) ! N ! M ! 0:

Here the modules HomAA; I) I and N are Gorenstein injective over 0 AÂj C by lemma 3.3ii), respectively, by construction. Hence M is also

Gorenstein injective over AÂj C because the class of Gorenstein injective modules is injectively resolving by [9, thm. 2.6]. iii) By construction, the upper sequence in the diagram stays exact if

j one applies to it the functor HomAÂj CHomAAÂ C; J); À) for any injective A-module J. It follows that the same holds for the lower row. But taking H0 of the isomorphism in lemma 3.4 shows

j HomAÂj CHomAAÂ C; J); À) ' HomAHomAC; J); À); so the lower row in the diagram also stays exact if one applies to it the functor HomAHomAC; J); À) for any injective A-module J. p

The special case C A of the following lemma was first proved by Frankild and Holm using the octahedral axiom; the present proof is simpler. In the lemma, CAA)andCBA) denote the Auslander and Bass classes of the semi-dualizing module C, as introduced in [4, def. 4.1)]. By S is denoted suspension of complexes in the derived category, and by id is denoted injective dimension of complexes as defined e.g. in [3, def. A.3.8)]. Cohen-Macaulay Homological Dimensions 99

LEMMA 4.2. Let C be a semi-dualizing module for A. Let M be a complex in CAA) which has non-zero homology and satisfies that

GidAÂj C M < 1. Write s supf i j HiM 6 0 g. Then there is a distinguished triangle in DA), SsH ! Y ! M !; where H is an A-module which is Gorenstein injective over AÂj C and where id C L Y)4Gid M: A A AÂj C

PROOF. By [4, prop. 4.8)] we know

L supf i j HiC A M) 6 0 gsupf i j HiM 6 0 gs;

L so we can pick an injective resolution of C A M of the form

J ÁÁÁ!0 ! Js ! JsÀ1 !ÁÁÁ: Then

L M RHomAC; C A M) HomAC; J) where the first is because M is in CAA). Now, HomAC; J) consists of A-modules which are Gorenstein injective over AÂj C by lemma 3.3ii), and if we write

n GidAÂj C M for the finite Gorenstein injective dimension of M over AÂj C, it follows from [3, thm. 6.2.4)] or [5, thm. 2.5)] that the soft truncation

ÁÁÁ!0 ! HomAC; Js) !ÁÁÁ!HomAC; JÀn1) ! G ! 0 !ÁÁÁ has G Gorenstein injective over AÂj C. The truncation remains quasi-isomorphic to M. By iterating lemma 4.1, the module G can be embedded into the complex

ÁÁÁ!0!HomAC; Is1)!HomAC; Is)!ÁÁÁ!HomAC; IÀn1)!G!0!ÁÁÁ; where the Ii are injective A-modules by 4.1i). This complex can only have non-zero homology in degree s 1; let us call the s 1)'st homology module H so the complex is quasi-isomorphic to Ss1H. Note that the A- module H is Gorenstein injective over AÂj C by 4.1ii). 100 Henrik Holm - Peter Jùrgensen

By 4.1iii), the identity on G can be lifted to a chain map

where we have omitted HomA everywhere for typographical reasons. Note that this is actually just a version of Auslander's pitchfork construction.) If we construct the mapping cone of the chain map 8), it is a standard observation that the null homotopic subcomplex ÁÁÁÀ!0 À! G G À! 0 À!ÁÁÁ splits off as a direct summand. The remaining complex Q consists of modules of the form HomAC; K) where K is an injective A-module, and it sits in homological degrees s 1; ...; Àn 1. In the derived category DA), we can replace a complex with any quasi- isomorphic complex. Hence 8) can be viewed as a morphism M ! Ss1H. Given that the mapping cone of 8) is Q up to a null-homotopic summand, this gives that there is a distinguished triangle M ! Ss1H ! Q ! in DA). Rotation gives a distinguished triangle SsH ! SÀ1Q ! M !

À1 in DA) where H is Gorenstein injective over AÂj C, and where S Q consists of modules of the form HomAC; K) where K is an injective A- module, and sits in homological degrees s; ...; Àn. We claim that with Y SÀ1Q, this is the lemma's triangle. It only remains to check id C L Y)4Gid M.ButY has the form A A AÂj C

Y ÁÁÁ!0 ! HomAC; Ks) !ÁÁÁ!HomAC; KÀn) ! 0 !ÁÁÁ where the Ki are injective A-modules, and [4, prop. 4.4) and obs. 4.10)] give that modules of the form HomAC; K) are acyclic for the functor C A À whence

L C A Y C A Y:

Moreover, [4, prop. 4.4) and obs. 4.10)] show C A HomAC; Ki) Ki for each i, so

C A Y ÁÁÁ!0 ! Ks !ÁÁÁ!KÀn ! 0 !ÁÁÁ: Cohen-Macaulay Homological Dimensions 101

The last two equations imply id C L Y)4n Gid M A A AÂj C as desired. p

LEMMA 4.3. Let C be a semi-dualizing module for A. Let M be a complex in CAA) with non-zero homology. Write s supf i j HiM 6 0 g and suppose that

s1 ExtA M; H) 0 for each A-module H which is Gorenstein injective over AÂj C. Then id C L M) Gid M: A A AÂj C

PROOF. To prove the lemma's equation, let us first prove the inequality

4. We may clearly suppose GidAÂj C M < 1. By lemma 4.2 there is a distinguished triangle in DA), 9 SsH ! Y ! M ! Ss1H; where H is an A-module which is Gorenstein injective over AÂj C, and where 10 id C L Y)4Gid M: A A AÂj C But

s1 s1 HomDA)M; S H) ExtA M; H) 0 by assumption, so the connecting morphism M ! Ss1H in 9) is zero, whence 9) is a split distinguished triangle with Y SsH È M. This implies

L L s L C A Y C A S H) È C A M) from which clearly follows

L L 11 idAC A M)4idAC A Y): Combining the inequalities 11) and 10) shows id C L M)4Gid M A A AÂj C as desired. L Let us next prove the inequality 5. Let t supf i j HiC A M) 6 0 g 102 Henrik Holm - Peter Jùrgensen

L and n idAC A M). We may clearly suppose n < 1 . Let

J ÁÁÁ!0 ! Jt !ÁÁÁ!ÁÁÁ!JÀn ! 0 !ÁÁÁ

L be an injective resolution of C A M. The complex M is in CAA) by assumption, so we get the first in

L M RHomAC; C A M) HomAC; J):

Lemma 3.3ii) implies that HomAC; J) is a complex of Gorenstein injective modules over AÂj C. Since HomAC; J)` HomAC; J`) 0 for `<Àn,we see Gid M4n id C L M) AÂj C A A as desired. p

The following lemma provides some complexes to which lemma 4.3 applies. By pd is denoted projective dimension of complexes as defined e.g. in [3, def. A.3.9)].

LEMMA 4.4. Let C be a semi-dualizing module for A. Let M be a complex of A-modules which has non-zero homology and satisfies that

HiM 0 for i 0 and that pdA M < 1. Write s supf i j HiM 6 0 g.IfH is an A-module which is Gorenstein injective over AÂj C, then

s1 ExtA M; H) 0:

PROOF. The conditions on M imply that it has a bounded projective resolution P, and clearly

s1 1 ExtA M; H) ExtAQ; H) when Q is the s'th cokernel of P. Since

ÁÁÁ!Ps1 ! Ps ! Q ! 0 is a projective resolution of Q and since P is bounded, we have pdAQ < 1. Hence it is enough to show

1 ExtAQ; H) 0 for each A-module Q with pdAQ < 1. The case Q 0 is clear, so we assume Q 6 0. To prove this, we first argue that if I is any injective A-module then

i 12 ExtAQ; HomAC; I)) 0 Cohen-Macaulay Homological Dimensions 103 for i > 0. For this, note that we have

RHomAQ; HomAC; I)) RHomAQ; RHomAC; I))

a) L RHomAQ A C; I) b) RHomAC; RHomAQ; I))

RHomAC; HomAQ; I)) where a) and b) are by adjunction, and consequently,

i i 13 ExtAQ; HomAC; I)) ExtAC; HomAQ; I)) for each i. The condition pdAQ < 1 implies idAHomAQ; I) < 1, and therefore HomAQ; I) belongs to CBA) by [4, prop. 4.4)]. Thus [4, obs. 4.10)] implies that the right hand side of 13) is zero for i > 0, proving equation 12).

Now set n pdAQ. Repeated use of lemma 4.1 shows that there is an exact sequence of A-modules

0 14 0 ! H ! HomAC; InÀ1) !ÁÁÁ!HomAC; I0) ! H ! 0; where I0; ...; InÀ1 are injective A-modules. Applying HomAQ; À) to 14) and using equation 12), we obtain

1 n1 0 ExtAQ; H) ExtA Q; H ) 0 as desired. Here the last equality holds because pdAQ n. p

The following proposition uses some homological invariants for complexes of modules. In particular, injective, projective and flat dimension of complexes are used. They are denoted id, pd, and fd, and the definition can be found e.g. in [3, defs. A.3.8), A.3.9), and A.3.10)]. Recall from [3, A.5.7.4)] that when A is local with residue class field k and C is a complex of A-modules with bounded finitely generated homology, then

idAC Àinff i j HiRHomAk; C) 6 0 g: Recall also from [3, A.6.3)] that the width of a complex of A-modules M is defined as

L widthA M inff i j HiM A k) 6 0 g: The following is the main result of this section. 104 Henrik Holm - Peter Jùrgensen

PROPOSITION 4.5. Assume that the ring A is local and let C be a semi- dualizing module for A. Let M be a complex of A-modules which has non- zero homology and satisfies HiM 0 for i 0 and fdA M<1. Then

idAC4GidAÂj C M widthA M:

PROOF. Denote by k the residue class field of A. Observe that HiM 0 L for i 0 implies HiM A k) 0 for i 0, whence widthA M > À1. Also, we can assume widthA M < 1, because the proposition is trivially true if widthA M 1. Since fdA M < 1, the isomorphism [3, A.4.23)] gives

L L RHomAk; C A M) RHomAk; C) A M: This implies a) in

L inff i j HiRHomAk; C A M) 6 0 g

a) L inff i j HiRHomAk; C) A M) 6 0 g

b) L inff i j HiRHomAk; C) 6 0 ginff i j HiM A k) 6 0 g

ÀidAC widthA M; where b) is by [3, A.7.9.2)]. Consequently,

L idAC Àinff i j HiRHomAk; C A M) 6 0 gwidthA M

L 4idAC A M) widthA M Ã ):

The condition fdA M < 1 implies pdA M < 1 by [11, Seconde partie, cor. 3.2.7)], and hence if we write s supf i j HiM 6 0 g, lemma 4.4 gives s1 ExtA M; H) 0 when H is an A-module which is Gorenstein injective over AÂj C.But fdA M < 1 also implies M 2 CAA) by [4, prop. 4.4)], and altogether, lemma 4.3 applies to M and gives

Ã ) GidAÂj C M widthA M as desired. p

We will also need the dual of proposition 4.5. First an easy lemma. Cohen-Macaulay Homological Dimensions 105

LEMMA 4.6. Let C be an A-module, let I be a faithfully injective A- module, and let M be a complex of A-modules with HiM 0 for i 0. Then

GidAÂj C HomA M; I) GfdAÂj C M:

PROOF. From lemma 3.1i) follows that E HomAAÂj C; I)isa faithfully injective AÂj C)-module. Hence

GidAÂj CHomAÂj C M; E) GfdAÂj C M follows from [3, thm. 6.4.2)]. But equation 2) in the proof of lemma 3.1 shows

HomAÂj C M; E) HomA M; I); so accordingly,

GidAÂj C HomAM; I) GfdAÂj C M: p

PROPOSITION 4.7. Assume that the ring A is local and let C be a semi-dualizing module for A.LetN be a complex of A-modules which has non-zero homology and satisfies HiN 0 for i 0 and idAN<1. Then

idAC4GfdAÂj C N depthAN:

PROOF. Apply Matlis duality and lemma 4.6 to proposition 4.5. p

5. Properties of the Cohen-Macaulay dimensions

This section contains our main results on the Cohen-Macaulay dimensions, as announced in the introduction. From now on, A is assumed to be local with residue class field k.

THEOREM 5.1. The following conditions are equivalent. CM) A is a Cohen-Macaulay ring with a dualizing module.

I1) CMidA M < 1 holds when M is any complex of A-modules with bounded homology. I2) There is a complex M of A-modules with bounded homology,

CMidA M < 1, fdA M < 1, and widthA M < 1. I3) CMidA k < 1. 106 Henrik Holm - Peter Jùrgensen

P1) CMpdA M < 1 holds when M is any complex of A-modules with bounded homology. P2) There is a complex M of A-modules with bounded homology,

CMpdA M < 1, idA M < 1, and depthA M < 1. P3) CMpdA k < 1. F1) CMfdA M < 1 holds when M is any complex of A-modules with bounded homology. F2) There is a complex M of A-modules with bounded homology,

CMfdA M < 1, idA M < 1, and depthA M < 1. F3) CMfdA k < 1.

PROOF. Let us first prove that conditions CM), I1), I2), and I3) are equivalent.

CM) ) I1) Let A be Cohen-Macaulay with dualizing module C. Then

AÂj C is Gorenstein by [7, thm. 5.6]. If M is a complex of A-modules with bounded homology, then M is also a complex of AÂj C)-modules with bounded homology, so

GidAÂj C M < 1 by [3, thm. 6.2.7)]. As C is in particular a semi-dualizing module, the definition of CMid then implies

CMidA M < 1: I1) ) I2) and I1) ) I3) Trivial.

I2) ) CM) When CMidA M < 1 then the definition of CMid implies that A has a semi-dualizing module C with

GidAÂj C M < 1: When width M < 1 then M has non-zero homology. And finally, when fd M < 1 also holds, then proposition 4.5 implies

idA C < 1: So C is a dualizing module for A, and hence A is Cohen-Macaulay by [3, A.8.5.3)] and has a dualizing module. Cohen-Macaulay Homological Dimensions 107

I3) ) CM) When CMidAk < 1, then A has a semi-dualizing module C with

GidAÂj C k < 1:

j Denoting by EAÂj Ck) the injective hull of k over AÂ C, it follows from [9, thm. 2.22] that

RHomAÂj CEAÂj Ck); k) has bounded homology. Denoting Matlis duality over AÂj C by _, we have

_ _ RHomAÂj CEAÂj Ck); k) RHomAÂj Ck ; EAÂj Ck) )

dj RHomAÂj Ck; AÂ C)

a) j dj RHomAÂj Ck; AÂ C) AÂj C AÂ C; d where a) is by [3, A.4.23)]. Since the completion AÂj C is faithfully flat over

j j AÂ C, it follows that also RHomAÂj Ck; AÂ C) has bounded homology, whence AÂj C is a Gorenstein ring. But then C is a dualizing module for A by [7, thm. 5.6] and so again, A is Cohen-Macaulay with a dualizing module. Similar proofs give that also CM), P1), P2), and P3) as well as CM), F1), F2), and F3) are equivalent. For this, proposition 4.7 should be used instead of proposition 4.5. p

REMARK 5.2. In condition I2) of theorem 5.1, one could consider for M either the ring A itself, or the Koszul complex Kx1; ...; xr) on any sequence of elements x1; ...; xr in the maximal ideal. These complexes satisfy fdA M < 1 and widthA M < 1, and so either of the conditions

CMidA A < 1 and CMidA Kx1; ...; xr) < 1 is equivalent to A being a Cohen-Macaulay ring with a dualizing module. Similarly, in conditions P2) and F2), one could consider for M either the injective hull of the residue class field, EAk), or a dualizing complex D if one is known to exist). These complexes satisfy idA M < 1 and depthA M < 1, and so either of the conditions

CMpdA EAk) < 1 and CMpdA D < 1 and

CMfdA EAk) < 1 and CMfdA D < 1 is equivalent to A being a Cohen-Macaulay ring with a dualizing module. 108 Henrik Holm - Peter Jùrgensen

REMARK 5.3. Our viewpoint of emphasizing the Cohen-Macaulay homological dimensions makes us think of theorem 5.1 as a main result. However, proposition 4.5 actually implies a more precise result, namely, if there is a complex M with non-zero homology, HiM 0 for i 0, and

GidAÂj C M < 1; fdA M < 1; and widthA M < 1; then A is Cohen-Macaulay with dualizing module C. Similarly, proposition 4.7 implies that if there is a complex N with non- zero homology and HiN 0 for i 0, then either of Gpd N < 1; id N < 1; and depth N < 1 AÂj C A A and

GfdAÂj C N < 1; idAN < 1; and depthAN < 1 implies that A is Cohen-Macaulay with dualizing module C.

THEOREM 5.4. Let M be an A-module. Then

CMidA M < 1,CMidA M dim A;

CMpdA M < 1,CMpdA M dim A;

CMfdA M < 1,CMfdA M dim A:

PROOF. The implications are trivial. To see the implication ) for CMpd, observe that when M is given there exists a semi-dualizing module C with CMpd M Gpd M: A AÂj C When this is finite, we have

Gpd M FPDAÂj C) AÂj C by [9, thm. 2.28], where FPD denotes the big) finitistic projective dimension. But

FPDAÂj C) dim AÂj C by [11, Seconde partie, thm. 3.2.6)], and A and AÂj C are finitely generated as modules over each other, so

dim AÂj C dim A: Together, this establishes the implication ) for CMpd. The implication ) for CMfd follows by a similar argument, using Cohen-Macaulay Homological Dimensions 109

that the finitistic flat dimension satisfies FFDAÂj C) FPDAÂj C) because of [11, Seconde partie, cor. 3.2.7)]. Finally, the implication ) for CMid follows from the one for CMfd using Matlis duality and lemma 4.6. p

The following results use CMdim, the Cohen-Macaulay dimension introduced by Gerko in [8], and Gdim, the G-dimension originally introduced by Auslander and Bridger in [1].

LEMMA 5.5. Let C be a semi-dualizing module for A and let M be a finitely generated A-module. If Gpd M < 1 AÂj C then CMdim M Gpd M: A AÂj C

PROOF. Combining [8, proof of thm. 3.7] with [8, def. 3.2'] shows CMdim M4Gpd M: A AÂj C So Gpd M < 1 implies CMdim M < 1 and hence AÂj C A

CMdimA M depthAA À depthA M by [8, thm. 3.8]. On the other hand, Gpd M G-dim M AÂj C C by [10, prop. 3.1], where G-dimC M is the homological dimension introduced in [4, def. 3.11)]. So G-dimC M is finite and hence

G-dimC M depthAA À depthA M by [4, thm. 3.14)]. Combining the last three equations shows CMdim M Gpd M A AÂj C as desired. p

THEOREM 5.6. Let M be a finitely generated A-module. Then

CMdimA M4CMpdA M4GdimA M; and if one of these numbers is finite then the inequalities to its left are equalities. 110 Henrik Holm - Peter Jùrgensen

PROOF. The first inequality is clear from lemma 5.5, since CMpdA M is defined as the infimum of all Gpd M. AÂj C For the second inequality, note that the ring A is itself a semi-dualizing module, so the definition of CMpd gives 4 in CMpd M4Gpd M Gpd M Gdim M; A AÂj A A A where the first is by [10, cor. 2.17], and the second holds by [3, cor. 4.4.6)] or [5, prop. 2.11)b)] because M is finitely generated.

Equalities: If GdimA M < 1 then CMdimA M < 1 by [8, thm. 3.7]. But GdimA M < 1 implies

GdimA M depthA A À depthA M by [3, thm. 2.3.13)], and similarly, CMdimA M < 1 implies

CMdimA M depthAA À depthA M by [8, thm. 3.8]. So it follows that CMdimA M GdimA M, and hence both inequalities in the theorem must be equalities.

If CMpdA M < 1 then by the definition of CMpd there exists a semi- dualizing module C over A with Gpd M < 1. But by lemma 5.5, any AÂj C such C has CMdim M Gpd M; A AÂj C so the first inequality in the theorem is an equality. p

We finish with two clasically flavoured results.

PROPOSITION 5.7 [Auslander-Buchsbaum formula]. Let M be a finitely generated A-module. If CMpdA M<1, then

CMpdA M depthAA À depthA M:

PROOF. By the definition of CMpd, we can pick a semi-dualizing A- module C such that CMpd M Gpd M: A AÂj C But M is finitely generated, so Gpd M Gdim M AÂj C AÂj C by [3, cor. 4.4.6)] or [5, prop. 2.11)b)]. The Auslander-Bridger formula gives

Gdim M depth AÂj C À depth M: AÂj C AÂj C AÂj C Cohen-Macaulay Homological Dimensions 111

Finally, it remains to note that

depth AÂj C depth A AÂj C A and depth M depth M AÂj C A because AÂj C and A are finitely generated over each other. p

PROPOSITION 5.8 [Bass formula]. Assume that A has a dualizing complex and let N 6 0 be a finitely generated A-module. If

CMidAN<1, then

CMidA N depthA:

PROOF. By the definition of CMid, we can pick a semi-dualizing A- module C such that CMidA N GidAÂj C N. By [5, thm. 6.4)], finiteness of

GidAÂj C N implies that

Gid N depth AÂj C; AÂj C AÂj C since N 6 0 is finitely generated over AÂj C, and since AÂj C is finitely generated over A and so has a dualizing complex because A does. Finally, we have

depth AÂj C depth A AÂj C A again. p

Acknowledgments. We wish to thank the referee for a number of ex- cellent suggestions.

REFERENCES

[1] M. AUSLANDER -M.BRIDGER, Stable module theory, Mem. Amer. Math. Soc., Vol. 94, American Mathematical Society, Providence, R.I., 1969. [2] L. L. AVRAMOV -V.N.GASHAROV -I.V.PEEVA, Complete intersection dimension, Inst. Hautes EÂ tudes Sci. Publ. Math. 86 1997), pp. 67-114. [3] L. W. CHRISTENSEN, Gorenstein dimensions, Lecture Notes in Math., Vol. 1747, Springer, Berlin, 2000. [4] L. W. CHRISTENSEN, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 2001), pp. 1839-1883. [5] L. W. CHRISTENSEN -A.FRANKILD -H.HOLM, On Gorenstein projective, injective and flat dimensions - a functorial description with applications,J. Algebra 302 2006), pp. 231-279. 112 Henrik Holm - Peter Jùrgensen

[6] D. FERRAND -M.RAYNAUD, Fibres formelles d'un anneau local noetheÂrien, Ann. Sci. EÂ cole Norm. Sup. 4) 3 1970), pp. 295-311. [7] R. M. FOSSUM -P.A.GRIFFITH -I.REITEN, Trivial extensions of abelian categories. Homological algebra of trivial extensions of abelian categories with applications to ring theory, Lecture Notes in Math., Vol. 456, Springer, Berlin, 1975. [8] A. A. GERKO, On homological dimensions, Sb. Math. 192 2001), pp. 1165-1179. [9] H. HOLM, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 2004), pp. 167-193. [10] H. HOLM,P.JéRGENSEN, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 2006), pp. 423-445. [11] M. RAYNAUD -L.GRUSON, CriteÁres de platitude et de projectiviteÂ. Techniques de ``platification'' d'un module, Invent. Math. 13 1971), pp. 1-89.

Manoscritto pervenuto in redazione il 30 settembre 2005