Cohen-Macaulay homological dimensions

Cohen-Macaulay Homological Dimensions.

HENRIKHOLM(*) - PETERJéRGENSEN(**)

ABSTRACT - We define three new homological dimensions: Cohen-Macaulay in- jective, projective, and flat dimension, and show that they inhabit a theory si- milar 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, thenAis 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: holm@imf.au.dk

(**) Indirizzo dell'A.: School of Mathematics and Statistics, Merz-Court New- castle University, Newcastle-Upon-Tyne NE1 7RU.

E-mail: popjoerg@maths.leeds.ac.uk

2000Mathematics Subject Classification. 13D05, 13D25.

However, recent years have seen much work on the Gorenstein theory which now contains both Gorenstein injective, projective, and flat dimen- sion. These dimensions inhabit a nice theory similar to classical homo- logical 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 in- habit a theory with a number of desirable features. The main result is the following, whereby finiteness of the Cohen-Macaulay dimensions char- acterizes 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) CMid_AM<1for each A-module M.

(iii) CMpd_AM<1for each A-module M.

(iv) CMfdAM_ <1for each A-module M.

Another main result is a bound for the Cohen-Macaulay dimensions in terms of the Krull dimension dimAof the ring.

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

CMid_AM<1 , CMid_AMdimA;

CMpd_AM<1 , CMpd_AMdimA;

CMfd_AM<1 , CMfd_AMdimA:

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. IfAis a ring with a semi-dualizing moduleC(as defined in [4]), then we can consider the trivial extension ringA^j C, and ifMis a complex ofA- modules, then we can considerMas a complex of (A^j C)-modules and take

Gorenstein homological dimensions of Mover A^j C. The infima of these over all semi-dualizing modulesCdefine 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 di- mensions 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 A^j C when C is a semi-dualizing module for A, and giving some bounds on the injective dimension ofC. And finally, section 5 proves all the main results as described.

Let us end the introduction with a few blanket items. Throughout,Ais a commutative noetherian ring, and complexes ofA-modules have the form

!M_i‡1!M_i!M_{i 1}! :

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.

DEFINITION2.1. LetCbe anA-module. The direct sumACcan be equipped with the product

(a;c)(a1;c1)ˆ(aa1;ac1‡ca1):

This turnsACinto a ring which is called the trivial extension ofAbyC and denotedA^j C.

There are ring homomorphisms

A ! A^j C ! A;

a 7 ! (a;0);

(a;c) 7 ! a

whose composition is the identity onA. These homomorphisms allow us to view anyA-module as an (A^j C)-module and any (A^j C)-module as anA- 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 overA^jC,

Gid_AC^j M; Gpd_ACj M; and Gfd_AC^j M;

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 complexNwith H_iNˆ0 fori0 has GpdNnif it is quasi-isomorphic to a complexGwhich has the form

!0!G_n!G_{n 1}!

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 Hom(P;Q) is also exact for each projective moduleQ.

Dualizing this gives the definition of Gorenstein injective dimension.

Finally, Gorenstein flat dimension is defined as follows. A complexNwith HiNˆ0 fori0 has GfdNnif it is quasi-isomorphic to a complexH which has the form

!0!H_n!H_{n 1}!

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.

DEFINITION2.2. A semi-dualizing moduleCforAis a finitely generated module for which the canonical mapA!Hom_A(C;C) is an isomorphism, while Extⁱ_A(C;C)ˆ0 for eachi51.

A semi-dualizing module is called dualizing if it has finite injective di- mension.

Note that equivalently, a finitely generated moduleCis semi-dualizing if the canonical morphism A!RHom_A(C;C) is an isomorphism in the derived category D(A). An example of a semi-dualizing module isAitself, soAalways has at least one semi-dualizing module. On the other hand, ifA has a dualizing module, then it is necessarily Cohen-Macaulay, as follows for instance by [3, (A.8.5.3)].

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

DEFINITION 2.3. LetM andN be complexes of A-modules such that H_iMˆ0 fori0 and H_iNˆ0 fori0.

The Cohen-Macaulay injective, projective, and flat dimensions ofMand N overAare

CMid_AMˆinffGid_ACj MjC is a semi-dualizing moduleg;

CMpd_ANˆinffGpd_ACj NjC is a semi-dualizing moduleg;

CMfdANˆinffGfdAC^j NjC is a semi-dualizing moduleg:

REMARK2.4. 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 di- mension 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 hullE(k) of the residue class field is a dualizing module. By [7, thm. 5.6], the trivial extension A^j E(k) 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 ifMis anA-module and we view it as anA^j E(k)-module, then

Gid_AE(k)j Mˆ0; Gpd_AE(k)j Mˆ0; and Gfd_AjE(k)Mˆ0;

and hence

CMidAMˆ0; CMpd_AMˆ0; and CMfdAMˆ0:

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

3. Lemmas on the trivial extension

This section studies the homological properties of the trivial extension A^j C. This is essential for subsequent developments since our Cohen- Macaulay dimensions are defined in terms ofA^j C.

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 (A^j C)- module.

(i) If I is an injective A-module, thenHom_A(F;I), with the(A^j C)- structure coming from the first variable, is an injective(A^jC)-module. If I is faithfully injective, then so isHom_A(F;I).

(ii) Each injective(A^j C)-module is a direct summand in a mod- ule HomA(A^j C;I), with the (A^j C)-structure coming from the first variable, where I is an injective A-module.

PROOF. (i) This holds because adjunction gives

Hom_ACj ( ;Hom_A(F;I))'Hom_A(F_ACj ;I) …1†

on (A^j C)-modules.

(ii) Note first the handy special case of equation (1) withFˆA^j C, Hom_ACj ( ;Hom_A(A^jC;I))'Hom_A( ;I):

…2†

To see that an injective (A^j C)-moduleJis a direct summand in a module of the form HomA(A^j C;I), it is enough to embed it into such a module. For this, first viewJas anA-module and choose an embeddingJ,!Iinto an in- jectiveA-moduleI. Then use equation (2) to convert the morphism ofA- modulesJ,!Ito a morphism of (A^j C)-modulesJ!Hom_A(A^jC;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.

LEMMA3.2. Let C be a semi-dualizing module for A.

(i) There is an isomorphism

A^j CRHom_A(A^j C;C)

in the derived categoryD(A^j C), where the(A^j C)-structure of theRHom comes from the first variable.

(ii) There is a natural equivalence

RHom_ACj ( ;A^j C)'RHom_A( ;C) of functors fromD(A)toD(A).

(iii) If M is inD(A)then the biduality morphisms M!RHom_A(RHom_A(M;C);C);

M!RHom_AC^j (RHom_A^j_C(M;A^j C);A^j C) are identified under the equivalence from part(ii).

(iv) There is an isomorphism

RHom_AC^j (A;A^j C)C inD(A^j C).

PROOF. (i) SinceCis semi-dualizing, there is a canonical isomorphism in D(A),

AC!RHomA(CA;C):

…3†

Let us write out more carefully how this arises. Let C!Ibe an in- jective resolution ofC. Then the canonical mapA!Hom(C;I) is a quasi- isomorphism, and there is clearly a quasi-isomorphismC!Hom_A(A;I).

Combining these maps gives a quasi-isomorphism AC!Hom_A(C;I)Hom_A(A;I);

that is, a quasi-isomorphism

AC!HomA(CA;I):

…4†

The right-hand side here is

Hom_A(CA;I)RHom_A(CA;C);

so the quasi-isomorphism (4) represents the isomorphism (3) in D(A).

But it is easy to check that in fact, the morphism (4) respects the action ofA^jCon both sides, so

A^j CRHom_A(A^j C;C) in D(A^j C).

(ii) This is a computation,

RHom_A^j_C( ;A^j C)'^(a)RHom_AC^j ( ;RHom_A(A^j C;C))

(b)'

RHom_A((A^jC)^L_ACj ;C) 'RHomA( ;C);

where (a) is by part (i) and (b) is by adjunction.

(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 A^j C.

(ii) Hom_A(A;I)I and Hom_A(C;I)are Gorenstein injective over A^j C.

PROOF. (i) To see thatA is Gorenstein projective overA^j C, by [3, prop. (2.2.2)] we need to see that the homology of

RHom_AC^j (A;A^j C)

is concentrated in degree zero and that the biduality morphism A!RHom_AjC(RHom_ACj (A;A^j C);A^j C)

is an isomorphism. The first of these conditions follows from lemma 3.2(ii).

The second condition holds because the biduality morphism can be identi- fied with

A!RHom_A(RHom_A(A;C);C)

by lemma 3.2(iii), and this in turn can be identified with the canonical morphism

A!RHomA(C;C) which is an isomorphism.

To see thatCis Gorenstein projective overA^j Ccan be done by the same method.

(ii) We will prove that Hom_A(C;I) is Gorenstein injective overA^j C, the case of Hom_A(A;I)Ibeing similar.

Since C is Gorenstein projective over A^j C, it has a complete pro- jective resolutionP; see [3, def. (4.2.1)]. In particular,Pis exact and the zeroth cycle module Z₀(P) is isomorphic to C. Since C is finitely gen- erated, we can assume that P consists of finitely generated (A^j C)- modules by [3, thms. (4.1.4) and (4.2.6)].

Lemma 3.1(i) shows thatJˆHomA(P;I) is a complex of injective (A^j C)- modules. It is clear that J is exact and that Z0(J)HomA(Z0(P);I)

Hom_A(C;I), so if we can prove that Hom_AC^j (K;J) is exact for each injective (A^j C)-moduleK, it will follow thatJ is a complete injective re-

solution of HomA(C;I) overA^j Cwhence HomA(C;I) is Gorenstein injective overA^j C; see [3, def. (6.1.1)].

But

Hom_AjC(K;J)ˆHom_ACj (K;Hom_A(P;I))Hom_A(K_ACj P;I);

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

Hom_A(A^j C;L)_A^j_CPHom_A(Hom_AC^j (P;A^j C);L);

and this is exact because L is an injective A-module while the complex Hom_ACj (P;A^j C) is exact becausePis 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 onD(A^j C),

RHom_AC^j (Hom_A(A^j C;I); )'RHom_A(Hom_A(C;I); );

where the (A^j C)-structure of Hom_A(A^j C;I)comes from the first vari- able.

PROOF. First note that

Hom_A(A^j C;I)'RHom_A(A^j C;I) '(a)

RHom_A(RHom_A(A^j C;C);I) '(b)

(A^j C)^L_ARHom_A(C;I) where (a) is by lemma 3.2(i) and (b) is by [3, (A.4.24)]. Hence

RHom_AjC(Hom_A(A^j C;I); )

'RHom_ACj ((A^j C)^L_ARHom_A(C;I); ) '(c)RHom_A(RHom_A(C;I); );

'RHom_A(Hom_A(C;I); )

where (c) is by adjunction. p

4. Bounds on the injective dimension ofC

This section studies the injective dimension of a semi-dualizing module C. If this dimension is finite, thenCis a dualizing module, and this forces the ringAto 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.

LEMMA4.1. Let C be a semi-dualizing module for A and let M be an A- module which is Gorenstein injective over A^j C. Then there exists a short exact sequence of A-modules,

0!M⁰!Hom_A(C;I)!M!0;

such that

(i) The A-module I is injective.

(ii) The A-module M⁰is Gorenstein injective over A^j C.

(iii) For each injective A-module J, the sequence stays exact if one applies to it the functorHomA(HomA(C;J); ).

PROOF. SinceMis Gorenstein injective overA^j C, it has a complete injective resolution; see [3, def. (6.1.1)]. From this can be extracted a short exact sequence of (A^j C)-modules,

0!N!K!M!0;

whereK is injective andN Gorenstein injective overA^j C, which stays exact if one applies to it the functor Hom_ACj (L; ) for any injective (A^j C)- moduleL.

In particular, the sequence stays exact if one applies to it the functor HomAC^j (HomA(A^j C;J); ) for any injective A-module J, because Hom_A(A^j C;J) is an injective (A^j C)-module by lemma 3.1(i).

By lemma 3.1(ii), the injective (A^j C)-moduleKis a direct summand in Hom_A(A^j C;I) for some injective A-module I. If KK⁰

Hom_A(A^jC;I), then by adding K⁰ to both the first and the second module in the short exact sequence, we may assume that the sequence has the form

0!N!Hom_A(A^j C;I)!^h M!0:

The module Nis still Gorenstein injective over A^j C, and the sequence

still stays exact if one applies to it the functor Hom_AC^j (Hom_A(A^j C;J); ) for any injective A-moduleJ.

Now let us consider in detail the homomorphism h. Elements of the source HomA(A^j C;I) have the form (a;g) whereA!^a I and C!^g I are homomorphisms of A-modules. The (A^j C)-module structure of Hom_A(A^j C;I) comes from the first variable, and one checks that it takes the form

(a;c)(a;g)ˆ(aa‡x_g(c);ag);

wherex_g(c)is the homomorphismA!Igiven bya7!ag(c).

The target ofhisMwhich is anA-module. When viewed as an (A^j C)- module,Mis annihilated by the ideal 0^j C, so

0ˆ(0;c)h(a;g)ˆh((0;c)(a;g))ˆh(x_g(c);0);

…5†

where the lastˆfollows from the previous equation.

In fact, this implies

h(a;0)ˆ0 …6†

for eachA!^a I. To see so, note that there is a surjectionF!Hom_A(C;I) withF free, and hence a surjectionCAF!CAHomA(C;I). The tar- get here is isomorphic toIby [4, prop. (4.4) and obs. (4.10)], so there is a surjectionC_AF!I. AsC_AFis a direct sum of copies ofC, this means that, given an element i in I, it is possible to find homomorphisms g₁;. . .;g_t :C!Iand elements c₁;. . .;c_t inC withiˆg₁(c₁)‡ ‡g_t(c_t).

Hence the homomorphismA!^a Igiven bya7!aisatisfies aˆx_{g1(c1)‡‡gt(ct)} ˆx_g1(c1)‡ ‡x_gt(ct); and so equation (5) implies equation (6).

To make use of this, observe that the canonical exact sequence of (A^j C)-modules

0 ! C ! A^j C ! A ! 0;

c 7 ! (0;c);

(a;c) 7 ! a …7†

induces an exact sequence

0!HomA(A;I)!HomA(A^j C;I)!HomA(C;I)!0

because I is injective. Equation (6) means that the composition of

HomA(A;I)!HomA(A^j C;I) and HomA(A^j C;I)!^h M is zero, so HomA(A^j C;I)!^h M factors through the surjection HomA(A^j C;I)!

!Hom_A(C;I). This means that we can construct a commutative dia- gram of (A^j C)-modules with exact rows,

We will show that if we view the lower row as a sequence ofA-modules, then it is a short exact sequence with the properties claimed in the lemma.

(i) TheA-moduleIis 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!HomA(A;I)!N!M⁰!0:

Here the modules Hom_A(A;I)I and N are Gorenstein injective over A^j C by lemma 3.3(ii), respectively, by construction. Hence M⁰ is also Gorenstein injective overA^j Cbecause 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 one applies to it the functor Hom_AjC(Hom_A(A^j C;J); ) for any injective A-moduleJ. It follows that the same holds for the lower row. But taking H0

of the isomorphism in lemma 3.4 shows

Hom_A^j_C(Hom_A(A^j C;J); )'Hom_A(Hom_A(C;J); );

so the lower row in the diagram also stays exact if one applies to it the functor Hom_A(Hom_A(C;J); ) for any injectiveA-moduleJ. p The special case CˆA of the following lemma was first proved by Frankild and Holm using the octahedral axiom; the present proof is simpler. In the lemma,_CA(A) and_CB(A) denote the Auslander and Bass classes of the semi-dualizing module C, as introduced in [4, def. (4.1)].

BySis 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)].

LEMMA 4.2. Let C be a semi-dualizing module for A. Let M be a complex in _CA(A) which has non-zero homology and satisfies that Gid_AC^j M<1. Write sˆsupfijH_iM6ˆ0g. Then there is a distinguished triangle inD(A),

S^sH!Y!M!;

where H is an A-module which is Gorenstein injective over A^j C and where

idA(C^L_AY)4Gid_AC^j M:

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

supfijHi(C^L_AM)6ˆ0g ˆsupfijHiM6ˆ0g ˆs;

so we can pick an injective resolution ofC^L_AMof the form Jˆ !0!Js!Js 1! : Then

MRHom_A(C;C^L_AM)Hom_A(C;J) where the firstis becauseMis in_CA(A).

Now, Hom_A(C;J) consists ofA-modules which are Gorenstein injective overA^j Cby lemma 3.3(ii), and if we write

nˆGid_AC^j M

for the finite Gorenstein injective dimension ofMoverA^j C, it follows from [3, thm. (6.2.4)] or [5, thm. (2.5)] that the soft truncation

!0!Hom_A(C;Js)! !Hom_A(C;J _n‡1)!G!0! hasGGorenstein injective overA^jC. The truncation remains quasi-iso- morphic toM.

By iterating lemma 4.1, the moduleGcan be embedded into the com- plex

!0!HomA(C;Is‡1)!HomA(C;Is)! !HomA(C;I n‡1)!G!0! ; where theI_iare injectiveA-modules by 4.1(i). This complex can only have non-zero homology in degree s‡1; let us call the (s‡1)'st homology moduleHso the complex is quasi-isomorphic toS^s‡1H. Note that theA- moduleHis Gorenstein injective overA^j Cby 4.1(ii).

By 4.1(iii), the identity onGcan be lifted to a chain map

where we have omitted Hom_A everywhere for typographical reasons.

(Note that this is actually just a version of Auslander's pitchfork con- struction.)

If we construct the mapping cone of the chain map (8), it is a standard observation that the null homotopic subcomplex

!0 !G G !0 !

splits off as a direct summand. The remaining complexQconsists of mod- ules of the form HomA(C;K) whereKis an injectiveA-module, and it sits in homological degreess‡1;. . .; n‡1.

In the derived category D(A), we can replace a complex with any quasi- isomorphic complex. Hence (8) can be viewed as a morphismM!S^s‡1H.

Given that the mapping cone of (8) isQup to a null-homotopic summand, this gives that there is a distinguished triangle

M!S^s‡1H!Q! in D(A). Rotation gives a distinguished triangle

S^sH!S ¹Q!M!

in D(A) where H is Gorenstein injective over A^j C, and where S ¹Q consists of modules of the form HomA(C;K) where K is an injective A- module, and sits in homological degreess;. . .; n.

We claim that withY ˆS ¹Q, this is the lemma's triangle. It only re- mains to check id_A(C^L_AY)4Gid_ACj M. B utYhas the form

Y ˆ !0!Hom_A(C;K_s)! !Hom_A(C;K _n)!0! where theK_iare injectiveA-modules, and [4, prop. (4.4) and obs. (4.10)]

give that modules of the form Hom_A(C;K) are acyclic for the functor C_A whence

C^L_AYC_AY:

Moreover, [4, prop. (4.4) and obs. (4.10)] showCAHomA(C;Ki)Kifor each i, so

C_AY  !0!K_s! !K _n!0! : (8)

The last two equations imply

id_A(C^L_AY)4nˆGid_ACj M

as desired. p

LEMMA4.3. Let C be a semi-dualizing module for A. Let M be a com- plex in CA(A)with non-zero homology. Write sˆsupfijHiM6ˆ0gand suppose that

Ext^s‡1_A (M;H)ˆ0

for each A-module H which is Gorenstein injective over A^jC. Then id_A(C^L_AM)ˆGid_ACj M:

PROOF. To prove the lemma's equation, let us first prove the inequality 4. We may clearly suppose GidAC^j M<1. By lemma 4.2 there is a dis- tinguished triangle in D(A),

S^sH!Y!M!S^s‡1H;

…9†

where H is an A-module which is Gorenstein injective over A^j C, and where

id_A(C^L_AY)4Gid_ACj M:

…10†

But

Hom_D(A)(M;S^s‡1H)Ext^s‡1_A (M;H)ˆ0

by assumption, so the connecting morphism M!S^s‡1H in (9) is zero, whence (9) is a split distinguished triangle withY S^sHM.

This implies

C^L_AY (C^L_AS^sH)(C^L_AM) from which clearly follows

idA(C^L_AM)4idA(C^L_AY):

…11†

Combining the inequalities (11) and (10) shows id_A(C^L_AM)4Gid_ACj M as desired.

Let us next prove the inequality5. LettˆsupfijH_i(C^L_AM)6ˆ0g

andnˆidA(C^L_AM). We may clearly supposen<1. Let Jˆ !0!Jt ! ! !J n!0!

be an injective resolution ofC^L_AM. The complexM is in_CA(A) by as- sumption, so we get the firstin

MRHom_A(C;C^L_AM)Hom_A(C;J):

Lemma 3.3(ii) implies that HomA(C;J) is a complex of Gorenstein injective modules overA^j C. Since Hom_A(C;J)<sub>`</sub>ˆHom<sub>A</sub>(C;J`)ˆ0 for` < n, we see

Gid_AC^j M4nˆid_A(C^L_AM)

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)].

LEMMA4.4. Let C be a semi-dualizing module for A. Let M be a com- plex of A-modules which has non-zero homology and satisfies that HiMˆ0for i0and thatpd_AM<1. Write sˆsupfijHiM6ˆ0g. If H is an A-module which is Gorenstein injective over A^jC, then

Ext^s‡1_A (M;H)ˆ0:

PROOF. The conditions onM imply that it has a bounded projective resolutionP, and clearly

Ext^s‡1_A (M;H)  Ext¹_A(Q;H) whenQis thes'th cokernel ofP. Since

!Ps‡1!Ps !Q!0

is a projective resolution ofQand sincePis bounded, we have pd_AQ<1.

Hence it is enough to show

Ext¹_A(Q;H)ˆ0

for eachA-moduleQwith pd_AQ<1. The caseQˆ0 is clear, so we assume Q6ˆ0.

To prove this, we first argue that ifIis any injectiveA-module then Extⁱ_A(Q;Hom_A(C;I))ˆ0

…12†

fori>0. For this, note that we have

RHom_A(Q;Hom_A(C;I))RHom_A(Q;RHom_A(C;I))

(a)

RHom_A(Q^L_AC;I)

(b)

RHom_A(C;RHom_A(Q;I))

RHomA(C;HomA(Q;I)) where (a) and (b) are by adjunction, and consequently,

Extⁱ_A(Q;HomA(C;I))Extⁱ_A(C;HomA(Q;I)) …13†

for each i. The condition pd_AQ<1 implies idAHomA(Q;I)<1, and therefore HomA(Q;I) belongs to _CB(A) by [4, prop. (4.4)]. Thus [4, obs. (4.10)] implies that the right hand side of (13) is zero fori>0, proving equation (12).

Now setnˆpd_AQ. Repeated use of lemma 4.1 shows that there is an exact sequence ofA-modules

0!H⁰!HomA(C;In 1)! !HomA(C;I0)!H!0;

…14†

where I₀;. . .;I_{n 1} are injectiveA-modules. Applying Hom_A(Q; ) to (14) and using equation (12), we obtain

Ext¹_A(Q;H)  Ext^n‡1_A (Q;H⁰)ˆ0

as desired. Here the last equality holds because pd_AQˆn. p The following proposition uses some homological invariants for com- plexes 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 whenAis local with residue class fieldk andCis a complex ofA-modules with bounded finitely generated homol- ogy, then

id_ACˆ inffijH_iRHom_A(k;C)6ˆ0g:

Recall also from [3, (A.6.3)] that the width of a complex ofA-modulesMis defined as

width_AMˆinffijH_i(M^L_Ak)6ˆ0g:

The following is the main result of this section.

PROPOSITION4.5. Assume that the ringAis local and letCbe a semi- dualizing module forA. LetMbe a complex ofA-modules which has non- zero homology and satisfiesHiM ˆ0fori0andfd_AM <1. Then

idAC4GidAC^j M‡widthAM:

PROOF. Denote bykthe residue class field ofA. Observe that H_iMˆ0 fori0 implies H_i(M^L_Ak)ˆ0 fori0, whence width_AM> 1. Also, we can assume width_AM<1, because the proposition is trivially true if widthAMˆ 1.

Since fd_AM<1, the isomorphism [3, (A.4.23)] gives RHomA(k;C^L_AM)RHomA(k;C)^L_AM:

This implies (a) in

inffijH_iRHom_A(k;C^L_AM)6ˆ0g

ˆ(a)inffijH_i(RHom_A(k;C)^L_AM)6ˆ0g

(b)ˆinffijHiRHomA(k;C)6ˆ0g ‡inffijHi(M^L_Ak)6ˆ0g

ˆ idAC‡widthAM;

where (b) is by [3, (A.7.9.2)]. Consequently,

id_ACˆ inffijH_iRHom_A(k;C^L_AM)6ˆ0g ‡width_AM 4id_A(C^L_AM)‡width_AM

ˆ():

The condition fd_AM<1implies pd_AM<1by [11, Seconde partie, cor. (3.2.7)], and hence if we writesˆsupfijHiM6ˆ0g, lemma 4.4 gives

Ext^s‡1_A (M;H)ˆ0

when H is an A-module which is Gorenstein injective over A^j C. B ut fdAM<1 also implies M2_CA(A) by [4, prop. (4.4)], and altogether, lemma 4.3 applies toMand gives

()ˆGid_ACj M‡width_AM

as desired. p

We will also need the dual of proposition 4.5. First an easy lemma.

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 HiMˆ0 for i0.

Then

Gid_ACj Hom_A(M;I)ˆGfd_ACj M:

PROOF. From lemma 3.1(i) follows that EˆHom_A(A^j C;I) is a faithfully injective (A^j C)-module. Hence

Gid_AC^j Hom_AC^j (M;E)ˆGfd_AC^j M follows from [3, thm. (6.4.2)].

But equation (2) in the proof of lemma 3.1 shows Hom_AC^j (M;E)HomA(M;I);

so accordingly,

Gid_ACj Hom_A(M;I)ˆGfd_ACj M:

p PROPOSITION 4.7. Assume that the ring A is local and let C be a semi-dualizing module for A. Let N be a complex of A-modules which has non-zero homology and satisfiesHiN ˆ0fori0 andidAN <1.

Then

idAC4GfdAC^j N‡depth_AN:

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 dimen- sions, as announced in the introduction. From now on,Ais assumed to be local with residue class fieldk.

THEOREM5.1. The following conditions are equivalent.

(CM) Ais a Cohen-Macaulay ring with a dualizing module.

(I1) CMid_AM<1holds when M is any complex of A-modules with bounded homology.

(I2) There is a complex M of A-modules with bounded homology, CMidAM<1,fdAM<1, andwidthAM<1.

(I3) CMid_Ak<1.

(P1) CMpd_AM<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, CMpd_AM<1,id_AM<1, anddepth_AM<1.

(P3) CMpd_Ak<1.

(F1) CMfdAM<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, CMfd_AM<1,id_AM<1, anddepth_AM<1.

(F3) CMfd_Ak<1.

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

(CM))(I1) LetAbe Cohen-Macaulay with dualizing moduleC. Then A^j Cis Gorenstein by [7, thm. 5.6]. IfMis a complex ofA-modules with bounded homology, then M is also a complex of (A^j C)-modules with bounded homology, so

Gid_ACj M<1

by [3, thm. (6.2.7)]. AsCis in particular a semi-dualizing module, the de- finition of CMid then implies

CMid_AM<1:

(I1))(I2) and (I1))(I3) Trivial.

(I2))(CM) When CMid_AM<1then the definition of CMid implies thatAhas a semi-dualizing moduleCwith

Gid_AC^j M<1:

When

widthM<1 thenMhas non-zero homology. And finally, when

fdM<1 also holds, then proposition 4.5 implies

id_AC<1:

SoC is a dualizing module forA, and henceAis Cohen-Macaulay by [3, (A.8.5.3)] and has a dualizing module.

(I3))(CM) When CMidAk<1, thenAhas a semi-dualizing moduleC

with Gid_ACj k<1:

Denoting byE_AjC(k) the injective hull ofkoverA^j C, it follows from [9, thm. 2.22] that

RHom_A^j_C(E_AC^j (k);k)

has bounded homology. Denoting Matlis duality overA^j Cby_, we have RHom_ACj (E_ACj (k);k)RHom_ACj (k^_;E_ACj (k)^_)

RHom_ACj (k;Ad^j C)

(a)

RHom_AjC(k;A^j C)_ACj Ad^j C;

where (a) is by [3, (A.4.23)]. Since the completionAd^j Cis faithfully flat over A^j C, it follows that also RHom_ACj (k;A^j C) has bounded homology, whenceA^j Cis a Gorenstein ring.

But thenCis a dualizing module forAby [7, thm. 5.6] and so again,Ais 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

REMARK5.2. In condition (I2) of theorem 5.1, one could consider forM either the ring A itself, or the Koszul complex K(x1;. . .;xr) on any se- quence of elementsx1;. . .;xrin the maximal ideal. These complexes satisfy fd_AM<1and width_AM<1, and so either of the conditions

CMid_AA<1 and CMid_AK(x₁;. . .;x_r)<1

is equivalent toAbeing a Cohen-Macaulay ring with a dualizing module.

Similarly, in conditions (P2) and (F2), one could consider forMeither the injective hull of the residue class field,E_A(k), or a dualizing complexD (if one is known to exist). These complexes satisfy id_AM<1 and depth_AM<1, and so either of the conditions

CMpd_AEA(k)<1 and CMpd_AD<1 and

CMfd_AE_A(k)<1 and CMfd_AD<1

is equivalent toAbeing a Cohen-Macaulay ring with a dualizing module.

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 complexMwith non-zero homology, H_iMˆ0 fori0, and

Gid_AjCM<1; fd_AM<1; and width_AM<1;

thenAis Cohen-Macaulay with dualizing moduleC.

Similarly, proposition 4.7 implies that if there is a complexNwith non- zero homology and H_iNˆ0 fori0, then either of

Gpd_ACj N<1; id_AN<1; and depth_AN<1 and

Gfd_ACj N<1; id_AN<1; and depth_AN<1 implies thatAis Cohen-Macaulay with dualizing moduleC.

THEOREM5.4. Let M be an A-module. Then CMid_AM<1 ,CMid_AMdimA;

CMpd_AM<1 ,CMpd_AMdimA;

CMfdAM<1 ,CMfdAMdimA:

PROOF. The implications(are trivial.

To see the implication)for CMpd, observe that whenMis given there exists a semi-dualizing moduleCwith

CMpd_AMˆGpd_ACj M:

When this is finite, we have

Gpd_ACj MFPD(A^j C)

by [9, thm. 2.28], where FPD denotes the (big) finitistic projective di- mension. But

FPD(A^j C)ˆdimA^j C

by [11, Seconde partie, thm. (3.2.6)], andAandA^j C are finitely gener- ated as modules over each other, so

dimA^j CˆdimA:

Together, this establishes the implication)for CMpd.

The implication ) for CMfd follows by a similar argument, using

that the finitistic flat dimension satisfies FFD(A^j C)FPD(A^j 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 in- troduced 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_ACj M<1 then

CMdim_AMˆGpd_ACj M:

PROOF. Combining [8, proof of thm. 3.7] with [8, def. 3.2'] shows CMdim_AM4Gpd_ACj M:

So Gpd_ACj M<1implies CMdim_AM<1and hence CMdim_AMˆdepth_AA depth_AM by [8, thm. 3.8].

On the other hand,

Gpd_ACj MˆG-dim_CM

by [10, prop. 3.1], where G-dim_CM is the homological dimension in- troduced in [4, def. (3.11)]. SoG-dimCMis finite and hence

G-dim_CMˆdepth_AA depth_AM by [4, thm. (3.14)].

Combining the last three equations shows CMdimAMˆGpd_ACj M

as desired. p

THEOREM5.6. Let M be a finitely generated A-module. Then CMdimAM4CMpd_AM4GdimAM;

and if one of these numbers is finite then the inequalities to its left are equalities.

PROOF. The first inequality is clear from lemma 5.5, since CMpd_AMis defined as the infimum of all Gpd_ACj M.

For the second inequality, note that the ringAis itself a semi-dualizing module, so the definition of CMpd gives4in

CMpd_AM4Gpd_AAj MˆGpd_AMˆGdim_AM;

where the first ˆ is by [10, cor. 2.17], and the second ˆ holds by [3, cor. (4.4.6)] or [5, prop. (2.11)(b)] becauseMis finitely generated.

Equalities: If Gdim_AM<1then CMdim_AM<1by [8, thm. 3.7]. But Gdim_AM<1implies

GdimAMˆdepth_AA depth_AM by [3, thm. (2.3.13)], and similarly, CMdim_AM<1implies

CMdim_AMˆdepth_AA depth_AM

by [8, thm. 3.8]. So it follows that CMdim_AMˆGdim_AM, and hence both inequalities in the theorem must be equalities.

If CMpd_AM<1then by the definition of CMpd there exists a semi- dualizing moduleC over A with Gpd_ACj M<1. But by lemma 5.5, any suchChas

CMdim_AMˆGpd_ACj M;

so the first inequality in the theorem is an equality. p We finish with two clasically flavoured results.

PROPOSITION5.7 [Auslander-Buchsbaum formula]. LetM be a finitely generatedA-module. IfCMpd_AM <1, then

CMpd_AM ˆdepth_AA depth_AM:

PROOF. By the definition of CMpd, we can pick a semi-dualizing A- moduleCsuch that

CMpd_AMˆGpd_ACj M:

ButMis finitely generated, so

Gpd_ACj MˆGdim_ACj M

by [3, cor. (4.4.6)] or [5, prop. (2.11)(b)]. The Auslander-Bridger formula gives

Gdim_AC^j Mˆdepth_ACj A^j C depth_ACj M:

Finally, it remains to note that

depth_ACj A^j Cˆdepth_AA and

depth_ACj Mˆdepth_AM

becauseA^j CandAare finitely generated over each other. p PROPOSITION 5.8 [Bass formula]. Assume that A has a dualizing complex and let N 6ˆ0 be a finitely generated A-module. If CMidAN <1, then

CMidAN ˆdepthA:

PROOF. By the definition of CMid, we can pick a semi-dualizing A- moduleCsuch that CMid_ANˆGid_AjC N. By [5, thm. (6.4)], finiteness of Gid_ACj Nimplies that

Gid_ACj Nˆdepth_ACj A^j C;

since N6ˆ0 is finitely generated over A^j C, and since A^j C is finitely generated overAand so has a dualizing complex becauseAdoes. Finally, we have

depth_ACj A^j Cˆdepth_AA

again. p

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

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Manoscritto pervenuto in redazione il 30 settembre 2005