View of Homological aspects of semidualizing modules

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HOMOLOGICAL ASPECTS OF SEMIDUALIZING MODULES

RYO TAKAHASHI and DIANA WHITE^∗ (Dedicated to the memory of Anders J. Frankild)

Abstract

We investigate the notion of theC-projective dimension of a module, whereCis a semidualizing module. WhenC=R, this recovers the standard projective dimension. We show that three natural definitions of finiteC-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finiteC-projective dimension. In parallel, we develop the dual theory for injective dimension andC-injective dimension.

Introduction

Grothendieck [6] introduced dualizing modules as tools for investigating cohomology theories in algebraic geometry. In this paper, we investigatesemi- dualizingmodules and associatedrelativecohomology functors. Foxby [4], Vasconcelos [12] and Golod [5] independently initiated the study of semidualizing modules. Over a noetherian ringR, a finitely generatedR-module C is semidualizingif the natural homothety map R → Hom_R(C, C) is an isomorphism and Ext_R¹(C, C)=0; see 1.2 for a more general definition. Ex- amples include a dualizing module, when one exists, and all finitely generated rank 1 projective modules.

Throughout this introduction, letRbe a commutative ring andCa semidu- alizingR-module. The class ofC-projectives, denotedPC, consists of those R-modules of the formC⊗RP for some projectiveR-moduleP. These form the building blocks of the so-calledGC-projectives, which are studied in depth in [13]. EveryR-moduleM admits anaugmented properPC-projective resol- ution. That is, there exists a complex

X⁺= · · · →C⊗RPn→ · · · →C⊗RP⁰→M →0

∗This research was conducted in part while R. T. visited the University of Nebraska in August 2006, partly supported by NSF grant DMS 0201904.

Received 7 November 2007.

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such that the complex

Hom_R(C, X⁺)= · · · →Pn→ · · · →P⁰→Hom_R(C, M)→0 is exact. Despite the fact that these augmented properPC-resolutions may not be exact, they still have particularly good lifting properties. In particular, they give rise to well-defined cohomology modules Extⁱ_P

C(M, N)for allR-modules M andN; see 1.5. In the caseC = R, these notions recover the projectives, projective resolutions and the “absolute” cohomology Extⁱ_R(M, N), respectively.

Because augmented proper PC-projective resolutions need not be exact, it is not immediately clear how best to define thePC-projective dimension of a module. For instance, should one consider arbitrary properPC-projective resolutions or only exact ones? Or should it be defined in terms of the vanishing of the functors Extⁿ_P

C(M,−)? The next result, proved in Corollary 2.10, shows that each of these approaches gives rise to the same invariant.

Theorem A. LetM be anRmodule. The following quantities are equal.

(1) inf

sup{n|Xn=0}

X⁺is an augmented proper PC-projective resolution ofM

(2) inf

sup{n|Xn=0}

X⁺is an exact augmented proper PC-projective resolution ofM

(3) sup

n|Extⁿ_P

C(M,−)=0

The proof of this result uses so-called Auslander and Bass class techniques;

see 1.8.

Our investigation demonstrates a strong connection between the modules of finitePC-projective dimension and modules of finite projective dimension, which is the focus of Section 2. For example, the following is part of The- orem 2.11.

Theorem B. IfM is anR-module, thenPC-pd_R(M) = pd_R(Hom_R(C, M)).

IfRis Cohen-Macaulay and local with dualizing moduleD, then a result of Sharp shows that the modules of finitePD-projective dimension are precisely the modules of finite injective dimension. Thus, this theorem recovers part of the Foxby equivalence from [1], namely thatMhas finite injective dimension if and only if Hom_R(D, M)has finite projective dimension.

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Section 4 explores the connection between the cohomology functors Ext_P_C and Ext_R. These connections are used in [9] to distinguish between several different relative cohomology theories. For example, the following is part of Theorem 4.1 and Corollary 4.2. See 1.8 for the definition of the Bass class.

Theorem C. Let M and N be R-modules. There is an isomorphism Extⁱ_P

C(M, N) ∼= Extⁱ_R(Hom_R(C, M),Hom_R(C, N)) for alli. If M and N are in the Bass class with respect toC, thenExtⁱ_P

C(M, N)∼=Extⁱ_R(M, N)for alli.

This result and several others from this work have already proved valuable for other investigations. For instance, they are used by the second author and her collaborators in [11] to analyze the structure of certain categories naturally associated to semidualizing modules, and in [9], [10] to study balance properties of relative cohomology theories and to distinguish between them. We expect this line of inquiry to continue to shed new light on the relationship between classical and relative homological algebra.

Finally, in Section 5 of the paper we use results from the previous sections to demonstrate the depth of the connection between modules of finite PC- projective dimension and modules of finite projective dimension.

1. Preliminaries

Throughout this paper, letRbe a commutative ring.

1.1. AnR-complexis a sequence ofR-module homomorphisms X= · · ·−−−→^∂ⁿ⁺¹^X Xn−−→∂n^X Xn−1−−−→ · · ·∂n−1^X

such that∂_n−^X1∂_n^X = 0 for each integern; thenth homology moduleof Xis Hn(X)= Ker(∂_n^X)/Im(∂_n+^X1). A morphism of complexesα:X →Y induces homomorphisms Hn(α): Hn(X)→Hn(Y ), andαis aquasiisomorphismwhen each H_n(α)is bijective. The complexXisboundedifXn =0 for|n| 0. It isdegreewise finiteif eachXi is finitely generated.

1.2. AnR-moduleCissemidualizingif

(a) Cadmits a degreewise finite projective resolution,

(b) The natural homothety mapR →Hom_R(C, C)is an isomorphism, and (c) Ext_R¹(C, C)=0.

A freeR-module of rank 1 is semidualizing, and indeed this is the only semidualizing module over a Gorenstein local ring. IfR is noetherian, local and admits a dualizing moduleD, thenDis semidualizing.

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The next two classes of modules have been studied in numerous papers, see e.g. [2] and [7].

1.3. The classes ofC-projectiveandC-injectivemodules are defined as PC = {C⊗RP |P is projective},

IC = {Hom_R(C, I )|I is injective}.

WhenC =R, we omit the subscript and recover the classes of projective and injectiveR-modules.

The next four paragraphs provide the necessary background on relative homological algebra. The reader is encouraged to consult [3] for details.

1.4. The classPCisprecoveringby [7, (5.10)]. That is, given anR-module M, there exists a projective moduleP and a homomorphismφ:C⊗RP →M such that, for every projectiveQ, the induced map

Hom_R(C⊗RQ, C⊗RP )−−−−−−−−−−→^Hom^R^(C⊗^R^Q,φ) Hom_R(C⊗RQ, M) is surjective. Dually, the classIC ispreenveloping.

1.5. Since the classPCis precovering, for anyR-moduleMone can iterat- ively take precovers to construct anaugmented properPC-projective resolution ofM, that is, a complex

X⁺= · · ·−−→^∂²^X C⊗RP1−−→∂1^X C⊗RP0−−→∂0^X M →0

such that Hom_R(C⊗RQ, X⁺)is exact for all projectiveR-modulesQ. The truncated complex

X= · · ·−−−→^∂²^X⁺ C⊗RP1−−−→∂1^X⁺ C⊗RP0→0 is aproperPC-projective resolutionofM. Forn0, set

^X_n⁺ =

M ifn=0, Ker(∂_n−^X⁺1) ifn1.

Note thatX⁺need not be exact unlessC =R.

Dually, letNbe anR-module withaugmented properIC-injective resolu- tion

Y⁺=0→N →Hom_R(C, I⁰) ^∂

0

−−→Y Hom_R(C, I¹) ^∂

1

−−→ · · ·Y

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For an integern0, set ⁿ_Y⁺ =

N ifn=0, Im(∂_Yⁿ⁻⁺¹) ifn1.

ProperPC-projective resolutions are unique up to homotopy equivalence;

see e.g. [8, (1.8)]. Accordingly, thenth relative cohomology modules Extⁿ_P

C(M, N)=HⁿHom_R(X, N)

whereXis a properPC-projective resolution ofM are well-defined for each integern. The cohomology modules Extⁿ_I

C(M, N)are defined dually.

1.6. ThePC-projective dimensionofM is PC- pd(M)=inf

sup{n|Xn=0}

Xis a properPC-projective resolution ofM

The modules ofPC-projective dimension zero are the non-zero modules inPC. TheIC-injective dimension, denotedIC- id(−)is defined dually.

1.7 (Dimension Shifting). Let N be an R-module. For any augmented properPC-projective resolutionX⁺(as above) ofM, there are isomorphisms

Extⁱ_P

C(M, N)∼=Extⁱ⁻_P¹

C(^X1⁺, N)

∼=Extⁱ⁻_P²

C(^X2⁺, N)∼= · · · ∼=Extⁱ⁻ⁿ_P

C(^X_n⁺, N) for integers 1n < i.

1.8. TheBass class with respect toC, denotedBCorBC(R), consists of allR-modulesM satisfying

(a) Ext_R¹(C, M)=0=Tor^R₁(C,Hom_R(C, M))=0, and

(b) The natural evaluation map νCCM:C ⊗R Hom_R(C, M) → M is an isomorphism. We will writeνM =νCCM if there is no confusion.

Dually, theAuslander class with respect toC, denotedACorAC(R), consists of allR-modulesM satisfying

(c) Tor^R₁(C, M)=0=Ext_R¹(C, C⊗RM), and

(d) The natural mapμCCM:M →Hom_R(C, C⊗RM)is an isomorphism.

We will writeμM =μCCM if there is no confusion.

We now state some basic results about the classes AC and BC. These facts are well-known whenRis noetherian. In this generality, the first follows

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from [7, (6.5)], the second follows from the first, and the third assertion is routine to check.

1.9. The following hold.

(a) If any twoR-modules in a short exact sequence are inAC, respectively BC, then so is the third.

(b) The classACcontains all modules of finite flat dimension. The classBC

contains all modules of finite injective dimension.

(c) IfMis inAC, thenC⊗RMis inBC. IfM is inBC, then Hom_R(C, M) is inAC.

2. Relative dimensions and Auslander and Bass classes

This section has two interwoven themes. First, we explore the interplay be- tweenPC-projective dimension and projective dimension. Second, we investigate the exactness of augmented properPC-projective resolutions for modules of finitePC-projective dimension.

We begin with the following lemma, which follows from the definitions of semidualizing modules and augmented proper resolutions, using 1.9(b).

Lemma2.1.LetCbe a semidualizingR-module andManR-module.

(a) If X⁺ is an augmented proper PC-projective resolution of M, then Hom_R(C, X⁺)is an augmented projective resolution ofHom_R(C, M). (b) IfY⁺is an augmented properIC-injective resolution ofM, thenC⊗RY⁺

is an augmented injective resolution ofC⊗RM.

We now investigate exactness of augmented properPC-projective resolutions.

Proposition2.2. Let C be a semidualizing R-module, M anR-module andna nonnegative integer.

(a) The following are equivalent.

(i) There exists an augmented properPC-projective resolution ofM which is exact in degree less thann;

(ii) Every augmented properPC-projective resolution ofMis exact in degree less thann;

(iii) The natural homomorphismνM :C⊗RHom_R(C, M)→Mis an isomorphism andTor^R_i (C,Hom_R(C, M))=0for0< i < n. (b) The following are equivalent.

(i) There exists an augmented proper IC-injective resolution of M which is exact in degree less thann;

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(ii) All augmented properIC-injective resolutions ofM are exact in degree less thann;

(iii) The natural homomorphismμM :C →Hom_R(C, C⊗RM)is an isomorphism andExtⁱ_R(C, C⊗RM)=0for0< i < n.

Proof. We prove only part(a). LetX⁺be an augmented properPC-projective resolution of M. By Lemma 2.1, the complex Hom_R(C, X) is a projective resolution of Hom_R(C, M). Hence, there is an isomorphism Tor^R_i (C, Hom_R(C, M)) ∼= Hi(C⊗RHom_R(C, X))for alli 0. Since eachXi is in BC, the natural chain mapC⊗RHom_R(C, X)→Xis an isomorphism. The result follows

From the above proposition, we obtain the following criterion for a given module to possess exact augmented proper resolutions.

Corollary2.3.LetCbe a semidualizingR-module andManR-module.

(i) Madmits an exact augmented properPC-projective resolution;

(ii) All augmented properPC-projective resolutions ofM are exact;

(iii) The natural homomorphismνM:C⊗RHom_R(C, M)→M is an isomorphism andTor^R₁(C,Hom_R(C, M))=0.

(b) The following are equivalent.

(i) Madmits an exact augmented properIC-injective resolution;

(ii) All augmented properIC-injective resolutions ofM are exact;

(iii) The natural homomorphismμM:M →Hom_R(C, C⊗RM)is an isomorphism andExt_R¹(C, C⊗RM)=0.

From the definitions of the Auslander and Bass classes, we have the following.

(a) AssumeM is inBC. Then every augmented properPC-projective resol- ution ofMis exact. In particular, everyPC-precover ofM is surjective.

(b) AssumeMis inAC. Then every augmented properIC-resolution ofM is exact. In particular, everyIC-preenvelope ofM is injective.

The next few technical results build toward the following fact, which follows from Corollaries 2.3 and 2.10: ifM has finitePC-projective dimension, then everyaugmented properPC-resolution ofM is exact.

Lemma2.5.LetCbe a semidualizing module andM anR-module.

(a) The composition Hom_R(C, νM)◦ μHom_R(C,M) is the identity map on Hom_R(C, M). Hence, Hom_R(C, νM) is a split epimorphism and μ^HomR(C,M)is a split monomorphism.

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(b) Assume thatνMis injective. The compositionμHom_R(C,M)◦Hom_R(C, νM) is the identity map onHom_R(C, C⊗RHom_R(C, M)). Hence,Hom_R(C, νM)is an isomorphism andμ^HomR(C,M)is the inverse isomorphism.

(c) The composition νC⊗RM ◦(C ⊗R μM) is the identity map on C ⊗R

M. Hence, C ⊗R μM is a split monomorphism and νC⊗RM is a split epimorphism.

(d) Assume thatμM is surjective. The composition(C⊗RμM)◦νC⊗RM is the identity map onC⊗RHom_R(C, C⊗RM). Hence,C⊗RμM is an isomorphism andνC⊗RM is the inverse isomorphism.

Proof. Part (a) is straightforward to check. For part (b), set ρ = μ^HomR(C,M)◦Hom_R(C, νM). Note that ifξ ∈Hom_R(C, C⊗RHom_R(C, M)), thenρ(ξ )sendsz ∈ C to z⊗(νM ◦ξ ). Thus, νM(ξ(z)−z⊗(νM ·ξ )) = νM(ξ(z))−νM(ξ(z)) = 0. SinceνM is injective, there is an equalityξ(z) = z⊗(νM·ξ ), thereby showing thatρis the identity. Parts (c) and (d) are proved similarly.

Lemma2.6.LetCbe a semidualizingR-module andManR-module.

(a) AssumeExt¹_R(C, C⊗RHom_R(C, M))=0. IfνM is injective, then it is an isomorphism.

(b) Assume thatTor^R₁(C,Hom_R(C, C⊗RM))=0. IfμMis surjective, then it is an isomorphism.

Proof. We prove only part (a), as part (b) is dual. Set L = CokerνM. Applying Hom_R(C,−)to the exact sequence

0→C⊗RHom_R(C, M)−→^ν^M M →L→0 induces an exact sequence

0→Hom_R(C, C⊗Hom_R(C, M))

Hom(C,νM)

−−−−−−−→Hom_R(C, M)→Hom_R(C, L)→0 where right-exactness follows from the equality Ext¹_R(C, C ⊗R Hom_R(C, M))=0. By Lemma 2.5, Hom_R(C, νM)is an isomorphism. Hence, the above exact sequence implies that Hom_R(C, L) = 0, and it follows from [7, (3.6)]

thatL=0.

(a) SupposeHom_R(C, M)is inAC. IfνMis injective, then it is an isomorph- ism.

(b) SupposeC⊗RMis inBC. IfμMis surjective, then it is an isomorphism.

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We now prove our first theorem, which leads to some of the main results of this section.

Theorem2.8. LetCbe a semidualizingR-module andM anR-module.

Then the following hold.

(a) M ∈BCif and only ifHom_R(C, M)∈AC. (b) M ∈ACif and only ifC⊗RM ∈BC.

Proof. We prove only part (a), as part (b) is proved similarly. Accord- ing to 1.9(c), it is enough to assume Hom_R(C, M) ∈ AC and show M ∈ BC. The definition of AC implies that Tor^R₁(C,Hom_R(C, M)) = 0 and Ext_R¹(C, C⊗R Hom_R(C, M)) = 0. We will show that the evaluation map νM is an isomorphism. Using the above vanishings, it will then follow that Ext_R¹(C, M)=0 andM is inBC.

By Lemma 2.5(a), the composition Hom_R(C, νM)◦μ^HomR(C,M)is the identity map on Hom_R(C, M). Since Hom_R(C, M)is inAC, the mapμ^HomR(C,M)

is an isomorphism, and so Hom_R(C, νM) is also an isomorphism. Setting K= ker(νM), it follows that Hom_R(C, K)=0, and so by [7, (3.6)],K =0.

Thus, the mapνM is injective, hence an isomorphism by Corollary 2.7(a).

Recall that the classBC contains all modules of finite injective dimension and the classAC contains all modules of finite projective dimension; see 1.9.

By virtue of Theorem 2.8, we now obtain additional examples of modules in ACandBC.

(a) IfPC-pd_R(M)is finite, thenMis inBC. (b) IfIC-id_R(M)is finite, thenM is inAC.

Proof. We prove only part (a). AssumePC- pd_R(M)is finite, and letX⁺ be an augmented properPC-resolution ofM. By Lemma 2.1(a), the complex Hom_R(C, X⁺)is a bounded projective resolution of Hom_R(C, M). By 1.9(b) Hom_R(C, M)is inAC so Theorem 2.8(a) implies thatM is inBC.

This yields the following key result.

Corollary2.10. LetCbe a semidualizingR-module andManR-module.

(a) The inequalityPC-pd_R(M) nholds if and only if there is an exact sequence

0→C⊗RPn→ · · · →C⊗RP0→M →0 with eachPi a projectiveR-module.

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(b) The inequality IC-id_R(M) nholds if and only if there is an exact sequence

0→M →Hom_R(C, I⁰)→ · · · →Hom_R(C, Iⁿ)→0 with eachIⁱ an injectiveR-module.

Proof. The “only if” direction of each statement follows immediately from Corollaries 2.4 and 2.9. For the other implications, use dimension shifting 1.7 to see that any exact sequence of the given form is an augmented properPC- projective orIC-injective resolution ofM.

We now investigate howPC-projective and projective dimension relate.

Theorem2.11.LetCbe a semidualizingR-module. The following equal- ities hold.

(a) pd_R(M)=PC-pd_R(C⊗RM) (b) IC-id_R(M)=id_R(C⊗RM) (c) PC-pd_R(M)=pd_R(Hom_R(C, M)) (d) id_R(M)=IC-id_R(Hom_R(C, M))

Proof. We prove only part (a). Assume pd_R(M)= s < ∞and consider an augmented projective resolution of M

X= 0→Ps →Ps−1→ · · · →P0→M →0.

By 1.9(b), one hasM ∈ACand so Tor^R₁(C, M)=0. Thus, the complex C⊗RX= 0→C⊗RPs →C⊗RPt−1→ · · · →C⊗RP⁰→C⊗RM →0 is exact and thus an augmented properPC-projective resolution ofC⊗RM. Note that properness can be shown by using 1.7, or as a special case of [13, (4.4)]. This provides an inequalitys PC- pd_R(C⊗RM). Conversely, assume thatPC- pd_R(C⊗RM)=t <∞. By Corollary 2.10(a), there is an augmented exactproperPC-resolution ofC⊗RM

X⁺= 0→C⊗RPt →C⊗RPt−1→ · · · →C⊗RP0→C⊗RM →0. Thus, the complex Hom_R(C, X⁺)is exact. Corollary 2.9 implies thatC⊗R

M is in BC and Theorem 2.8 then implies that M is in AC. Thus, μM is an isomorphism. Since eachμPi is also an isomorphism, Hom_R(C, X⁺)is isomorphic to an exact sequence of the form

0→Pt →Pt−1→ · · · →P⁰→M →0.

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Thus, pd_R(M)t =PC- pd_R(C⊗RM).

Note that by assembling the information above, we get the following extension of the Foxby equivalence [1].

Theorem2.12 (Foxby equivalence).LetC be a semidualizingR-module, and letn be a non-negative integer. Set PC(R)n, P(R)n, IC(R)n, and I(R)n to be the classes of modules ofC-projective, projective, C-injective and injective dimension of at mostn, respectively. Then there are equivalences of categories

P(R) −−−−−−−−−−−−−−→←−−−−−−−−−−−−−−_Hom^C⊗_∼^R⁻

R(C,−) PC(R) P(R)n

C⊗R−

−−−−−−−−−−−−−−→_∼

←−−−−−−−−−−−−−−_Hom

R(C,−) PC(R)n

AC(R) −−−−−−−−−−−−−−→←−−−−−−−−−−−−−−_Hom^C⊗_∼^R⁻

R(C,−) BC(R) IC(R)n

C⊗R−

−−−−−−−−−−−−−−→_∼

←−−−−−−−−−−−−−−_Hom

R(C,−) I(R)n

IC(R) −−−−−−−−−−−−−−→←−−−−−−−−−−−−−−_Hom^C⊗_∼^R⁻

R(C,−) I(R) 3. Vanishing of relative cohomology and consequences

In this section, we investigate how vanishing of the relative cohomology functors Extⁱ_P

C(M,−)and Extⁱ_I

C(−, N), respectively, characterizes the finiteness ofPC- pd_R(M)andIC- id_R(N). Our proofs use specific properties of semidualizing modules and so do not carry over directly to other relative settings.

Theorem3.1.LetCbe a semidualizingR-module andManR-module.

(i) Ext¹_P

C(M,−)=0 (ii) Ext_P¹

C(M,−)=0 (iii) MisC-projective (a) The following are equivalent.

(i) Ext¹_I

C(−, M)=0

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(ii) Ext_I¹

C(−, M)=0 (iii) MisC-injective

Proof. We prove part (a); part (b) is dual.

(iii)⇒(ii): IfMisC-projective, then the complex

· · · →0→M −→⁼ M →0 is an augmented properPC-resolution ofMand so Ext_P¹

C(M,−)=0.

(ii)⇒ i) is immediate.

(i)⇒(iii): Let

X= · · ·−→^d² C⊗RP¹−→^d¹ C⊗RP⁰−→^d⁰ M →0

be an augmented proper PC-resolution of M. Let K⁰ be the kernel of d⁰, and letβ:K⁰ →C ⊗R P⁰be the inclusion map. There is a homomorphism α:C⊗RP¹ → K⁰ such thatd¹ = βα. Noting thatβαd² = d¹d² = 0, the injectivity ofβimplies thatαd2 = 0. Since Ext¹_P

C(M, K0)= 0, the induced sequence

Hom_R(C⊗RP⁰, K⁰)→Hom_R(C⊗RP¹, K⁰)→Hom_R(C⊗RP², K⁰) is exact. Hence, there existsξ ∈Hom_R(C⊗RP0, K0)such thatα = ξ d1 = ξβα. There is an equality Hom_R(C, α) = Hom_R(C, ξ ) ◦Hom_R(C, β)◦ Hom_R(C, α)and so Hom_R(C, ξ )◦Hom_R(C, β)=idHom_R(C,K⁰), as Hom_R(C, α)is surjective. Therefore, the exact sequence (which is exact by properness ofX)

0→Hom_R(C, K0)−−−−−−−→^Hom^R^(C,β) Hom_R(C, C⊗RP0)→Hom_R(C, M)→0 splits. Since Hom_R(C, C⊗RP0) ∼= P0 isR-projective, so is Hom_R(C, M). Theorem 2.11 and Corollary 2.10(a) now imply thatM isC-projective.

Using dimension shifting, we have the following extension of the previous result.

Theorem3.2.LetCbe a semidualizingR-module, letnbe a non-negative integer, and letM, NbeR-modules.

(i) Extⁿ⁺_P ¹

C (M,−)=0 (ii) Extⁿ⁺_P ¹

C (M,−)=0 (iii) PC-pd_R(M)n.

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(i) Extⁿ⁺_I ¹

C (−, N)=0 (ii) Extⁿ⁺_I ¹

C (−, N)=0 (iii) IC-id_R(N)n.

The preceding two results imply the following.

Corollary3.3. LetCbe a semidualizingR-module,ManR-module and n0.

(i) There exists an augmented properPC-projective resolutionX⁺of Msuch that^X_n⁺isC-projective.

(ii) Every augmented properPC-projective resolutionX⁺of M has the property that^X_n⁺isC-projective.

(i) There exists an augmented properIC-injective resolutionY⁺ of Msuch thatⁿ_Y⁺ isC-injective.

(ii) Every augmented properIC-injective resolutionY⁺ofM has the property thatⁿ_Y⁺ isC-injective.

We conclude this section by showing that if any two of three modules in a short exact sequence have finitePC-projective dimension then so does the third. Note that the standard constructions (using Horseshoe Lemmas, mapping cones, etc.) that show this result whenC = R can be used in this setting.

However, we offer a shorter proof that uses the classical result.

Proposition3.4. LetCbe a semidualizingR-module. Consider an exact sequence ofR-modules

0→M→M →M→0.

If any two of the modules have finite PC-projective dimension, respectively IC-injective dimension, then so does the third.

Proof. Assume that two of the modulesM, M, Mhave finitePC-projective dimension. By Corollary 2.9, these two modules are inBC. By 1.9(a), this forces all of the modulesM, M, Mto be inBC. Thus, the complex

0→Hom_R(C, M)→Hom_R(C, M)→Hom_R(C, M)→0 is exact. By Theorem 2.11(c), two of the above Hom modules have finite projective dimension and hence so does the third. Theorem 2.11(c) implies that all ofM, M, Mhave finitePC-projective dimension. The other assertion is dual.

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4. Comparing relative and absolute cohomology

In this section we investigate the interplay between absolute Ext and the relative cohomologies Ext_P_Cand Ext_I_C. Under certain circumstances, they agree with their corresponding absolute counterparts.

Theorem4.1.Let C be a semidualizingR-module, and letM andN be R-modules. There exist isomorphisms.

Extⁱ_P

C(M, N)∼=Extⁱ_R(Hom_R(C, M),Hom_R(C, N)) Extⁱ_I

C(M, N)∼=Extⁱ_R(C⊗RM, C⊗RN)

Proof. We prove only part (a). Let X⁺ be an augmented proper PC- resolution ofM. By Lemma 2.1, the complex Hom_R(C, X⁺)is an augmented projective resolution of Hom_R(C, M). Thus, the equalities below hold by definition

Extⁱ_R(Hom_R(C, M),Hom_R(C, N))

=Hⁱ(Hom_R(Hom_R(C, X),Hom_R(C, N)))

∼=Hⁱ(Hom_R(C⊗RHom_R(C, X), N))

∼=Hⁱ(Hom_R(X, N))

=Extⁱ_P

C(M, N),

while the isomorphisms follow from adjunction and the containmentPC ⊆ BC.

With appropriate Auslander and Bass class assumptions, the aforementioned relative cohomology modules agree precisely with the absolute Ext.

Corollary 4.2. Let C be a semidualizing R-module, and let M, N be R-modules.

(a) IfM andNare inBC, thenExtⁱ_P

C(M, N)∼=Extⁱ_R(M, N)for alli. (b) IfM andNare inAC, thenExtⁱ_I

C(M, N)∼=Extⁱ_R(M, N)for alli. Proof. We prove only part(a). Let TotX denote the total complex of a double complexX. LetP⁺be an augmented projective resolution of Hom_R(C, M), and letI⁺ be an augmented injective resolution ofN. SinceM andN are inBC, the complexesC⊗RP⁺and Hom_R(C, I⁺)are exact. There is an isomorphism

Hom_R(C⊗RP , I )∼=Hom_R(P ,Hom_R(C, I ))

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of double complexes. This provides the second isomorphism below Extⁱ_R(M, N)∼=Hⁱ(Tot Hom_R(C⊗RP , I ))

∼=Hⁱ(Tot Hom_R(P ,Hom_R(C, I )))

∼=Extⁱ_R(Hom_R(C, M),Hom_R(C, N))

∼=Extⁱ_P

C(M, N),

while the last isomorphism follows from Theorem 4.1.

5. Further parallels between the classical and relative theories

The results of the previous sections demonstrate that there is a tight connection between modules of finitePC-projective dimension and modules of finite projective dimension. In this section we indicate how the machinery developed above allows us to extend many classical results to this new setting. We begin by showing thatPC-projective dimension has the ability to detect when a ring is regular.

Proposition5.1.Let(R,ᒊ, k)be a noetherian, local ring andCa semi- dualizingR-module. The following are equivalent.

(i) PC-pd_R(M)is finite for allR-modulesM. (ii) PC-pd_R(k)is finite.

(iii) Ris regular.

Proof. (i)⇒(ii) is trivial.

(ii)⇒(iii) SincePC- pd_R(k)is finite, Lemma 2.1 implies pd_R(Hom_R(C, k))is finite. Since Hom_R(C, k)is a nonzerok-vector space, pd_R(k)is finite.

Thus,Ris regular.

(iii)⇒(i) SinceRis regular, the only semidualizingR-module isRitself.

Thus,C =Rso this follows from the Auslander-Buchsbaum-Serre theorem.

Our methods also apply to bounded complexes ofC-projective modules, as the next result shows.

Proposition5.2 (New Intersection Theorem for complexes ofC-projective modules). Let(R,^ᒊ)be a noetherian local ring andC a semidualizingR- module. If there exists a non-exact complex

X=0→C^α^s →C^α^s−1 → · · · →C^α¹ →C^α⁰ →0 withR(Hi(X))finite for alli, thensdim(R).

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Proof. First, note that the complex

Hom_R(C, X)=0→R^α^s →R^α^s−1 → · · · →R^α¹ →R^α⁰ →0 is non-exact. Indeed, if it were exact, then it would split, forcing the complex C⊗RHom_R(C, X)∼=Xto be exact, a contradiction.

Now fix a primeᒍ=ᒊ. Since the homology of the complexXhas support equal toᒊ, the complexXᒍis exact. This forces the complex Hom_R_ᒍ(Cᒍ, Xᒍ)to be exact, as Ext_R¹(C,C)=0. This forces H_i(Hom_R(C,X))ᒍ∼= H_i(Hom_R_ᒍ(Cᒍ, Xᒍ)) = 0 for all i. Thus, R(Hi(Hom_R(C, X))) < ∞ for all i. The New Intersection Theorem now implies thats dim(R).

Next, we extend Bass’ result that a ring is noetherian if and only if the class of injectiveR-modules is closed under direct sums.

Proposition5.3.LetRbe a commutative ring andCa semidualizingR- module. The ringR is noetherian if and only if the classIC is closed under direct sums.

Proof. AssumeR is noetherian. Let {Iλ} be a collection of injectiveR- modules. SinceCis finitely presented, there is an isomorphism

λ

Hom_R(C, Iλ)∼=Hom_R C,

λ

Iλ

and the desired result follows by Bass’ result.

Conversely, assume the classIC is closed under direct sums. Let{Iλ}be a collection of injective modules so that{Hom_R(C, Iλ)}is a collection ofC- injective modules. By assumption, there is an isomorphism

Hom_R(C, Iλ)∼=Hom_R(C, J )

for some injectiveJ. This provides the third isomorphism below. The first and fourth isomorphisms follow from the fact that any injective module is inBC. The second is by the commutativity of tensor products and coproducts.

Iλ ∼=

C⊗RHom_R(C, Iλ)

∼=C⊗R

(Hom_R(C, Iλ))

∼=C⊗RHom_R(C, J )

∼=J.

In particular,

Iλis injective.

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Finally, we provide an example in which this technique does not seem to provide a straightforward way in which to extend a classical result. Consider the following:

Question5.4. Let (R,ᒊ, k)be a local, Cohen-Macaulay ring admitting a dualizing moduleD. LetC be a semidualizing R-module. If there exists anR-moduleM of finite depth with finitePC-projective dimension and finite IC-injective dimension, mustRbe Gorenstein?

Note that, as we try to apply the aforementioned techniques, we see that Hom_R(C, M)has finite projective dimension, whileC⊗RMhas finite injective dimension. To apply the classical result, we need a single module that has both finite projective dimension and finite injective dimension.

Acknowledgments. The authors would like to thank Lars Winther Chris- tensen, Sean Sather-Wagstaff and the referee for helpful comments and sug- gestions.

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DEPARTMENT OF MATHEMATICAL SCIENCES FACULTY OF SCIENCE

SHINSHU UNIVERSITY 3-1-1 ASAHI, MATSUMOTO NAGANO 390-8621 JAPAN

E-mail:takahasi@math.shinshu-u.ac.jp

DEPARTMENT OF MATHEMATICS UNIVERSITY OF COLORADO DENVER CAMPUS BOX 170

P.O. BOX 173364 DENVER, CO 80217-3364 USA

E-mail:diana.white@ucdenver.edu