Embedding modules of finite homological dimension

0.2 MB·English

by  Sean Sather-Wagstaff

EMBEDDING MODULES OF FINITE HOMOLOGICAL
DIMENSION
2
1 SEANSATHER-WAGSTAFF
0
2
Abstract. ThispaperbuildsonworkofHochsterandYaothatprovidesnice

n embeddingsforfinitelygeneratedmodulesoffiniteG-dimension,finiteprojec-
a tive dimension, or locally finite injective dimension. We extend these results
J by providing similar embeddings in the relative setting, that is, for certain

4 modules offiniteGC-dimension,finitePC-projectivedimension,locallyfinite
1 GIC-injectivedimension,orlocallyfiniteIC-injectivedimensionwhereC isa
semidualizingmodule. Alongthe way, weextend someresults formodules of

] finitehomologicaldimensiontomodulesoflocallyfinitehomologicaldimension
C
intherelativesetting.
A
.
h
t
a
m 1. Introduction
[ Throughout this note, R is a commutative noetherian ring with identity. The

purpose of this note is to extend some results of Hochster and Yao, beginning
1
v with the following, wherein z[i] =z1,...,zi and G-dimR(M) is the G-dimension of
7 Auslander and Bridger [1, 2].1
3
0 Theorem 1.1 ([14, Theorem 2.3]). Let M be a non-zero finitely generated R-
3 module. Then
.
1
(1) IfG-dim (M)=r <∞,thenthereareanR-regularsequencez =z ,...,z ,
0 R 1 r
integers n ,n ,...,n >0 with n >1, and an exact sequence
2 0 1 r r
1
0→M →Z →N →0
:
v
Xi with Z =⊕ri=0(R/(z[i]))ni and G-dimR(N)6r.
(2) If pd (M) = r < ∞, then there exist an R-regular sequence z = z ,...,z ,
R 1 r
r integers n ,n ,...,n >0 with n >1, and an exact sequence
a 0 1 r r
0→M →Z →N →0
with Z =⊕r (R/(z ))ni and pd (N)6r.
i=0 [i] R
Date:January17,2012.
2000 Mathematics Subject Classification. 13D02,13D05,13D07.
Key words and phrases. Auslanderclasses, Bass classes,canonical modules, G-dimension, G-
injectivedimension,injectivedimension,projectivedimension,semidualizingmodules.
This material is based on work supported by North Dakota EPSCoR and National Science
FoundationGrantEPS-0814442. Theauthor wassupportedinpartbyagrantfromtheNSA.
1This is also now called “Gorenstein projective dimension” after the work of Enochs, Jenda,
andothers. However, this isageneralization ofAuslanderand Bridger’soriginalnotionfornon-
finitely-generatedmodules. Sincewearefocusedonfinitelygeneratedmoduleshere,wecontinue
withAuslanderandBridger’sterminology.
1
2 SEANSATHER-WAGSTAFF
ThepointofTheorem1.1,asHochsterandYaoindicate,isthatthegivenmodule
of finite G-dimension or projective dimension embeds in a module that obviously
has finite projective dimension, with some extra control on the G-dimension or
projective dimension of the cokernel.
The first of our results, stated next, is for modules of finite G -dimension and
C
modules of finite P -projective dimension, where C is a semidualizing R-module,
C
after Golod [12], Holm and Jørgensen [15] and White [21].2 (See Section 2 for
definitions and background material.) On the one hand, this result is a corollary
to Theorem1.1. On the other hand, Theorem1.1 is the specialcase C =R,so our
result generalizes this one. This result is proved in Proofs 2.8 and 2.13, which are
similar to the proof of [14, Theorem 4.2].
Theorem 1.2. Let M be a non-zero finitely generated R-module, and let C be a
semidualizing R-module. Then
(1) If G -dim (M) = r < ∞ and M is in the Bass class B (R), then there
C R C
exist an R-regular sequence z = z ,...,z , integers n ,n ,...,n > 0 with
1 r 0 1 r
n >1, and an exact sequence
r
0→M →Z →N →0
with Z =⊕r (C/(z )C)ni and G -dim (N)6r and N ∈B (R).
i=0 [i] C R C
(2) If P -pd (M) = r < ∞, then there exist an R-regular sequence z =
C R
z ,...,z , integers n ,n ,...,n >0 with n >1, and an exact sequence
1 r 0 1 r r
0→M →Z →N →0
with Z =⊕r (C/(z )C)ni and P -pd (N)6r.
i=0 [i] C R
The next result we extend is the following:
Theorem 1.3 ([14, Theorem 4.2]). Let R be a Cohen-Macaulay ring with a point-
wise dualizing module ω. Then, for any non-zero finitely generated R-module M
withlocallyfiniteinjectivedimension,thereexistanintegerr =pd (Hom (ω,M)),
R R
an R-regular sequence z = z ,...,z , non-negative integers n ,n ,...,n with
1 r 0 1 r
n >1, and an exact sequence
r
0→M →Z →N →0
with Z =⊕r (ω/(z )ω)ni, such that N has locally finite injective dimension and
i=0 [i]
pd (Hom (ω,N))6r.
R R
WeextendthisintwodirectionsinTheorems1.4and1.5. First,wehavethever-
sionfor G-injective dimension. Second, we giveversionsrelativeto a semidualizing
module. These are proved in Proofs 3.5, 3.12, and 3.16.
Theorem 1.4. Let R be a Cohen-Macaulay ring with a pointwise dualizing module
ω. For any non-zero finitely generated R-module M with locally finite G-injective
dimension,thereexistanintegerr =G-dim (Hom (ω,M)),anR-regularsequence
R R
z = z ,...,z , non-negative integers n ,n ,...,n with n > 1, and an exact
1 r 0 1 r r
sequence
0→M →Z →N →0
2It is worth noting that these notions appeared (more or less) implicitly in several places.
However, to the best of our knowledge these are the first places where GC-dimension and PC-
projectivedimensionwerestudiedexplicitly.
EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION 3
with Z = ⊕r (ω/(z )ω)ni such that N has locally finite G-injective dimension
i=0 [i]
and G-dim (Hom (ω,N))6r.
R R
Theorem 1.5. Let R be a Cohen-Macaulay ring with a pointwise dualizing module
ω. Let M a non-zero finitely generated R-module, and let C be a semidualizing
R-module. Set C† =Hom (C,ω). Then
R
(1) If M has locally finite GI -id and M is in the Auslander class A (R), then
C C
there exist an integer r =G-dim(Hom (ω,C⊗ M)), an R-regular sequence
R R
z =z ,...,z , non-negative integers n ,n ,...,n with n >1, and an exact
1 r 0 1 r r
sequence
0→M →Z →N →0
such that N has locally finite GI -id, and Z = ⊕r (C†/(z )C†)ni, N ∈
C i=0 [i]
A (R) and G-dim(Hom (ω,C⊗ N))6r.
C R R
(2) If M has locally finite I -id, then there exist an integer
C
r =pd (Hom (ω,C⊗ M))
R R R
an R-regular sequence z = z ,...,z , non-negative integers n ,n ,...,n
1 r 0 1 r
with n >1, and an exact sequence
r
0→M →Z →N →0
such that N has locally finite I -id, and Z = ⊕r (C†/(z )C†)ni and
C i=0 [i]
pd (Hom (ω,C⊗ N))6r.
R R R
We summarize the organization of this paper. Section 2 contains the proof of
Theorem 1.2, along with the necessary background material. Section 3 similarly
treats Theorems 1.4 and 1.5; this section also contains severallemmas about mod-
ules of locally finite G-injective dimension, locally finite G -injective dimension,
C
and locally finite I -injective dimension where C is a semidualizing module, as
C
these notions have not been developed in the literature as best we know.3 Note
that we do not develop the background material in the most optimal manner, fo-
cusing instead on accessibility. To be specific, we present background material as
it is needed in Sections 2 and 3 instead of putting it all in its own section. Also,
we feel that the definition of locally finite GI -id is a bit much to swallow on the
C
first bite, so we first discuss the special case of G-id and work our way up.
2. Proof of Theorem 1.2
Notation2.1. Weletpd (−)andid (−)denotetheclassicalprojectivedimension
R R
and the classical injective dimension, respectively.
The study of semidualizing modules was initiated independently (with different
names) by Foxby [9], Golod [12], and Vasconcelos [20]. For example, a finitely
generated projective R-module of rank 1 is semidualizing.
Definition 2.2. A finitely generated R-module C is semidualizing if the natural
map R → Hom (C,C) is an isomorphism, and Ext>1(C,C) = 0. An R-module
R R
C is pointwise dualizing if it is semidualizing and id (C ) < ∞ for all maximal
Rm m
ideals m⊂R.4 An R-module C is dualizing if it is semidualizing and id (C)<∞.
R
3Thscanbepartiallyexplainedbythefactthat,atthetimeofthewritingofthisarticle,the
localizationquestionforG-injectivedimensionisstillnotcompletelyanswered. Remark3.2.
4In[14]theterm“globalcanonical module”isusedinplaceof“pointwisedualizingmodule”.
OurchoicefollowstheterminologyofGrothendieck andHartshorne[13].
4 SEANSATHER-WAGSTAFF
Fact 2.3.
(a) It is straightforwardto show that the semidualizing property is local. That
is, if C is a semidualizing R-module, then for each multiplicatively closed subset
U ⊆ R the localization U−1C is a semidualizing U−1R-module; and conversely, if
M is a finitely generated R-module such that M is a semidualizing R -module
m m
for each maximal ideal m ⊂ R, then M is a semidualizing R-module. Similarly,
if C is a (pointwise) dualizing module for R, then for each multiplicatively closed
subset U ⊆ R the localization U−1C is a (pointwise) dualizing module for U−1R;
and conversely, if M is a finitely generated R-module such that M is a dualizing
m
module for R for each maximal ideal m ⊂ R, then M is a pointwise dualizing
m
module for R. Moreover, an R-module M is a dualizing module for R if and only
if it is a pointwise dualizing module for R and the Krull dimension of R is finite.
∼
(b) Let C be a semidualizing R-module. The isomorphism R = HomR(C,C)
impliesthatSupp (C)=Spec(R)andAss (C)=Ass(R). Thus,anelementx∈R
R R
isC-regularifandonlyifitisR-regular. WhenxisR-regular,thequotientC/xC is
a semidualizing R/xR-module. Thus, by induction, a sequence z = z ,...,z ∈ R
1 r
is R-regular if and only if it is C-regular; see [11, Theorem 4.5]. Also, from [16,
Proposition 3.1] we know that, for all R-modules M 6= 0, we have C⊗ M 6= 0 6=
R
Hom (C,M).
R
The next categories were introduced by Foxby [10] when C is dualizing, and by
Vasconcelos [20, §4.4] for arbitrary C, with different notation.
Definition 2.4. Let C be a semidualizing R-module. The Auslander class of C is
theclassA (R)ofR-modulesM suchthatTorR (C,M)=0=Ext>1(C,C⊗ M),
C >1 R R
and the natural map M → Hom (C,C ⊗ M) is an isomorphism. The Bass
R R
class of C is the class B (R) of R-modules N such that Ext>1(C,M) = 0 =
C R
TorR (C,Hom (C,M)), and the natural evaluation map C ⊗ Hom (C,N) →N
>1 R R R
is an isomorphism.
Remark 2.5. It is straightforward to show that the Auslander and Bass classes
for C =R are trivial: A (R) and B (R) both contain all R-modules.
R R
The following notion was introduced and studied by Holm and Jørgensen [15]
and White [21]. Note that in the special case C = R, we recover the class of
projective R-modules and the classical projective dimension.
Definition2.6. LetC beasemidualizingR-module. LetP (R)denotetheclassof
C
modulesM ∼=P⊗RC withP projective. ModulesinPC(R)arecalledC-projective.
We let P -pd (−) denote the homologicaldimension obtainedfrom resolutionsin
C R
P (R), with the convention that P -pd (0)=−∞.5
C C R
Fact 2.7. Let C be a semidualizing R-module. The Bass class B (R) contains
C
all R-modules of finite P -pd; see [16, Lemmas 4.1 and 5.1, and Corollary 6.3].
C
GivenanR-moduleM,onehasP -pd (M)=pd (Hom (C,M))andpd (M)=
C R R R R
P -pd (C⊗ M)by [19,Theorem2.11]. Inparticular,onehasP -pd (M)<∞
C R R C R
if and only if pd (Hom (C,M)) < ∞, and one has pd (M) < ∞ if and only if
R R R
P -pd (C⊗ M)<∞.
C R R
Proof 2.8 (Proof of Theorem 1.2(2)). Assume that M is a non-zero finitely gen-
erated R-module such that P -pd (M) = r < ∞. Then Fact 2.7 implies that
C R
5Weobservethesameconventionforanyhomologicaldimensionofthezeromodule.
EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION 5
pd (Hom (C,M)) = r < ∞. Since M 6= 0, we have Hom (C,M) 6= 0, by
R R R
Fact2.3(b), so Theorem1.1(2) providesanR-regularsequence z =z ,...,z , inte-
1 r
gers n ,n ,...,n >0 with n >1, and an exact sequence
0 1 r r
(2.8.1) 0→Hom (C,M)→Z′ →N′ →0
R
with Z′ = ⊕r (R/(z ))ni and pd (N′)6 r. In particular, we have N′ ∈ A (R)
i=0 [i] R C
which implies that TorR(C,N′) = 0. Thus, an application of C ⊗ − to the
1 R
sequence (2.8.1) yields the next exact sequence:
(2.8.2) 0→C⊗ Hom (C,M)→C⊗ Z′ →C⊗ N′ →0.
R R R R
Fact2.7impliesthatM ∈BC(R),sowehaveC⊗RHomR(C,M)∼=M. Theequality
Z′ =⊕r (R/(z ))ni implies that Z :=C⊗ Z′ =⊕r (C/(z )C)ni. Because of
i=0 [i] R i=0 [i]
Fact2.7,theconditionpd (N′)6rimpliesthatP -pd (C⊗ N′)6r. Thus,with
R C R R
N :=C⊗ N′, the sequence (2.8.2) satisfies the conclusionof Theorem 1.2(2). (cid:3)
R
The remainder of the proof of Theorem 1.2 is similar to the above proof, but it
requires a bit more technology.
Fact 2.9. Let C be a semidualizing R-module.
(a)TheAuslanderclassA (R)containsallR-modulesoffiniteprojectivedimen-
C
sion,andtheBassclassB (R)containsallR-modulesoffinite injectivedimension;
C
see [16, Lemmas 4.1 and 5.1, and Corollary 6.3]. Moreover, it is straightforward
to show that the Bass class satisfies the following local-global principal: For an
R-module M, the following conditions are equivalent:
(i) M ∈B (R);
C
(ii) U−1M ∈BU−1C(U−1R) for each multiplicatively closed subset U ⊆R;
(iii) M ∈B (R ) for each p∈Spec(R); and
p Cp p
(iv) M ∈B (R ) for each maximal ideal m∈Supp (M).
m Cm m R
ItfollowsthatB (R)containseveryR-moduleoflocallyfinite injectivedimension.
C
The Auslander class satisfies an analogous local-global principal.
(b)TheAuslanderandBassclassesalsosatisfythetwo-of-threepropertyby[16,
Corollary 6.3]. That is, given a short exact sequence 0 → M′ → M → M′′ → 0
of R-module homomorphisms,if two of the modules in the sequence are in A (R),
C
then so is the third module, and similarly for B (R).
C
(c) From [19, (2.8)] we know that an R-module M is in B (R) if and only if
C
Hom (C,M) ∈ A (R), and that M ∈ A (R) if and only if C ⊗ M ∈ B (R).
R C C R C
This is known as Foxby equivalence.
Definition 2.10. Let C be a semidualizing R-module. A finitely generated R-
module G is totally C-reflexive if the natural map G → Hom (Hom (G,C),C)
R R
is an isomorphism, and Ext>1(G,C) = 0 = Ext>1(Hom (G,C),C). Let G (R)
R R R C
denote the class of totally C-reflexive R-modules, and set G(C)=G (R)∩B (R).
C C
In the case C = R we use the more common terminology “totally reflexive” and
the notation G(R)=G (R)=G(R)∩B (R). We abbreviate as follows:
R R
G -dim (−)=the homological dimension obtained from resolutions in G (R)
C R C
G(C)-pd (−)=the homological dimension obtained from resolutions in G(C)
R
G-dim (−)=the homological dimension obtained from resolutions in G(R).
R
The following facts are included for perspective.
6 SEANSATHER-WAGSTAFF
Fact 2.11. Let C be a semidualizing R-module, and let M be a finitely gen-
erated R-module. Because of the containments P (R) ⊆ G(C) ⊆ G (R), one has
C C
G -dim (M)6G(C)-pd (M)6P -pd (M)withequalitytotheleftofanyfinite
C R R C R
quantity. In particular, the case C =R says that G-dim (M)=G(R)-pd (M)6
R R
pd (M) since G(R)=G (R), with equality holding when pd (M)<∞.
R R R
The next lemma explains how these homologicaldimensions are connected.
Lemma2.12([18,Lemma2.9]). LetC beasemidualizing R-module. Forafinitely
generated R-module M, the following conditions are equivalent:
(i) G(C)-pd (M)<∞;
R
(ii) G -dim (M)<∞ and M ∈B (R); and
C R C
(iii) G-dim (Hom (C,M))<∞ and M ∈B (R).
R R C
When these conditions are satisfied, we have
G(C)-pd (M)=G -dim (M)=G-dim (Hom (C,M)).
R C R R R
Now we are in position to complete the proof of Theorem 1.2.
Proof 2.13 (Proof of Theorem 1.2(1)). Assume that M is a non-zero finitely
generated R-module such that G -dim (M) = r < ∞ and M ∈ B (R). Then
C R C
Lemma 2.12 implies that G-dim (Hom (C,M)) = r < ∞. Since M 6= 0, we
R R
have Hom (C,M) 6= 0, by Fact 2.3(b), so Theorem 1.1(1) provides an R-regular
R
sequence z = z ,...,z , integers n ,n ,...,n > 0 such that n > 1, and an exact
1 r 0 1 r
sequence
(2.13.1) 0→Hom (C,M)→Z′ →N′ →0
R
with Z′ =⊕r (R/(z ))ni and G-dim (N′)6r.
i=0 [i] R
The condition M ∈ B (R) implies that Hom (C,M) ∈ A (R) by Fact 2.9(c),
C R C
and Fact 2.9(a) implies that Z′ ∈ A (R). Also from Fact 2.9(a)–(b), we conclude
C
that N′ ∈ A (R), which implies that TorR(C,N′) = 0. Thus, an application of
C 1
C⊗ − to the sequence (2.13.1) yields the next exact sequence:
R
(2.13.2) 0→C⊗ Hom (C,M)→C⊗ Z′ →C⊗ N′ →0.
R R R R
The assumption M ∈BC(R) gives an isomorphism C⊗RHomR(C,M)∼=M. The
equality Z′ =⊕r (R/(z ))ni implies that Z :=C⊗ Z′ =⊕r (C/(z )C)ni.
i=0 [i] R i=0 [i]
The condition N′ ∈A (R) implies that N :=C⊗ N′ ∈B (R) by Fact 2.9(c),
C R C
and N′ ∼= HomR(C,N) by the definition of what it means for N′ to be in AC(R).
Thus, because of Lemma 2.12, we have
G -dim (N)=G-dim(Hom (C,N))=G-dim (N′)6r.
C R R R
Thus, the sequence (2.13.2) satisfies the conclusion of Theorem 1.2(1). (cid:3)
3. Proofs of Theorems 1.4 and 1.5
We continue with a definition due to Enochs and Jenda [7].
Definition 3.1. Acomplete injective resolution is anexactcomplexY ofinjective
R-modules such that Hom (J,Y) is exact for each injective R-module J. An R-
R
module N is G-injective if there exists a complete injective resolution Y such that
N ∼=Ker(∂Y), andY isacomplete injective resolution of N. LetGI(R)denotethe
0
classofG-injective R-modules,andlet Gid (−)denote the homologicaldimension
R
obtained from coresolutions in GI(R). We say that an R-module M has locally
EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION 7
finite G-injective dimension provided that Gid (M ) < ∞ for each maximal
Rm m
ideal m⊂R.
Remark 3.2. Using existing technology, the modules of finite G-injective dimen-
sion behave best when R is Cohen-Macaulay with a dualizing module.6 For in-
stance, in general it is not known whether the G-injective dimension localizes; but
itis knownto localizewhenR has a dualizingmodule. This is due to the following
connection with Bass classes, the local case of which is from [8].7
Lemma 3.3. Assume that R is Cohen-Macaulay with a pointwise dualizing mod-
ule. Let M be an R-module. Then an R-module M has locally finite G-injective
dimension if and only if M ∈B (R).
ω
Proof. If R is local, then the result follows from [8, Proposition 1.4 and Theo-
rems 1.6 and 2.5]. By definition, the condition “M has locally finite G-injective
dimension”isalocalcondition. Fact2.9(b)showsthatthecondition“M ∈B (R)”
ω
is also a local condition. Note that Fact 2.3(a) implies that for each maximal ideal
m ⊂ R, the localization ω is a dualizing module for the Cohen-Macaulay local
m
ring R . Thus, the general result follows from the local case. (cid:3)
m
The next result shows that the G-dimension of a finitely generated module be-
haves like its projective dimension.
Lemma 3.4. Let M be a finitely generated R-module. The following conditions
are equivalent:
(i) G-dim (M)<∞;
R
(ii) G-dim (M )<∞ for each maximal ideal m⊂R.
Rm m
When R is Cohen-Macaulay with a pointwise dualizing module ω, these conditions
are equivalent to the following:
(iii) M ∈A (R).
ω
Proof. The implication (i) =⇒ (ii) follows from the inequality G-dim (M ) 6
Rm m
G-dim (M), which is straightforward to verify. The implication (ii) =⇒ (i) is
R
from[4,Theorem3.3]. WhenRisCohen-Macaulaywithapointwisedualizingmod-
uleω,theequivalence(ii) ⇐⇒(iii)followsfromthelocalresults[8,Proposition1.3
and Theorems 1.6 and 2.1] as in the proof of Lemma 3.3.8 (cid:3)
6Most good behavior is also known when R has a dualizing complex, but we restrict our
attention to the Cohen-Macaulay case. For instance, the conclusion of Lemma 3.3 holds when
R is only assumed to have a pointwise dualizing complex; the proof is the same, using results
from[6]inplaceoftheresultsfrom[8].
7It is worth noting that the results in [8] assume that R is local, hence with finite Krull
dimension; also, the results of [6] assume implicitly that R has finite Krull dimension since the
definitionofadualizingcomplexusedthereincludesanassumptionoffiniteinjectivedimension.
Contrastthiswithourdefinitionofpointwisedualizingmodule,andwithGrothendieck’sdefinition
ofapointwisedualizingcomplexfrom[13].
8Note that this uses the characterization of totally reflexive modules in terms of “complete
resolutions”foundin[3,(4.4.4)];seealso[5,Theorem3.1]. Specifically,anR-moduleM istotally
reflexive if and only if there is an exact complex F = ··· −∂−1F→ F0 −∂−0F→ F−1 −∂−−F−→1 ··· such that
HomR(F,R) is exact and M ∼= Im(∂0F). Note that the local result was also announced in [10,
Corollary3.3].
8 SEANSATHER-WAGSTAFF
Proof 3.5 (Proof of Theorem 1.4). Assume that R is Cohen-Macaulay with a
pointwisedualizingmoduleωandthatM isanon-zerofinitelygeneratedR-module
with locallyfinite G-injective dimension. Lemma 3.3implies that M ∈B (R). Us-
ω
ing Foxby equivalence (from Fact 2.9(c)) we conclude that Hom (ω,M)∈A (R),
R ω
so Lemma 3.4 implies that r := G-dim (Hom (ω,M)) < ∞. The fact that R is
R R
noetherianimpliesthatHom (ω,M)isfinitely generated. (The proofconcludesas
R
in Proof 2.13. We include the details for the sake of thoroughness.)
Since M 6= 0, we have Hom (ω,M) 6= 0, by Fact 2.3(b), so Theorem 1.1(1)
R
impliesthatthereareanR-regularsequencez =z ,...,z ,integersn ,n ,...,n >
1 r 0 1 r
0 with n >1, and an exact sequence
r
(3.5.1) 0→Hom (ω,M)→Z′ →N′ →0
R
with Z′ = ⊕r (R/(z ))ni and G-dim (N′) 6 r. Fact 2.9(a) implies that Z′ ∈
i=0 [i] R
A (R). From Fact 2.9(b), we conclude that N′ ∈ A (R), which implies that
ω ω
TorR(ω,N′)=0. Thus,anapplicationofω⊗ − to the sequence (3.5.1)yields the
1 R
next exact sequence:
(3.5.2) 0→ω⊗ Hom (ω,M)→ω⊗ Z′ →ω⊗ N′ →0.
R R R R
The assumption M ∈ Bω(R) gives an isomorphism ω⊗RHomR(ω,M) ∼= M. The
equality Z′ =⊕r (R/(z ))ni implies that Z :=ω⊗ Z′ =⊕r (ω/(z )ω)ni.
i=0 [i] R i=0 [i]
The condition N′ ∈A (R) implies that N :=ω⊗ N′ ∈B (R) by Fact 2.9(c),
ω R ω
and N′ ∼= HomR(ω,N) by the definition of what it means for N′ to be in Aω(R).
Thus, we have G-dim(Hom (ω,N)) = G-dim (N′) 6 r. So, the sequence (3.5.2)
R R
satisfies the conclusion of Theorem 1.4. (cid:3)
Forournextresults,weneedabetterunderstandingoftherelationshipbetween
semidualizing modules and a pointwise dualizing module.
Lemma 3.6. Assumethat R is Cohen-Macaulay with a pointwise dualizing module
ω. If C is a semidualizing R-module, then so is Hom (C,ω).
R
Proof. Inthelocalcase,thisresultisstandard;see,e.g.,[17,Facts1.18–1.19]. Since
the semidualizingand(pointwise)dualizingconditionsarelocalby Fact2.3(a),the
general case of the result follows. (cid:3)
The next two lemmas elaborate on the local-global behavior for our invariants.
Lemma 3.7. Let M be a finitely generated R-module, and let C be a semidualizing
R-module. Then P -pd (M) < ∞ if and only if P -pd (M ) < ∞ for each
C R Cm Rm m
maximal ideal m⊂R.
Proof. The forward implication follows from the inequality P -dim (M ) 6
Cm Rm m
P -pd (M), which is straightforward to verify. For the converse, assume that
C R
P -pd (M ) < ∞ for each maximal ideal m ⊂ R. It follows that the module
HCommR(CR,mM)mm∼= HomRm(Cm,Mm) has finite projective dimension over Rm for
all m. Hence, the finitely generated R-module Hom (C,M) has finite projective
R
dimension, so Fact 2.7 implies that P -pd (M) is finite. (cid:3)
C R
The nextlemma is asouped-up versionof a specialcaseofa resultof Takahashi
and White that is documented in [18].
Lemma 3.8. Assumethat R is Cohen-Macaulay with a pointwise dualizing module
ω. For each finitely generated R-module M, one has P -pd (M)<∞ if and only
ω R
if M has locally finite injective dimension.
EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION 9
Proof. Whenω isadualizingmoduleforR,i.e.,whenRhasfiniteKrulldimension,
this result is from [18, Lemma 2.7]. In particular, this takes care of the local case.
The general case follows from the local case, by Lemma 3.7. (cid:3)
Here are some more notions from [15].
Definition 3.9. Let C be a semidualizing R-module, and let I (R) denote the
C
class of modules N ∼= HomR(C,I) with I injective. Modules in IC(R) are called
C-injective. We let I -id (−) denote the homological dimension obtained from
C R
coresolutions in I (R). We say that an R-module M has locally finite I -id pro-
C C
vided that I -id (M )<∞ for each maximal ideal m⊂R.
Cm Rm m
Fact 3.10. Let C be a semidualizing R-module. The Auslander class A (R)
C
contains every R-module of finite I -injective dimension; see [16, Lemmas 4.1 and
C
5.1]. Given an R-module M, one has I -id (M) = id (C ⊗ M) and id (M) =
C R R R R
I -id (Hom (C,M)) by[19, Theorem2.11]. Inparticular,one hasI -id (M)<
C R R C R
∞ if and only if id (C ⊗ M) < ∞, and one has id (M) < ∞ if and only if
R R R
I -id (Hom (C,M))<∞. It follows that M has locally finite I -id if and only
C R R C
if C⊗ M has locally finite injective dimension, and M has locally finite injective
R
dimension if and only if Hom (C,M) has locally finite I -id.
R C
Lemma 3.11. Assume that R is Cohen-Macaulay with a pointwise dualizing mod-
ule ω. Let C be a semidualizing R-module, and let x = x ,...,x ∈ R be an R-
1 i
regular sequence. Then HomR(C,ω/(x)ω)∼=C†/(x)C† where C† =HomR(C,ω).
Proof. Let K be the Koszul complex KR(x), and let K+ denote the augmented
Koszul complex
K+ =(0→R→···→R→ R/(x) →0).
| {z }
deg. −1
Since x is R-regular, K+ is an exact sequence of R-modules of finite projective
dimension. Thus, each module K+ is in A (R), so the induced complex
j ω
ω⊗ K+ =(0→ω →···→ω →ω/(x)ω →0)
R
| {z }
deg. −1
isanexactsequenceofR-moduleslocallyoffiniteinjectivedimension. Inparticular,
themodulesinthisexactsequenceareinB (R),sothenextsequenceisalsoexact:
C
Hom (C,ω⊗ K+)
R R
=(0→Hom (C,ω)→···→Hom (C,ω)→Hom (C,ω/(x)ω)→0).
R R R
| {z }
deg. −1
Itisstraightforwardtoshowthatthedifferentialonthissequenceinpositivedegrees
is the same as the differential on C†⊗ K. It follows that
R
HomR(C,ω/(x)ω)∼=H0(C†⊗RK)∼=C†/(x)C†
as desired. (cid:3)
Proof 3.12 (Proof of Theorem 1.5(2)). Assume that R is Cohen-Macaulay with
a pointwise dualizing module ω and that M is a non-zero finitely generated R-
module with locally finite I -id. Fact 3.10 implies that C⊗ M has locally finite
C R
injective dimension. Since M 6= 0, we have C ⊗ M 6= 0, by Fact 2.3(b), so
R
Theorem 1.3 provides an integer r =pd (Hom (ω,C⊗ M)), a proper R-regular
R R R
10 SEANSATHER-WAGSTAFF
sequence z = z ,...,z , non-negative integers n ,n ,...,n with n > 1, and an
1 r 0 1 r r
exact sequence
(3.12.1) 0→C⊗ M →Z′ →N′ →0
R
with Z′ = ⊕r (ω/(z )ω)ni, where N′ has locally finite injective dimension and
i=0 [i]
pd (Hom (ω,N′))6r.
R R
As C ⊗ M has locally finite injective dimension, we have C ⊗ M ∈ B (R)
R R C
by Fact 2.9(b), so Ext1R(C,C ⊗RM) = 0 and HomR(C,C ⊗RM) ∼= M. Thus, an
applicationofHom (C,−)to the sequence(3.12.1)yields the nextexactsequence:
R
(3.12.2) 0→M →Hom (C,Z′)→Hom (C,N′)→0.
R R
SinceN′ haslocallyfiniteinjectivedimension,Fact3.10impliesthattheR-module
N := Hom (C,N′) has locally finite I -id. With Z := Hom (C,Z′), it remains
R C R
to show that Z ∼=⊕ri=0(C†/(z[i])C†)ni and pdR(HomR(ω,C⊗RN))6r. The first
of these follows from the next sequence of isomorphisms
Z ∼=HomR(C,⊕ri=0(ω/(z[i])ω)ni)
∼=⊕ri=0HomR(C,ω⊗R(R/(z[i]))ni
∼=⊕r (C†/(z )C†)ni
i=0 [i]
where the last isomorphism is from Lemma 3.11. To complete the proof, observe
that since N′ has locally finite injective dimension, it is in B (R), so we have
C
N′ ∼=C⊗RN. ThisimpliesthatpdR(HomR(ω,C⊗RN))=pdR(HomR(ω,N′))6r,
as desired. (cid:3)
In the next definition, the special case C = R recovers the complete injective
resolutions and Gorenstein injective modules.
Definition 3.13. Let C be a semidualizing R-module. A complete I I-coresolu-
C
tion is a complex Y of R-modules such that Y is exact and Hom (U,Y) is exact
R
for eachU ∈I (R), Y is injective for i60, and Y is C-injective for i>0. An R-
C i i
module N is G -injective if there exists a complete I I-coresolution Y such that
C C
N ∼= Ker(∂0Y), and Y is a complete ICI-coresolution of N. Let GIC(R) denote
the class of G -injective R-modules, and set G(I ) = GI (R)∩A (R). We let
C C C C
GI -id (−) and G(I )-id (−) denote the homologicaldimensions obtained from
C R C R
coresolutions in GI (R) and G(I ), respectively.
C C
An R-module M has locally finite G(I )-id providedthat G(I )-id (M )<
C Cm Rm m
∞ for each maximal ideal m ⊂ R. An R-module M has locally finite GI -id
C
provided that GI -id (M )<∞ for each maximal ideal m⊂R.
Cm Rm m
The next lemma explains the relationbetweenthe different Gorensteininjective
dimensions.
Lemma 3.14 ([18, Lemma 2.10]). Let C be a semidualizing R-module. For an
R-module M, the following conditions are equivalent:
(i) G(I )-id (M)<∞;
C R
(ii) GI -id (M)<∞ and M ∈A (R); and
C R C
(iii) Gid (C⊗ M)<∞ and M ∈A (R).
R R C
When these conditions are satisfied, we have
G(I )-id (M)=GI -id (M)=Gid (C⊗ M).
C R C R R R