Table Of ContentFOXBY EQUIVALENCE, LOCAL DUALITY AND GORENSTEIN HOMOLOGICAL
DIMENSIONS
FATEMEHMOHAMMADIAGHJEHMASHHADANDKAMRANDIVAANI-AAZAR
Abstract. Let (R,m)be alocal ringand (−)∨ denote the Matlis duality functor. We investigate the
2 relationshipbetweenFoxbyequivalenceandlocaldualitythroughgeneralizedlocalcohomologymodules.
1 AssumethatRpossessesanormalizeddualizingcomplexDandXandY aretwohomologicallybounded
0
complexes of R-modules with finitely generated homology modules. We present several duality results
2
form-sectioncomplexRΓm(RHomR(X,Y)). Inparticular,ifG-dimensionofX andinjectivedimension
n
ofY arefinite,thenweshowthat
a
J RΓm(RHomR(X,Y))≃(RHomR(Y,D⊗LRX))∨.
0
1 Wededuce severalapplications ofthesedualityresults. Inparticular,weestablishGrothendieck’s non-
vanishingTheoreminthecontext ofgeneralizedlocalcohomology modules.
]
C
A
.
h 1. Introduction
t
a
m Let(R,m)be a Cohen-Macaulaylocalring with a canonicalmodule ω andP (resp. I) denote the full
[ subcategory of finitely generatedR-modules of finite projective (resp. injective) dimension. By virtue of
3 [Sh, Theorem 2.9], there is the following equivalence of categories
v
0 ω⊗R− //
7 P oo I .
7 HomR(ω,−)
5
. LetM andN betwofinitelygeneratedR-modulesandianon-negativeinteger. DenotetheMatlisduality
6
0 functor Hom (−,E(R/m)) by (−)∨. If M has finite projective dimension, then by Suzuki’s Duality
R
10 Theorem[Su,Theorem3.5],thereisanaturalisomorphismHmi (M,N)∼=ExtRdimR−i(N,ω⊗RM)∨.Also,
: if N has finite injective dimension, then the Herzog-Zamani Duality Theorem [HZ, Theorem 2.1 b)]
v
Xi asserts that Hmi(M,N)∼=ExtRdimR−i(HomR(ω,N),M)∨. (These results can be considered as variants of
the Local Duality Theorem [BS, 11.2.8] in the context of generalized local cohomology modules.) Hence
r
a
theequivalencebetweentwosubcategoriesP andI canbeconnectedtolocaldualitythroughgeneralized
local cohomology modules.
Now, assume that (R,m) is a local ring with a normalized dualizing complex D. By [CFrH, Theorem
4.1andPrposition3.8b)](resp. [CFrH,Theorem4.4])AuslandercategoryAf(R)(resp. Bf(R))consists
exactlyofallhomologicallyboundedcomplexesofR-moduleswithfinitelygeneratedhomologymodulesof
finite G-dimension(resp. Gorensteininjective dimension). By Foxbyequivalence,there is anequivalence
2010 Mathematics Subject Classification. 13D45,13D05,13D07,13C14.
Key words and phrases. Auslandercategories, Foxbyequivalence, G-dimension,generalizedlocalcohomology modules,
Gorensteininjectivedimension,localduality,normalizeddualizingcomplexes, totallyreflexivemodules.
ThesecondauthorwassupportedbyagrantfromIPM(No. 89130118).
1
2 F.MOHAMMADIAGHJEHMASHHADANDK.DIVAANI-AAZAR
of categories
D⊗LR− //
Af(R) oo Bf(R) ;
RHomR(D,−)
see e.g. [C, Theorem 3.3.2 a), b), e) and f)]. Foxby equivalence between the two categories Af(R) and
Bf(R) is a natural generalization of the above mentioned equivalence between the subcategories P and
I. Inview ofwhat we sawin the firstparagraph,itis naturalto ask whether Foxby equivalence canalso
be connected to local duality through generalizedlocal cohomology modules. Assume that X and Y are
two homologically bounded complexes of R-modules with finitely generated homology modules and let i
be an integer. The following natural questions arise:
Question 1.1. Suppose that G-dimension of X is finite. Is Hi (X,Y)∼=Ext−i(Y,D⊗L X)∨?
m R R
Question 1.2. Suppose that Gorenstein injective dimension of Y is finite. Is
Hmi (X,Y)∼=Ext−Ri(RHomR(D,Y),X)∨?
Our main aim in this paper is to answer these questions. Example 3.6 below shows that the answers
of these questions are negative in general, but by adding some extra assumptions on the complexes X
and Y, we can deduce our desired natural isomorphisms. Consider the following assumptions:
a) Projective dimension of X is finite.
b) Projective dimension of Y is finite.
b’) Both G-dimension of X and projective dimension of Y are finite.
c) Both G-dimension of X and injective dimension of Y are finite.
d) Injective dimension of Y is finite.
e) Injective dimension of X is finite.
e’) Both Gorenstein injective dimension of Y and injective dimension of X are finite.
f) Both Gorenstein injective dimension of Y and projective dimension of X are finite.
We show that each of a), b) and c) implies the natural isomorphism
RΓ (RHom (X,Y))≃(RHom (Y,D⊗L X))∨,
m R R R
and each of d), e) and f) implies the natural isomorphism
RΓ (RHom (X,Y))≃(RHom (RHom (D,Y),X))∨.
m R R R
These immediately yield our desired isomorphisms Hi(X,Y) ∼= Ext−i(Y,D⊗L X)∨ and Hi (X,Y) ∼=
m R R m
Ext−i(RHom (D,Y),X)∨,respectively. ThesedualityresultsarefarreachinggeneralizationsofSuzuki’s
R R
Duality Theorem and the Herzog-ZamaniDuality Theorem.
We presentsome applications of the aboveduality results. Firstof all, we improvethe main results of
[HZ];seePropositions4.3and4.4below. ThenweestablishananalogueofGrothendieck’snon-vanishing
Theorem in the context of generalized local cohomology modules. Let (R,m) be a local ring and M,N
two finitely generated R-modules such that X := M,Y := N satisfy one of the above assumptions a),
b’), d), and e’). When R is Cohen-Macaulay,we show that
cd (M,N)=dimR−depth(Ann N,M).
m R
Finally, we give a partial generalization of the Intersection inequality; see Proposition 4.7 below.
FOXBY EQUIVALENCE, LOCAL DUALITY AND ... 3
2. Prerequisites
Throughout this paper, R is a commutative Noetherian ring with nonzero identity. The m-adic com-
pletion of an R-module M over a local ring (R,m) will be denoted by M.
c
(2.1) Hyperhomology. We will work within D(R), the derived categoryof R-modules. The objects
inD(R)arecomplexesofR-modulesandsymbol≃denotesisomorphismsinthiscategory. Foracomplex
X =···−→Xn+1 ∂−nX→+1 Xn −∂→nX Xn−1 −→···
in D(R), its supremum and infimum are defined, respectively, by supX := sup{i ∈ Z|H (X) 6= 0} and
i
infX := inf{i ∈ Z|H (X) 6= 0}, with the usual convention that sup∅ = −∞ and inf∅ = ∞. For an
i
integer ℓ, ΣℓX is the complex X shifted ℓ degrees to the left. Modules will be considered as complexes
concentratedindegreezeroandwedenotethefullsubcategoryofcomplexeswithhomologyconcentrated
indegreezerobyD (R). Thefullsubcategoryofcomplexeshomologicallyboundedtotheright(resp. left)
0
is denoted by D❂(R) (resp. D❁(R)). Also, the full subcategories of homologically bounded complexes
and of complexes with finitely generated homology modules will be denoted by D✷(R) and Df(R),
respectively. Throughout for any two properties ♯ and ♮ of complexes, we set D♮(R) := D (R)∩D♮(R).
♯ ♯
So for instance, D✷f(R) stands for the full subcategory of homologically bounded complexes with finitely
generated homology modules.
For any complex X in D❂(R) (resp.D❁(R)), there is a bounded to the right (resp. left) complex
P (resp. I) consisting of projective (resp. injective) R-modules which is isomorphic to X in D(R).
A such complex P (resp. I) is called a projective (resp. injective) resolution of X. A complex X is
said to have finite projective (resp. injective) dimension, if X possesses a bounded projective (resp.
injective) resolution. Similarly, a complex X is said to have finite flat dimension if it is isomorphic
L
(in D(R)) to a bounded complex of flat R-modules. The left derived tensor product functor −⊗ ∼
R
is computed by taking a projective resolution of the first argument or of the second one. The right
derived homomorphismfunctor RHom (−,∼) is computed by taking a projective resolution of the first
R
argument or by taking an injective resolution of the second one. For any two complexes X and Y
and any integer i, set ExtiR(X,Y) := H−i(RHomR(X,Y)). Let X be a complex and a an ideal of
R. Recall that SuppRX := ∪l∈ZSuppRHl(X), depth(a,X) := −supRHomR(R/a,X) and dimRX :=
sup{dimR/p− infX |p ∈ SpecR}. For any complexes X,Y and Z, there are the following natural
p
isomorphisms in D(R).
Shifts: Leti,j betwointegers. ThenΣiX⊗LΣjY ≃Σj+i(X⊗LY)andRHom (ΣiX,ΣjY)≃
R R R
Σj−iRHom (X,Y).
R
L L
Commutativity: X ⊗ Y ≃Y ⊗ X.
R R
Adjointness: Let S be an R-algebra. If X ∈D❂(S), Y ∈D(S) and Z ∈D❁(R), then
L
RHom (X ⊗ Y,Z)≃RHom (X,RHom (Y,Z)).
R S S R
Tensor evaluation: Assume thatX ∈D❂f(R), Y ∈D✷(R)andZ ∈D❂(R). If either projective
dimension of X or flat dimension of Z is finite, then
L L
RHom (X,Y)⊗ Z ≃RHom (X,Y ⊗ Z).
R R R R
4 F.MOHAMMADIAGHJEHMASHHADANDK.DIVAANI-AAZAR
Hom evaluation: Assume that X ∈ D❂f(R), Y ∈ D✷(R) and Z ∈ D❁(R). If either projective
dimension of X or injective dimension of Z is finite, then
L
X ⊗ RHom (Y,Z)≃RHom (RHom (X,Y),Z).
R R R R
(2.2)Gorenstein homological dimensions. AnR-moduleM issaidtobe totally reflexive ifthere
exists an exact complex P of finitely generated projective R-modules such that M ∼= im(P0 −→ P−1)
andHom (P,R)isexact. Also,anR-moduleN issaidtobe Gorenstein injective ifthereexistsanexact
R
complex I of injective R-modules such that N ∼=im(I1 −→I0) and HomR(E,I) is exact for all injective
R-modulesE;see[EJ]. Obviously,anyfinitelygeneratedprojectiveR-moduleistotallyreflexiveandany
injective R-moduleis Gorensteininjective. A complexX ∈D✷f(R)issaidtohavefinite G-dimensionifit
isisomorphic(inD(R))toaboundedcomplexoftotallyreflexiveR-modules. Also,acomplexX ∈D✷(R)
is said to have finite Gorenstein injective dimension if it is isomorphic (in D(R)) to a bounded complex
of Gorenstein injective R-modules.
(2.3) Auslander categories. Let (R,m) be a local ring. A normalized dualizing complex for R is
a complex D ∈ D✷f(R) such that the homothety morphism R −→ RHomR(D,D) is an isomorphism
in D(R), D has finite injective dimension and supD = dimR. Assume that R possesses a normalized
dualizingcomplexD. TheAuslandercategoryAf(R)(withrespecttoD)isthefullsubcategoryofD✷f(R)
whose objects are exactly those complexes X ∈ D✷f(R) for which D ⊗LR X ∈ D✷f(R) and the natural
L
morphism η : X −→ RHom (D,D⊗ X) is an isomorphism in D(R). Also, the Auslander category
X R R
Bf(R) (with respect to D) is the full subcategory of D✷f(R) whose objects are exactly those complexes
X ∈D✷f(R)forwhichRHomR(D,X)∈D✷f(R)andthenaturalmorphismεX :D⊗LRRHomR(D,X)−→
X is an isomorphism in D(R). By [CFrH, Theorem 4.1 and Prposition 3.8 b)], Af(R) precisely consists
ofallcomplexesX ∈D✷f(R)whoseG-dimensionsarefinite. Also,[CFrH,Theorem4.4]yieldsthatBf(R)
consists of all complexes X ∈D✷f(R) whose Gorenstein injective dimensions are finite.
(2.4) Local cohomology. Let a be an ideal of R. The right derived functor of a-section functor
Γa(−) = l−i→mnHomR(R/an,−) is denoted by RΓa(−). For any complex X ∈ D❁(R), the complex
RΓa(X)∈D❁(R)isdefinedbyRΓa(X):=Γa(I),whereI isan(every)injectiveresolutionofX. Also,for
any twocomplexes X ∈D❂(R) andY ∈D❁(R), the generalizeda-sectioncomplex RΓa(X,Y) is defined
by RΓa(X,Y) := RΓa(RHomR(X,Y)); see [Y]. For any integer i, set Hai(X,Y) := H−i(RΓa(X,Y))
and denote sup{i ∈ Z|Hi(X,Y) 6= 0} by cd (X,Y). Let M and N be two R-modules. The notion of
a a
generalized local cohomology modules Hi(M,N) := lim Exti (M/anM,N) was introduced by Herzog
a −→n R
in his Habilitationsschrift [He]. When M is finitely generated, [Y, Theorem 3.4]yields that Hi(M,N)∼=
a
H−i(RΓa(M,N)) for all integers i.
LetCˇ(a)denotetheCˇechcomplexonaseta={x ,x ,...,x }ofgeneratorsofa. So,bythedefinition,
1 2 n
Cˇ(a)=Cˇ(x )⊗ ···⊗ Cˇ(x ),whereforeachi,Cˇ(x )isthecomplex0−→R−→R −→0concentrated
1 R R n i xi
in degrees 0 and −1 in which homomorphisms are the natural ones. For any complex X ∈ D❁(R), [Sc,
Theorem 1.1 iv)] implies that RΓ (X)≃X ⊗L Cˇ(a).
a R
3. Duality Results
We start by proving two lemmas which are needed in the proof of the main result of this section.
Lemma 3.1. Let (R,m) be a local ring possessing a normalized dualizing complex D and X,Y ∈D✷f(R).
FOXBY EQUIVALENCE, LOCAL DUALITY AND ... 5
i) Assume that one of the following conditions holds:
a) either projective dimension X or Y is finite,
b) both G-dimension of X and injective dimension of Y are finite.
Then
L L
X ⊗ RHom (Y,D)≃RHom (Y,D⊗ X).
R R R R
ii) Assume that one of the following conditions holds:
a) either injective dimension Y or X is finite,
b) both Gorenstein injective dimension of Y and projective dimension of X are finite.
Then
L
X ⊗ RHom (Y,D)≃RHom (RHom (D,Y),X).
R R R R
L
Proof. i) The case a) follows immediately by using commutativity of the bivariant functor −⊗ ∼ and
R
tensor evaluation. Assume that b) holds. Since Y ∈Bf(R), we have RHomR(D,Y)∈D✷f(R). As Y has
finiteinjectivedimension,[C,Theorem3.3.2d)]and[C,TheoremA.5.7.2]implythatRHom (D,Y)has
R
finite projective dimension. Next, as Y ∈ Bf(R) and RHom (D,D) ≃ R, [C, Lemma 3.3.3 b)] yields
R
that RHom (Y,D) ≃ RHom (RHom (D,Y),R). Now, tensor evaluation, the fact X ∈ Af(R) and
R R R
[C, Lemma 3.3.3 b)] yield that:
L L
X⊗ RHom (Y,D) ≃X ⊗ RHom (RHom (D,Y),R)
R R R R R
L
≃RHom (RHom (D,Y),R)⊗ X
R R R
≃RHom (RHom (D,Y),X)
R R
L
≃RHom (RHom (D,Y),RHom (D,D⊗ X))
R R R R
L
≃RHom (Y,D⊗ X).
R R
ii)AssumethatGorensteininjectivedimensionofY isfinite. ThenY ∈Bf(R),andsoRHom (D,Y)∈
R
D✷f(R). As, we saw in i), we have
L L
X ⊗ RHom (Y,D)≃RHom (RHom (D,Y),R)⊗ X.
R R R R R
As we mentioned above,the finiteness ofinjective dimensionofY, implies that RHom (D,Y)has finite
R
projective dimension. Thus b) andthe firstcase of a) follow by tensor evaluation. It remains to consider
the second case of a). So, assume that X has finite injective dimension. Now, as X ∈ Bf(R), by using
tensor evaluation and Hom evaluation, we can deduce that:
L L L
X ⊗RRHomR(Y,D) ≃(cid:0)D⊗RRHomR(D,X)(cid:1)⊗RRHomR(Y,D)
L L
≃D⊗R(cid:0)RHomR(Y,D)⊗RRHomR(D,X)(cid:1)
L L
≃D⊗RRHomR(cid:0)Y,D⊗RRHomR(D,X)(cid:1)
L
≃D⊗ RHom (Y,X)
R R
≃RHom (RHom (D,Y),X).
R R
(cid:3)
ThefirstassertionofthenextresultwasalreadyprovedbyFoxby[Fo2,Proposition6.1]. Forcomplete-
ness’sake,we include aneasyprooffor it. Recallthatfor acomplex Y ∈D❁(R), its injective dimension,
id Y, is defined by
R
idRY :=inf{sup{l∈Z|I−l 6=0}|I is an injective resolution of Y}.
6 F.MOHAMMADIAGHJEHMASHHADANDK.DIVAANI-AAZAR
Lemma 3.2. Let a be an ideal of R. Let X ∈ D❂f(R) and Y ∈ D✷(R). Then RΓa(X,Y) ≃
RHom (X,RΓ (Y)). In particular, if X is homologically bounded and not homologically trivial, then
R a
cd (X,Y)≤id Y +supX.
a R
Proof. Letabeageneratingsetofa. AsCˇ(a)isaboundedcomplexofflatR-modules,tensorevaluation
property yields that
RΓ (X,Y)≃RHom (X,Y)⊗L Cˇ(a)≃RHom (X,RΓ (Y)).
a R R R a
Now, assume that X is homologically bounded and not homologically trivial. Since supX is an
integer,wemay anddoassume thatid Y <∞. So,there is a boundedcomplex I consistingofinjective
R
modules such that it is isomorphic to Y in D(R) and I = 0 for all j < −id Y. One has RΓ (X,Y)≃
j R a
RHom (X,Γ (I)). The complex Γ (I) is a bounded complex consisting of injective modules. Now by
R a a
[C, Corollary A.5.2], we have
id Y ≥id Γ (I)≥−supX −infRHom (X,Γ (I)).
R R a R a
Thus −infRΓ (X,Y)≤id Y +supX, as claimed. (cid:3)
a R
Now, we are ready to prove the main result of this section.
Theorem3.3. Let(R,m)bealocalringpossessinganormalizeddualizingcomplexDandX,Y ∈D✷f(R).
i) Assume that one of the following conditions holds:
a) either projective dimension X or Y is finite,
b) both G-dimension of X and injective dimension of Y are finite.
Then
L
RΓ (X,Y)≃Hom (RHom (Y,D⊗ X),E(R/m)).
m R R R
ii) Assume that one of the following conditions holds:
a) either injective dimension Y or X is finite,
b) both Gorenstein injective dimension of Y and projective dimension of X are finite.
Then
RΓ (X,Y)≃Hom (RHom (RHom (D,Y),X),E(R/m)).
m R R R
Proof. Denote the Matlis duality functor Hom (−,E(R/m)) by (−)∨. By Local Duality Theorem for
R
anycomplexZ ∈D✷f(R),weknowthatRΓm(Z)≃(RHomR(Z,D))∨,seee.g. [Ha,ChapterV,Theorem
6.2]. Using Lemma 3.2 and adjointness yields that:
RΓ (X,Y) ≃RHom (X,RΓ (Y))
m R m
≃RHom (X,RHom (Y,D)∨)
R R
≃(X ⊗L RHom (Y,D))∨.
R R
Hence Lemma 3.1 completes the proof. (cid:3)
Theorem 3.3 has the following immediate corollary.
Corollary 3.4. i) Let the situation be as in Theorem 3.3 i). Then
L
−infRΓ (X,Y)=supRHom (Y,D⊗ X)
m R R
FOXBY EQUIVALENCE, LOCAL DUALITY AND ... 7
and
L
−supRΓ (X,Y)=infRHom (Y,D⊗ X).
m R R
ii) Let the situation be as in Theorem 3.3 ii). Then
−infRΓ (X,Y)=supRHom (RHom (D,Y),X)
m R R
and
−supRΓ (X,Y)=infRHom (RHom (D,Y),X).
m R R
Thefirstpartofthe followingcorollaryextendsSuzuki’sDualityTheoremanditssecondpartextends
the Herzog-ZamaniDuality Theorem.
Corollary 3.5. Let (R,m) be a Cohen Macaulay local ring possessing a canonical module ω. Let M,N
be two finitely generated R-modules and i an integer.
i) Assume that one of the following conditions holds:
a) projective dimension of M is finite,
b) both G-dimension of M and injective dimension of N are finite.
Then
Hmi (M,N)∼=HomR(ExtRdimR−i(N,ω⊗RM),E(R/m)).
ii) Assume that one of the following conditions holds:
a) injective dimension of N is finite,
b) both Gorenstein injective dimension of N and projective dimension of M are finite.
Then
Hmi (M,N)∼=HomR(ExtRdimR−i(HomR(ω,N),M),E(R/m)).
L
Proof. i) If G-dimension of M is finite, then by [C, Theorem 3.4.6], one has ω⊗ M ≃ω⊗ M. Also,
R R
if Gorenstein injective dimension of N is finite, then [C, Theorem 3.4.9] asserts that RHom (ω,N) ≃
R
Hom (ω,N). Hence the conclusion is immediate by Theorem 3.3. Note that ΣdimRω is a normalized
R
dualizing complex of R. (cid:3)
Example 3.6. None of Questions 1.1 and 1.2 have positive answers. To see this, let (R,m,k) be a non-
regular Gorenstein local ring. Let d:=dimR, and as before, let (−)∨ denote the Matlis duality functor.
Since R is Gorenstein, by [C, Theorems 1.4.9and 6.2.7],both G-dimension of k and Gorensteininjective
dimension of k are finite. Assume that one of these questions has an affirmative answer. Then, by
Theorem 3.3, it turns out that
Exti (k,k)∼=limExti (k/mnk,k)∼=Hi(k,k)∼=Extd−i(k,k)∨
R −→ R m R
n
for all non-negative integers i. This yields that Exti (k,k) = 0 for all i ∈/ {0,1,...d}. So, R is regular
R
and we get a contradiction.
8 F.MOHAMMADIAGHJEHMASHHADANDK.DIVAANI-AAZAR
4. Applications
We start this section by proving a couple of lemmas.
Lemma 4.1. Let X ∈D✷f(R) and N ∈D0f(R). Then
−supRHom (N,X)=inf{depth X |p∈Supp N}=depth (Ann H (N),X).
R Rp p R R R 0
Proof. The assertion follows immediately by [Fo2, Proposition 3.4] and [FI, Proposition 2.10]. (cid:3)
G-dimension of a complex X ∈Df(R), G–dim X, is defined by
❂ R
G–dim X :=inf{sup{l∈Z|Q 6=0}|Q is a bounded to the right complex of
R l
totally reflexive R-modules and Q≃X}.
Lemma 4.2. Let (R,m) be a local ring possessing a normalized dualizing complex D, X ∈ D✷f(R) and
M,N two nonzero finitely generated R-modules.
i) If M has finite G-dimension and Supp M ∩Assh N 6=∅, then
R R
L
supRHom (N,D⊗ M)≥dim N.
R R R
ii) If N has finite Gorenstein injective dimension, then
supRHom (RHom (D,N),X)=depthR−depth (Ann N,X).
R R R R
Proof. i)Letpbe aprime idealofR. From[Fo1, 15.17c)]and[C,A.8.5.3],onehasinfD =dimR/p+
p
depthR . As M ∈ Af(R), by [C, Observation 3.1.7], it follows that M ∈ Af(R ). So, by applying [C,
p p p
Lemma A.6.4] and [C, A.6.3.2], we can deduce that
L
depth M =depth (RHom (D ,D ⊗ M ))
Rp p Rp Rp p p Rp p
=depth (D ⊗L M )+dimR/p+depthR .
Rp p Rp p p
Thus by Lemma 4.1 and the Auslander-Buchsbaum formula for G-dimension (see e.g. [C, Theorem
1.4.8]), one has:
supRHom (N,D⊗L M) =−inf{−dimR/p−G–dim M |p∈Supp N}
R R Rp p R
=sup{dimR/p+G–dim M |p∈Supp N}
Rp p R
=sup{dimR/p+G–dim M |p∈Supp M ∩Supp N}
Rp p R R
≥dim N.
R
ii) As N ∈Bf(R), one has
L
N ≃D⊗ RHom (D,N). (∗)
R R
By [C, A.8.5.3], we have infD = depthR. Set s := depthR. Then applying Nakayama’s Lemma for
complexes (see e.g. [C, Corollary A.4.16]) to (∗) yields that
infRHom (D,N)=−infD =−s.
R
On the other hand, by [C, Proposition A.4.6], we have
supRHom (D,N)≤supN −infD =−s.
R
FOXBY EQUIVALENCE, LOCAL DUALITY AND ... 9
Hence ΣsRHom (D,N) ∈Df(R). From (∗), one can conclude that ΣsRHom (D,N) and N have the
R 0 R
same support, and so Lemma 4.1 implies that
supRHom (RHom (D,N),X) =supΣsRHom (ΣsRHom (D,N),X)
R R R R
=s+supRHom (ΣsRHom (D,N),X)
R R
=s−depth (Ann H (ΣsRHom (D,N)),X)
R R 0 R
=s−depth (Ann N,X).
R R
(cid:3)
In the sequel, we establish a characterizationof Cohen-Macaulay modules. It partially improves [HZ,
Theorem 3.3]. To this end, for a complex Y ∈ D❁(R), we fix the notation Y⊥ for the full subcategory
of Df(R) whose objects are exactly those complexes X ∈ Df(R) for which Hi (X,Y) = 0 for all i 6=
0 0 m
depth Y.
R
Proposition 4.3. Let (R,m) be a local ring and N a nonzero finitely generated R-module. Consider the
following conditions.
i) N is Cohen-Macaulay.
ii) There is a nonzero R-module M ∈ N⊥ of finite projective dimension such that Supp M ∩
R
Assh N 6=∅.
R
iii) There is a nonzero R-module M ∈N⊥ of finite G-dimension such that Supp M∩Assh N 6=∅.
R R
Then i) and ii) are equivalent and clearly ii) implies iii). In addition, if either projective or injective
dimension of N is finite, then all these conditions are equivalent.
Proof. If a finitely generated R-module M has finite G-dimension, then it is easy to check that the
R-module M has finite G-dimension too. Also, if a finitely generated R-module M satisfies Supp M ∩
b c R
AsshRN 6=∅, then SuppRbMc∩AsshRbNb 6=∅. So, without loss of generality,we may and do assume that
R is complete. So, R possesses a normalized dualizing complex D.
i)⇒ii)Assume that N is Cohen-Macaulay. Then Hi (R,N)=Hi (N)=0 for all i6=depth N, and
m m R
so R∈N⊥.
ii)⇒iii) is clear.
Assume that either projective or injective dimension of N is finite. We show iii) implies i). Suppose
that there exists a nonzero R-module M ∈ N⊥ which has finite G-dimension. Then Hi (M,N) = 0 for
m
all i6=depth N. Hence from Corollary 3.4 i) and Lemma 4.2 i), we deduce that
R
L
depth N =−infRΓ (M,N)=supRHom (N,D⊗ M)≥dim N,
R m R R R
and so N is Cohen-Macaulay.
ii)⇒i) is similar to the proof of iii)⇒i). (cid:3)
Next,weestablishtheGorensteinanalogueofProposition4.3. Itisworthtopointoutthatitimproves
[HZ, Proposition 3.5 1)]. Recall that a non-homologically trivial complex Y ∈ D✷f(R) is said to be
Gorenstein if id Y =depth Y.
R R
Proposition 4.4. Let (R,m,k) be a local ring and Y ∈D✷f(R) a non-homologically trivial complex. The
following are equivalent:
10 F.MOHAMMADIAGHJEHMASHHADANDK.DIVAANI-AAZAR
i) Y is Gorenstein.
ii) Y⊥ =Df(R).
0
iii) k ∈Y⊥.
Proof. i)⇒ ii) Let X ∈ Df(R). By [Y, Theorem 2.7], inf{i∈ Z|Hi (X,Y) 6= 0} = depth Y. Since Y
0 m R
is Gorenstein, one has id Y = depth Y, and so by Lemma 3.2, it turns out that Hi(X,Y)=0 for all
R R m
i6=depth Y.
R
ii)⇒iii) is clear.
iii) ⇒ i) Since Supp Y ∩Supp k = {m}, by [Y, Lemma 2.4], one has Hi(k,Y) = Exti (k,Y) for
R R m R
all integers i. Thus Exti (k,Y) = 0 for all i 6= depth Y. By [C, A.5.7.4], this yields that id Y =
R R R
depth Y. (cid:3)
R
Lemma 4.5. Let (R,m) be a local ring possessing a normalized dualizing complex D and X ∈ D✷f(R).
Assume that X has finite G-dimension. Then
L
depth (a,X)−dimR≤depth (a,D⊗ X)≤depth (a,X)−depthR
R R R R
for all ideals a of R.
Proof. LetabeageneratingsetforagivenidealaofR. AsCˇ(a)isaboundedcomplexofflatR-modules,
X ∈Af(R) and infD =depthR, [CH, Theorem 4.7 i)] and [C, Proposition 3.3.7 a)] imply that
L L
supRΓ (X)+depthR≤supRΓ (D⊗ X)=sup(D⊗ RΓ (X))≤supRΓ (X)+dimR.
a a R R a a
But for any complex Z ∈ D✷f(R), by [Fo2, Proposition 3.14 c)], one has depthR(a,Z) = −supRΓa(Z).
This completes the proof. (cid:3)
The following result can be considered as Grothendieck’s non-vanishing Theorem in the context of
generalized local cohomology modules. It also improves [DH, Theorem 3.5].
Proposition 4.6. Let (R,m) be a local ring, N a nonzero finitely generated R-module and X ∈D✷f(R).
i) Assume that one of the following conditions is satisfied:
a) projective dimension of X is finite,
b) G-dimension of X and projective dimension of N are finite.
Then
depthR−depth(Ann N,X)≤cd (X,N)≤dimR−depth(Ann N,X).
R m R
ii) Assume that one of the following conditions is satisfied:
a) injective dimension of N is finite,
b) Gorenstein injective dimension of N and either projective dimension or injective dimension of
X are finite.
Then
cd (X,N)=depthR−depth(Ann N,X).
m R
Proof. LetZ ∈D✷f(R). Clearly,thenonehasZ⊗RR∈D✷f(R). Ifprojective(resp. injective)dimension
b b
of Z is finite, then Z ⊗ R has finite projective (resp. injective) dimension over R. Also, it is easy to
R
b b
check that if G-dimension of Z is finite, then so is G-dimension of Z ⊗ R over R. By [FF, Theorem
R
b b
