Stable cohomology over local rings

0.51 MB·English

by  Luchezar L. Avramov

STABLE COHOMOLOGY OVER LOCAL RINGS
7
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LUCHEZARL.AVRAMOVANDOANAVELICHE
0
2
Abstract. For a commutative noetherian ring R with residue field k stable
an cohomology modules EdxtnR(k,k) have been defined for each n ∈ Z, but their
J meaninghasremainedelusive. Itisprovedthat the k-rankofanyEdxtnR(k,k)
characterizesimportantpropertiesofR,suchasbeingregular,completeinter-
0
section,orGorenstein. Thesenumericalcharacterizationsarebasedonresults
1
concerning thestructure of Z-gradedk-algebracarriedby stablecohomology.

Itisshownthatinmanycasesitisdeterminedbyabsolutecohomologythrough
C] a canonical homomorphismof algebras ExtR(k,k)→EdxtR(k,k). Some tech-
niquesdevelopedinthepaperareapplicabletothestudyofstablecohomology
A
functorsovergeneral associativerings.
.
h
t
a
m
[ Contents

3 Introduction 1
v
1. Cohomology theories 3
1
2. Comparisons 6
2
0 3. Additional structures 8
8 4. Non-zero-divisors 11
0 5. Depth of cohomology modules 14
5 6. Finiteness of stable cohomology 18
0
7. Structure of stable cohomology algebras 21
/
h 8. Stable cohomology algebras of complete intersection rings 24
t
a 9. Stable cohomology algebras of Gorenstein rings 25
m 10. Stable cohomology algebras of Golod rings 31
: Appendix A. Depth over graded algebras 36
v
Acknowledgments 38
i
X References 39
r
a
Introduction
A stable cohomology theory over an associative ring R associates to every pair
(M,N) of R-modules groups Extn(M,N), which are defined for each n ∈ Z and
R
vanish for all n when M or N has finite projective dimension. Different construc-
tions, have been proposedby Bdenson and Carlson, Mislin, and Vogel, and all yield
canonicallyisomorphictheories. However,therehavebeenfewapplicationsoutside
Date:February2,2008.
2000 Mathematics Subject Classification. 13D07,13H10,20J06.
L.L.A.partlysupportedbyNSFgrantsDMS0201904.
1
2 L.L.AVRAMOVANDO.VELICHE
of group theory and Galois theory, for which the prototype—Tate cohomology for
finite groups—was created in the 1950s.
In the first four sections we develop general techniques for computing stable
cohomology. We approach it through a canonical transformation ι: Ext →Ext
R R
of absolute cohomology into stable cohomology, which we study by systematically
using the compatible multiplicative structures carried by the two theories. Adnew
feature are extensive applications of a third cohomological functor, the bounded
cohomology Ext , which appears in a long exact sequence measuring the kernel
R
and the cokernel of ι. By extending a construction of Eisenbud we show how to
track changes in stable cohomology under factorizations of non-zero-divisors.
The core of the paper is its second part, devoted to stable cohomology over
commutative noetherian local rings. One goal is to investigate if and how this
theoryreflectsordetectspropertiesofaringoramodule. Asecondgoalistostudy
the structure of the local cohomology functors themselves. Historical precedent in
commutative algebra points to the residue field k of a local ring R as the ultimate
test case, so the focus is kept on it for much of the second part of the paper.
When applying the general machinery to a local ring R with residue field k
we heavily use the fact that the absolute cohomology algebra E = Ext (k,k) is
R
the universal enveloping algebra of a graded Lie algebra. The existence of such a
structure underlies a well documented successful interaction between local algebra
and rational homotopy theory. F´elix, Halperin, and Thomas have transplanted
from algebra and systematically used in topology a notion of depth of cohomology
modules. We take the concept back into algebra and use it in a different manner.
Backgroundmaterial is developed in Section A and Section 5.
In Section 6 we give necessary and sufficient conditions for a local ring R to be
regular (respectively, complete intersection, Gorenstein) in terms of the vanishing
(respectively, size, finiteness) of rank Extn(k,k) for a single value n ∈ Z. The
k R
last result is surprising: unlike regularity or complete intersection, Gorensteinness
is not recognized even by the entire sequdence (rankkExtnR(k,k))n>0.
In Section 7 we start the study of the graded k-algebra S = Extn(k,k). A
R
result of Martsinkovsky, for which we give a short proof, shows that when R is
singular the map of graded algebras ι: E →S is injective. We deterdmine Coker(ι)
as a left E-module and prove that depthE ≥ 2 implies S = ι(E)⊕T where T is
the E-torsion submodule of S, and is the unique direct complement of ι(E) as a
left E-module. In Section 8 this information is used to produce a nearly complete,
explicit computation of the algebra S for complete intersection rings.
Itisnaturaltoaskwhethertheresultsforcompleteintersectionsextend,insome
form, to all singular Gorenstein rings R with codimR ≥ 2. In Section 9 we prove
that depthE ≥2 implies T is a two-sided ideal of S with T2 =0, is isomorphic to
a shift of Hom (E,k) as left E-module, and S = ι(E)⊕T. A similar relationship
k
between the Tate cohomology algebra H∗(G,k) of a finite group G and its coho-
mology algebra H∗(G,k) was discovered by Benson and Carlson. The parallel is
remarkable,asH∗(G,k)isgraded-commbutativeandfinitelygenerated,whileE may
be non-finitely generated and almost always is very far from commutative.
One has depthE ≥ 1 for all singular rings, so the condition depthE ≥ 2 is not
toospecial. WeprovethatitholdsforseveralclassesofGorensteinrings,including
those of codimension 2 or 3, those of minimal multiplicity, and the localizations
STABLE COHOMOLOGY 3
of Koszul algebras. We are not aware whether splitting occurs always: Gorenstein
rings with depthE =1 are hard to come by, and for the known ones E splits off S.
It was noted above that when R is not Gorenstein rank Extn(k,k) is infinite
k R
for each n, so over such rings a different structure of S may be expected. As a
test case in Section 10 we turn to Golod rings, whose homoldogical properties are
in many respects opposite to those of Gorenstein rings. We show that depthE =1
holdsforallGolodrings,andforasubclassofsuchringsweworkoutthe structure
of S in sufficient detail to prove that ι does not split as a map of left E-module.
1. Cohomology theories
We start by describing notions concerning complexes and, more generally, DG
(that is, differential graded) modules and algebras. The latter are used to describe
composition products carried by the absolute cohomology functors. We then intro-
duceabounded cohomology theorythathasnotbeensystematicallystudiedbefore.
Finally, we present Vogel’s construction of stable cohomology.
1.1. DGalgebras andDGmodules. TogradeacomplexC weusesubscriptsor
superscripts. Thus,C canbewritteneitherasasequenceofmaps∂C: C →C ,
n n n−1
or as a sequence of maps ∂−n: C−n → C−n+1, with ∂−n = ∂C. Accordingly, an
C C n
elementc∈C isassignedalower(orhomological)degreen,denotedand anupper
n
(or cohomological)degree −n; we write ⌊c⌋=n and ⌈c⌉=−n, respectively. When
the nature of degree does not matter we use |c| in place of either ⌊c⌋ or ⌈c⌉.
When z ∈C is a cycle cl(z) denotes its homology class.
For every s ∈ Z let ΣsC denote the complex with (ΣsC) = C and ∂ΣsC =
n n−s n
(−1)s∂C ; let σs: C →ΣsC be the bijective map C ∋c7→c∈(ΣsC) .
n−s n n+s
Bimodules have actions from the left and from the right, listed in that order. If
A and A′ are DG algebras and C is a DG A-A′-bimodule, then the formula
a·σs(c)·a′ =(−1)|a|sσs(a·c·a′)
turns ΣsC into a DG A-A′-bimodule and σs into a chain map of DG bimodules.
Furthermore, when B is a DG A-A-bimodule the map
B⊗ ΣsC −→Σs(B⊗ C)
A A
given by b⊗σs(c)7→(−1)|b|sσs(b⊗c) is an isomorphism of DG A-A′-modules.
For the rest of the section R denotes an associative ring, M and N are left
R-modules, and and F →M and G→N are projective resolutions.
1.2. Absolute Ext. Let Hom (F,G) denote the complex of abelian groups with
R
Hom (F,G) = Hom (F ,G )=Hom (F,G)−n
R n R i i+n R
i∈Z
Y
as component of homological degree n (cohomologicaldegree −n), and differential
∂(β)=∂Gβ−(−1)|β|β∂F .
The induced map Hom (F,G)→Hom (F,N) is a quasi-isomorphism, so one has
R R
H(HomR(F,G))∼=H(HomR(F,N))=ExtR(M,N).
4 L.L.AVRAMOVANDO.VELICHE
1.2.1. Composition of homomorphisms turns Hom (F,F) and Hom (G,G) into
R R
DGalgebras,andHom (F,G)intoaDGHom (G,G)-Hom (F,F)-bimodule. The
R R R
composition products induced in homology can be computed from any pair of pro-
jective resolutions. They turn Ext (M,M) and Ext (N,N) into graded algebras,
R R
and Ext (M,N) into a graded Ext (N,N)-Ext (M,M)-bimodule.
R R R
1.2.2. The DG algebra Hom (G,G) acts on the complex G by evaluation of ho-
R
momorphisms. For every complex C of right R-modules the map
HomR(G,G)⊗Z(C⊗RG)−→C⊗RG
α⊗(c⊗g)7−→(−1)|α||c|c⊗α(g)
endowsC⊗ GwithastructureofleftDGmoduleoverHom (G,G). Clearly,this
R R
structure is natural with respect to morphisms of complexes C →C′.
1.2.3. LetLbearightR-module. SettingC =Lin(1.2.2)oneobtainsamorphism
HomR(G,G)⊗Z(L⊗RG)−→L⊗RG.
In homology it induces for all l,n∈Z homomorphisms of abelian groups
ExtnR(N,N)⊗ZTorRl (L,N)−→TorRl−n(L,N)
that turn TorR(L,N) into a graded left module over Ext (N,N).
R
1.3. Bounded Ext. A homomorphism β ∈ Hom (F,G) is bounded if β = 0
R i
for all i ≫ 0. The subset Hom (F,G) of Hom (F,G), consisting of all bounded
R R
homomorphisms, is a subcomplex, with components
Hom (F,G) = Hom (F ,G )=Hom (F,G)−n.
R n R i i+n R
i∈Z
a
The graded abelian group Ext (M,N)=H(Hom (F,G)) with components
R R
Extn(M,N)=Hn(Hom (F,G)),
R R
is called the bounded cohomology of M and N over R.
1.3.1. It is easy to see that Hom (F,G) is a DG subbimodule of Hom (F,G) for
R R
the actions of Hom (F,F) and Hom (G,G) described in (1.2.1), so Ext (M,N)
R R R
becomes a graded Ext (N,N)-Ext (M,M)-bimodule.
R R
The elementary observation below plays a pivotal role in the paper. It should
be noted that the right-hand analog of this statement fails, see Example (10.9).
Lemma 1.3.2. For every τ∈Ext (M,N) there exists an integer j ≥0, such that
R
Ext>j(N,N)·τ =0.
R
Proof. By hypothesis, τ = cl(β) for some chain map β ∈ Hom (F,G) satisfying
R
β(F)⊆G for some j ≥0. For each γ ∈Hom (G,G) one then has
<j R n
(γβ)(F)=γ β(F) ⊆γ G ⊆G
<j <j+n
Since G =0 for n≤−j, this(cid:0)impli(cid:1)es Ex(cid:0)t>j(N(cid:1),N)·cl(β)=0. (cid:3)
<j+n R
Some of the DG module structures discussed so far are linked as follows:
STABLE COHOMOLOGY 5
Lemma 1.3.3. There is a morphism of DG Hom (G,G)-Hom (F,F)-bimodules
R R
ω: Hom (F,R)⊗ G−→Hom (F,G)
R R R
with actions on the source given by (1.2.1), (1.2.2), and on the target by (1.3.1).
If the R-module F is finite for each i, then ω is bijective.
i
Proof. It is easy to verify that ω′(φ⊗g)(f)=(−1)|g||f|φ(f)g defines a morphism
ω′: Hom (F,R)⊗ G−→Hom (F,G)
R R R
of DG bimodules. The image of ω′ lies in Hom (F,G), so it yields a morphism ω
R
with the desired source and target. For each n∈Z the following equalities
(Hom (F,R)⊗ G) = Hom (F ,R)⊗ G
R R n R i R j
(−ia)+j=n
Hom (F,G) = Hom (F ,G )
R n R i j
j−i=n
a
holdby definition. When eachR-module F is finite, ω restricts to anisomorphism
i
Hom (F ,R)⊗ G →Hom (F ,G ) for each pair (i,j). (cid:3)
R i R j R i j
1.4. Stable Ext. Using the subcomplex Hom (F,G) described in (1.3), set
R
H[om (F,G)=Hom (F,G)/Hom (F,G).
R R R
Following Pierre Vogel, we define the stable cohomology of M and N over R to be
the graded abelian group Ext (M,N)=H(H[om (F,G)) with components
R R
Extn(M,N)=Hn(H[om (F,G)).
dR R
The assignment(M,N)7→Ext (M,N) yieldsa cohomologicalfunctor,contravari-
d R
ant in M and covariant in N, from R-modules to graded Z-modules.
d
1.4.1. As Hom (F,G) is a DG subbimodule of Hom (F,G) for the left action
R R
of Hom (G,G) and the right action of Hom (F,F), see (1.3.1), one sees that
R R
Ext (M,M) and Ext (N,N) are graded algebras, and Ext (M,N) is a graded
R R R
Ext (N,N)-Ext (M,M)-bimodule.
R R
d d d
Stablecohomologyovergeneralassociativeringstookalongtimetoemerge,and
d d
then it appeared in several avatars. We give a short, incomplete list of sources.
1.4.2. HistoricallythefirstexampleofstablecohomologyisTate’scohomologythe-
oryHn(G,−)formodulesoverafinite groupG: OnehasHn(G,−)=Extn (Z,−),
ZG
where ZG is the group ring of G; see [13, Ch. XII]. Tate’s construction is based
on cbomplete resolutions of Z. Buchweitz [12] extended tbhe techniquedto define a
two-variable theory over two-sided noetherian Gorenstein rings.
The functors Extn(−,−) were introduced by Vogel in the mid-1980s. The first
R
published account appears only in [19], where it is called ‘Tate-Vogel cohomology’.
Different approacdhes were independently proposed by Benson and Carlson [8] and
by Mislin [26]; background and details can be found in Kropholler’s survey [22,
§4]1. We have settled on the name ‘stable cohomology’ to emphasize the fact
that Ext0(M,N) is a group of homomorphisms of objects in a stabilization of
R
the category of R-modules, see Beligiannis [7, §§3,5] for details.
d
1WhereDefinition(4.2.2)containsatypo: Ωi+nN shouldbechanged toΩi−nN.
6 L.L.AVRAMOVANDO.VELICHE
2. Comparisons
In this section R is an associative ring, M and N are left R-modules, F → M
and G → N denote projective resolutions. The objective is to describe important
links between the cohomology theories introduced in Section 1.
2.1. By construction, there is an exact sequence of DG bimodules
(2.1.1) 0−→Hom (F,G)−→Hom (F,G)−→H[om (F,G)−→0
R R R
that is unique up to homotopy. It defines an exact sequence
Ext (M,N) η //Ext (M,N) ι // Ext (M,N) ð //
R R R
(2.1.2)
ΣExt (M,N) Ση //ΣExt (M,N) d
R R
of graded Ext (N,N)-Ext (M,M)-bimodules. Thus, there is an exact sequence
R R
··· // Extn(M,N) ηn // Extn(M,N) ιn //Extn(M,N) ðn //
R R R
(2.1.3)
Extn+1(M,N) ηn+1//Extn+1(M,N) d// ···
R R
of abelian groups, and the latter is natural in both module arguments.
Furthermore, ι: Ext (M,M) → Ext (M,M) is a homomorphism of graded
R R
algebras, and ι: Ext (M,N) → Ext (M,N) is an equivariant homomorphism of
R R
graded Ext (N,N)-Ext (M,M)-bimoddules.
R R
d
It is easy to determine if η is an isomorphism; see also [22, (4.2.4)], [33, (4.5.1)].
d d
Proposition 2.2. The following conditions are equivalent.
(i) M has finite projective dimension.
(ii) Extn(M,−)=0 for every n∈Z.
R
(iii) Extn(−,M)=0 for every n∈Z.
R
(iv) Edxt0(M,M)=0.
R
(v) ηdn: Extn(M,−)→Extn(M,−) is an isomorphism for every n∈Z.
R R
(vi) ηdn: Extn(−,M)→Extn(−,M) is an isomorphism for every n∈Z..
R R
Proof. Choosing a finite resolution F → M one gets Hom (F,G) = Hom (F,G)
R R
for every resolution G. Thus, (i) implies (v) and (vi). The exact sequence (2.1.3)
shows that (v) implies (ii), and (vi) implies (iii). It is clear that (ii) or (iii) implies
(iv). If (iv) holds, then for some γ ∈ Hom (F,F)1 and some p ≥ 0 the morphism
R
β =idF −∂γ+γ∂: F →F satisfies β =0 for all i≥p. Thus, one gets
i
0=Hp(Hom (β,−))=Hp(Hom (idF,−))=idHp(HomR(β,−))
R R
hence Extp(M,−)=Hp(Hom (F,−))=0; that is, pd M <p, so (i) holds. (cid:3)
R R R
Next we give a criterion for ι to be an isomorphism in high degrees.
Theorem 2.2. For an integer m the following conditions are equivalent.
(i) ιn: Extn(M,−) → Extn(M,−) is an isomorphism for all n > m and ιm
R R
is an epimorphism.
(ii) Extn(M,P)=0 fordall n>m and every projective R-module P.
R
(iii) Extn(M,−)=0 for all n>m.
R
STABLE COHOMOLOGY 7
When M has a resolution by finite projective modules they are also equivalent to
(ii′) Extn(M,R)=0 for all n>m.
R
Proof. The exact sequence (2.1.3) shows that (i) and (iii) are equivalent. If (iii)
holds, then so does (ii), because Extn(M,−) ∼= Extn(M,−) by Proposition (2.2).
R R
It is clear that (ii) implies (ii′). The converse holds because the hypothesis on M
in (ii′) implies that the functor Extn(M,−) commutes with all direct sums.
R
To prove that (ii) implies (iii) fix an integer n > m and choose a chain map
α∈Hom (F,G)n. Thus, for some fixed s≥n and all j ≥s one has α =0, while
R j
(2.2.1) ∂G α −(−1)nα ∂F =0 holds for all j ∈Z.
j j+1−n j+1 j j+1
We need to find a homomorphism β ∈Hom (F,G)n−1 that satisfies
R
(2.2.2) ∂G β −(−1)n−1β ∂F =α for all j ∈Z.
j j+2−n j+1 j j+1 j+1
Set β = 0 for j ≥ s and assume by descending induction on j that we already
j
have maps β satisfying (2.2.2) for some integer i ∈ [n,s] and all j ≥ i. Set
j j
δh =(−1)n+1Hom (∂F ,G ) for each h. Using (2.2.1) and (2.2.2) we get
R h+1 h−n i i
δi α −∂G β =(−1)n+1α ∂F +(−1)n∂G β ∂F
i i+1−n i i i+1 i+1−n i i+1
(cid:0) (cid:1)=−∂iG+1−nαi+1+(−1)n∂iG+1−nβi∂iF+1
=−∂G α +(−1)n−1β ∂F
i+1−n i+1 i i+1
=−∂iG+1−n∂(cid:0)iG+2−nβi+1 (cid:1)
=0
On the other hand, since one has Exti (M,G )=0 the sequence
R i−n
δi−1 δi
Hom (F ,G )−−−−→Hom (F ,G )−−→Hom (F ,G )
R i−1 i−n R i i−n R i+1 i−n
is exact, so there exists a homomorphism β : F →G , such that
i−1 i−1 i−n
α −∂G β =δi−1(β )=−(−1)n−1β ∂F .
i i+1−n i i−1 i−1 i
Thus, β satisfies (2.2.2) , so the induction step is complete. As a result, for
i−1 i−1
each j ≥ n−1 we now have a homomorphism β : F → G satisfying the
j j j−(n−1)
equality (2.2.2) . As G = 0 for j < n−1, setting β = 0 we extend the
j j−(n−1) j
equality to all j ∈Z. We have proved Extn(M,N)=0, as desired. (cid:3)
R
2.3. A complete resolution of M is a morphism of complexes ν: T →F such that
ν is bijective for all i ≫ 0, each T is projective, and for all n ∈ Z and every
i i
projective R-module P one has H (T)=0=H (Hom (T,P)); see [14, (1.1)]. (In
n n R
some contexts it is assumed, in addition, that the R-modules T are also finite; no
n
such hypothesis is needed or made here.) When such a complete resolution exists,
H (Hom (M,N)) is calledthe nth Tate cohomology of M with coefficients in N.
−n R
Cornick and Kropholler [14, (1.2)] prove that when Tate cohomology is defined
itisnaturallyisomorphictostablecohomology. We deduce thisfromTheorem2.2:
Corollary 2.4. If ν: T →F is a complete resolution of M, then one has
Hn(HomR(T,N))∼=ExtnR(M,N) for each n∈Z.
d
8 L.L.AVRAMOVANDO.VELICHE
Proof. Fix n∈Z and set K =Coker(∂T). For each i≥1 one then has
n
(2.4.1) Hn−1+i(Hom (T,N))=Exti (K,N),
i R R
because Σ−(n−1)(T ) is a projective resolution of K. From the condition on T
>n−1
one gets Exti (K,P)=0 for all i≥1 and every projective R-module P, so
R
(2.4.2) Ext1(K,N)∼=Ext1(K,N).
R R
holds by the theorem. Choose p ≥ n with ν bijective for i ≥ p. The R-module
i
L=Ker(∂F ) is then isomorphic to Ker(∂dT ), so there exist exact sequences
p−1 p−1
0−→L−→T −→···−→T −→T −→K −→0
p−1 n n−1
0−→L−→F −→···−→F −→F −→M −→0
p−1 1 0
In view of (2.2), the iterated connecting maps defined by these sequences yield
(2.4.3) Ext1(K,N)∼=Extn−p(L,N)∼=Extn(M,N).
R R R
Tofinishtheproof,concatenatetheisomorphisms(2.4.1) ,(2.4.2),and(2.4.3). (cid:3)
1
d d d
3. Additional structures
In this section we discuss the existence of finer natural structures on stable
cohomology groups, such as rings of operators or internal gradings.
Proposition 3.1. Let R be an algebra over a commutative ring K.
(1) The exact sequence (2.1.2) is one of graded K-modules, and the various pair-
ings of cohomology groups are K-bilinear.
If, in addition, K is noetherian, R is finite as a K-module, M and N are finite
R-modules, and n is an integer, then the following assertions also hold.
(2) The K-modules Extn(M,N) and Extn+1(M,N) are finite simultaneously.
R R
(3) When Ext≫j(M,R)=0 theK-moduleExtn(M,N) is finitefor every n∈Z.
R R
d
Proof. (1)Thisisduetothefactthattherelevantmapsincohomologyareinduced
d
by morphisms of complexes or K-modules.
(2) By (1), the maps in the exact sequence (2.1.3) are K-linear, and under our
hypotheses the K-modules Extn(M,N) and Extn+1(M,N) are noetherian.
R R
(3) Let F → M and G → N be resolutions by finite projective R-modules.
Choose m ≥ 1 so that H (Hom (F,R)) = Extj (M,R) = 0 for all j > m; then
−j R R
Hom (F,R) is quasi-isomorphic to the complex C of right R-modules defined by
R
Hom (F ,R) for 0≥i≥−m;
R −i
C = ImHom (∂F ,R) for i=−m−1;
i  R m+1
0 for i<−m−1 or i>0.
Asaconsequence,HomR(F,R)⊗RGisquasi-isomorphictoC⊗RG. Foreachn∈Z
this gives the second isomorphism below; Lemma 1.3.3 provides the first one:
Extn(M,N)=Hn(Hom (F,G))
R R
∼=Hn(HomR(F,R)⊗RG)
∼=Hn(C⊗RG)
As (C ⊗ G)n = m Cj ⊗ Gn−j is a finite K-module for each n, we see that
R j=0 R
Extn(M,N) is a finite K-module; by (2), so is Extn(M,N). (cid:3)
R ` R
d
STABLE COHOMOLOGY 9
Stable cohomology behaves predictably under flat base change.
Proposition 3.2. Let R be a commutative noetherian ring, M an R-module that
admits a resolution by finite projective modules, and R → R′ a homomorphism of
rings such that the right R-module R′ is flat.
For each R-module N there is then a commutative diagram
R′⊗RExtR(M,N) R′⊗Rι //R′⊗RExtR(M,N)
∼= (cid:15)(cid:15) d(cid:15)(cid:15)∼=
ExtR′(R′⊗RM,R′⊗RN) ι //ExtR′(R′⊗RM,R′⊗RN)
When N =M all the maps in the diagram are homomorphisms of graded algebras.
d
Proof. Set (−)′ = (R′ ⊗ −). Let F → M be a resolution by finite projective
R
R-modules and G→N be a projective resolution. In the commutative square
R′⊗ Hom (F,G) ≃ //R′⊗ Hom (F,N)
R R R R
∼=
(cid:15)(cid:15) (cid:15)(cid:15)
HomR′(F′,G′) ≃ //// HomR′(F′,N′)
the isomorphism is due to the choice of F. Thus, the left vertical map is a quasi-
isomorphism. It appears in the following commutative diagram where the vertical
arrows are induced by the map α7→α′ and the rows are exact, see (2.1.1):
0 // R′⊗RHomR(F,G) // R′⊗RHomR(F,G) //R′⊗RH[omR(F,G) //0
∼= ≃
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //HomR′(F′,G′) // HomR′(F′,G′) // H[omR′(F′,G′) //0
TheflatnessofR′ impliesthatF′ →M′ andG′ →N′ areR′-projectiveresolutions,
and that the homology of the right hand square above is the desired diagram. (cid:3)
Next we turn to cohomology of graded objects.
3.3. We saythatthe ringR isinternally graded if R= ∞ R asabeliangroups,
i∈Z i
andR R ⊆R holdsforalli,j. InternalgradingsforM,N aredefinedsimilarly.
i j i+j
By convention, we allow M to be written also as M−Li. As usual, we let M(s)
i
denote the graded R-module with M(s) =M for all i∈Z.
i s+i
Assume R, M, and N are internally graded. A homomorphism β: M → N is
homogeneous of internal degree −j if β(M )⊆N holds for each i∈Z. All such
i i−j
mapsformanabeliansubgroupHomgr (M,N)j ofHom (M,N). Clearly,thesum
R R
of these subgroups is direct, so Hom (M,N) contains as a subgroup the group
R
Homgr (M,N)= Homgr (M,N)j.
R R
j∈Z
M
When M is finitely presented, one has Homgr (M,N)=Hom (M,N).
R R
EverygradedR-moduleM hasagraded free resolution F →M thatis,aresolu-
tion in which each F is a graded free R-module and differentials are homogeneous
i
of internal degree 0. It produces a subcomplex Homgr (F,N) of Hom (F,N),
R R
consisting of graded abelian groups and homomorphisms of internal degree 0.
10 L.L.AVRAMOVANDO.VELICHE
Assume that each R-module F is finite. One then has Homgr (F,N) =
i R
Hom (F,N), and hence the absolute Ext groups inherit an internal grading:
R
Extn(M,N)= Extn(M,N)j.
R R
j∈Z
M
For each n∈Z one also has equalities
Hom (F,G) = Homgr (F ,G )j.
R n R i i+n
i∈Zj∈Z
YM
However,thereisnoinducedinternalgradingonthe righthandside,soextrasteps
are needed to introduce such a grading on stable cohomology groups.
Proposition 3.4. Assume R is an internally graded ring, M,N are internally
graded R-modules, and M has a graded free resolution F → M where each F is
i
finite (as is the case, for example, when R is left noetherian and M is finite).
For each n ∈ Z the abelian groups Extn(M,N) and Extn(M,N) then have
R R
natural internal gradings, which are preserved by the homomorphisms in the exact
sequence (2.1.3) and are additive under tdhe various products.
If M has a complete resolution T by finite projective graded R-modules with
differentials ∂T of degree 0, then the internal gradings of Hn(Homgr (T,N)) and
R
Extn(M,N) are preserved by the isomorphisms of Corollary 2.4.
R
Proof. Let G→N be graded free resolution. For all j,n∈Z the subgroups
d
Homgr (F,G)j = Homgr (F ,G )j
R n R i i+n
i∈Z
Y
of Hom (F,G) form a subcomplex Homgr (F,G)j of Hom (F,G). The Com-
R n R R
parison Theorem for graded resolutions shows that the canonical morphism
Homgr (F,G)j →Homgr (F,N)j isaquasi-isomorphism. Itfollowsthatthecom-
R R
plex Homgr (F,G)= Homgr (F,G)j appears in a commutative diagram
R j∈Z R
LHomgr (F,G) ≃ //Homgr (F,N)
R R
(cid:15)(cid:15)
Hom (F,G) ≃ // Hom (F,N)
R R
where the horizontal maps are quasi-isomorphisms, and the equality is due to the
finiteness of the modules F . Thus, the left vertical map is a quasi-isomorphism.
i
Setting Homgr (F,G) = Homgr (F ,G )j for each n ∈ Z one
R n j∈Z i∈Z R i i+n
gets an internally gradedsubcomplex Homgr (F,G) of the internally graded com-
L ` R
plex Homgr (F,G). Thus, H\omgr (F,G) = Homgr (F,G)/Homgr (F,G) is an
R R R R
internally graded complex. On the other hand, one has equalities
Homgr (F,G) = Homgr (F ,G )j
R n R i i+n
j∈Zi∈Z
Ma
= Homgr (F ,G )j
R i i+n
i∈Zj∈Z
aM
= Hom (F ,G )=Hom (F,G)
R i i+n R n
i∈Z
a