Completion by Derived Double Centralizer

0.21 MB·English

by  Marco Porta

COMPLETION BY DERIVED DOUBLE CENTRALIZER
MARCOPORTA,LIRANSHAULANDAMNONYEKUTIELI
2
1 Abstract. LetAbeacommutativering,andletabeaweaklyproregularideal
0 inA. (IfAisnoetherianthenanyidealinitisweaklyproregular.) SupposeK
2 isacompactgeneratorofthecategoryofcohomologically a-torsioncomplexes.
l We provethatthederived double centralizerofK isisomorphictothea-adic
u completion ofA. The proofrelieson the MGM equivalence from[PSY]. Our
J
resultextendsearlierworkofDwyer-Greenlees-Iyengar[DGI]andEfimov[Ef].

3

]
C
A 0. Introduction
. Let A be a commutative ring. We denote by D(ModA) the derived category of
h
t A-modules. Given M ∈D(ModA) we define
a
m ExtA(M):=M HomD(ModA)(M,M[i]).
[ i∈Z

This is a graded A-algebra with the Yoneda multiplication. The derived endomor-
1
v phism DG algebra of M is a DG A-algebra B :=REndA(M), which is well-defined
2 up to quasi-isomorphism. The graded algebra H(B):= Hi(B) has the prop-
1 erty that H(B)∼=Ext (M). Li∈Z
A
6 Consider the derived category D˜(DGMB) of left DG B-modules. We can view
0
. M as an object of D˜(DGMB), and thus we get the graded A-algebra
7
0 Ext (M)= Hom (M,M[i]).
2 B M D˜(DGMB)
i∈Z
1
: This is the derived double centralizer algebra of M. It is well-defined up to iso-
v
morphism. The details on the derived endomorphism DG algebra and the derived
i
X double centralizer algebra are workedout in Section 2.
r Let a be an ideal in A. The a-torsion functor Γa can be right derived, giving a
a triangulated functor RΓa from D(ModA) to itself. A complex M ∈ D(ModA) is
calledacohomologically a-torsioncomplexifthecanonicalmorphismRΓa(M)→M
is an isomorphism. The full triangulated category on the cohomologically torsion
complexes is denoted by D(ModA)a-tor. It is known that when a is finitely gener-
ated,thecategoryD(ModA)a-tor iscompactlygenerated(forinstancebytheKoszul
complex K(A;a) associated to a finite generating sequence a of a).
The concept of weakly proregular sequence in A was introduced in [AJL, Cor-
rection] and [Sc]. We recall the definition in Section 1 (it is a bit technical). An
Date:3July2012.
Key words and phrases. Adiccompletion,derivedfunctors,derivedMoritatheory.
Mathematics Subject Classification 2010. Primary: 13D07;Secondary: 13B35,13C12, 13D09,
18E30.
This research was supported by the Israel Science Foundation and the Center for Advanced
StudiesatBGU.
1
2 MARCOPORTA,LIRANSHAULANDAMNONYEKUTIELI
idealainAiscalledweakly proregularifitcanbegeneratedbyaweaklyproregular
sequence. It is important to note that if A is noetherian, then any finite sequence
in it is weakly proregular, and any ideal in A is weakly proregular. But there are
some fairly natural non-noetherian examples (see [PSY, Example 3.35].
Here is our main result (repeated as Theorem 4.2 in the body of the paper).
Theorem 0.1. Let A be a commutative ring, let a be a weakly proregular ideal
in A, let K be a compact generator of D(ModA)a-tor, and let B := REndA(K).
Then Exti (K)=0 for all i6=0, and there is a unique isomorphism of A-algebras
B
Ext0(K)∼=A.
B
b
Our result extends earlier work of Dwyer-Greenlees-Iyengar [DGI] and Efimov
[Ef]; see Remark 4.9 for a comparison.
Let us say a few words on the proof of Theorem 0.1. We use derived Morita
theorytofindanisomorphismofgradedalgebrasbetweenExt (K)andExt (N)op,
B A
where N :=RΓa(A). The necessaryfacts about derivedMorita theory are recalled
in Section 3. We then use MGM equivalence (recalled in Section 1) to prove that
Ext (N)∼=Ext (A)∼=A.
A A
b b
Acknowledgments. We wish to thank BernhardKeller, JohnGreenlees, Alexan-
der Efimov and Maxim Kontsevich for helpful discussions.
1. Weak Proregularity and MGM Equivalence
LetAbeacommutativering,andletabeanidealinit. (Wedonotassumethat
A is noetherian or a-adically complete.) There are two operations on A-modules
associatedto this data: the a-adic completion and the a-torsion. For an A-module
M its a-adic completion is the A-module Λa(M) = M := lim←i M/aiM. An
element m ∈M is called an a-torsion element if aim =c0 for i≫0. The a-torsion
elements form the a-torsion submodule Γa(M) of M.
LetusdenotebyModAthecategoryofA-modules. Sowehaveadditivefunctors
Λa andΓa fromModA to itself. The functor Γa is left exact;whereasΛa is neither
leftexactnorrightexact. AnA-moduleiscalleda-adicallycompleteifthecanonical
homomorphism τM :M →Λa(M) is bijective; and M is a-torsion if the canonical
homomorphism σM : Γa(M) → M is bijective. If the ideal a is finitely generated,
thenthefunctorΛaisidempotent;namelyforanymoduleM,itscompletionΛa(M)
is a-adically complete. (There are counterexamples to that for infinitely generated
ideals – see [Ye, Example 1.8].)
The derived category of ModA is denoted by D(ModA). The derived functors
LΛa,RΓa :D(ModA)→D(ModA)
exist. The left derivedfunctor LΛa is constructed using K-flatresolutions,and the
right derived functor RΓa is constructed using K-injective resolutions. This means
that for any K-flat complex P, the canonical morphism ξP : LΛa(P) → Λa(P)
is an isomorphism; and for any K-injective complex I, the canonical morphism
ξI :Γa(I)→RΓa(I) is an isomorphism.
A complex M ∈ D(ModA) is called a cohomologically a-torsion complex if
the canonical morphism σMR : RΓa(M) → M is an isomorphism. The complex
M is called a cohomologically a-adically complete complex if the canonical mor-
phism τML : M → LΛa(M) is an isomorphism. We denote by D(ModA)a-tor and
COMPLETION BY DERIVED DOUBLE CENTRALIZER 3
D(ModA)a-com the full subcategories of D(ModA) consisting of cohomologicallya-
torsioncomplexesandcohomologicallya-adicallycompletecomplexes,respectively.
These are triangulated subcategories.
Very little can be said about the functors LΛa and RΓa, and about the cor-
responding triangulated categories D(ModA)a-tor and D(ModA)a-com, in general.
However we know a lot when the ideal a is weakly proregular.
Before defining weakproregularitywe haveto talk about Koszul complexes. Re-
call that for an element a∈A the Koszul complex is
a·
K(A;a):= ···→0→A−→A→0→··· ,
(cid:0) (cid:1)
concentrated in degrees −1 and 0. Given a finite sequence a = (a ,...,a ) of
1 n
elements in A, the Koszul complex associated to this sequence is
K(A;a):=K(A;a )⊗ ···⊗ K(A;a ).
1 A A n
This is a complex of finitely generated free A-modules, concentrated in degrees
−n,...,0. There is a canonical isomorphism of A-modules H0(K(A;a)) ∼= A/(a),
where (a) is the ideal generated by the sequence a.
For any i ≥ 1 let ai := (ai,...,ai). If j ≥ i then there is a canonical homo-
1 n
morphismof complexes p :K(A;aj)→K(A;ai), which in H0 correspondsto the
j,i
surjection A/(aj) → A/(ai). Thus for every k ∈ Z we get an inverse system of
A-modules
(1.1) Hk(K(A;ai)) ,
(cid:8) (cid:9)i∈N
with transition homomorphisms
Hk(p ):Hk(K(A;aj))→Hk(K(A;ai)).
j,i
Of course for k = 0 the inverse limit equals the (a)-adic completion of A. What
turns out to be crucial is the behavior of this inverse system for k < 0. For more
details please see [PSY, Section 3].
An inverse system {Mi}i∈N of abelian groups, with transition maps pj,i :Mj →
M , is called pro-zero if for every i there exists j ≥i such that p is zero.
i j,i
Definition 1.2. (1) Let a be a finite sequence in A. The sequence a is called
a weakly proregular sequence if for every k ≤−1 the inversesystem (1.1) is
pro-zero.
(2) An ideal a in A is called a a weakly proregular ideal if it is generated by
some weakly proregular sequence.
The etymology and history of related concepts are explained in [AJL], [AJL,
Correction]and [Sc]. If a is a regular sequence, then it is weakly proregular. More
important is the following result of Schenzel.
Theorem 1.3 ([Sc]). If A is noetherian, then every finite sequence in A is weakly
proregular, and hence every ideal in A is weakly proregular.
Here is another useful fact.
Theorem 1.4 ([PSY]). Let a be a weakly proregular ideal in a ring A. Then any
finite sequence that generates a is weakly proregular.
There are interesting examples of weakly proregular ideals in non-noetherian
rings (see [PSY, Example 3.35]).
We shall need the following theorem.
4 MARCOPORTA,LIRANSHAULANDAMNONYEKUTIELI
Theorem 1.5 (MGM Equivalence, [PSY]). Let A be a commutative ring, and a a
weakly proregular ideal in it.
(1) For any M ∈D(ModA) one has RΓa(M)∈D(ModA)a-tor and LΛa(M)∈
D(ModA)a-com.
(2) The functor
RΓa :D(ModA)a-com →D(ModA)a-tor
is an equivalence, with quasi-inverse LΛa.
We should mention that results related to Theorem 1.5 appeared previously in
[AJL, Sc, DG].
2. The Derived Endomorphism Algebra
Let K be a commutative ring, and let M = Mi and N = Ni be
DG A-modules. We denote by HomK(M,N)i theLsei∈tZof K-linear homLomi∈oZrphisms
φ:M →N of degree i. We get a DG K-module
HomK(M,N):= HomK(M,N)i
M
i∈Z
with differential
(2.1) d(φ):=d ◦φ−(−1)iφ◦d
N M
for φ∈HomK(M,N)i.
NowconsideranassociativeunitalnoncommutativeDGK-algebraA= Ai.
Li∈Z
SupposeM andN areleft DGA-modules. Ahomomorphismφ∈HomK(M,N)i is
A-linear(inthegradedsense)ifφ(a·m)=(−1)ija·φ(m)foralla∈Aj andm∈M.
The set of all such homomorphisms is denoted by Hom (M,N)i. In this way we
A
obtain a DG K-module HomA(M,N), which is a DG submodule of HomK(M,N).
The category of left DG A-modules is denoted by DGMA. The set of mor-
phisms HomDGMA(M,N) is precisely the set of 0-cocycles in the DG K-module
Hom (M,N). Note that DGMA is an abelian category.
A
For a DG A-module M there is a noncommutative DG K-algebra End (M):=
A
Hom (M,M). Since the left actions of A and End (M) on M commute, we see
A A
that M is a left DG module over the DG algebra A⊗KEndA(M).
For a DG A-module M = Mi and j ∈ Z, the j-th shift of M is the DG
Li
A-module M[j] defined as follows. The i-th homogeneouscomponent is (M[j])i :=
Mi+j. The action of A on M[j] is a· m := (−1)ija·m for a ∈ Ai and m ∈ M.
[j]
The differential is d :=(−1)jd . In this way the shift M 7→M[j] becomes an
M[j] M
automorphism of the category DGMA.
Given an A-linear homomorphism φ : M → N of degree i, there is an induced
A-linear homomorphism
(2.2) φ[j]:=(−1)ijφ:M[j]→N[j],
also of degree i. This determines an isomorphism of DG K-modules
≃
Hom (M,N)−→Hom (M[j],N[j]).
A A
When N =M we get a canonical isomorphism of DG K-algebras
≃
(2.3) End (M)−→End (M[j]),
A A
sending φ∈End (M)i to φ[j]=(−1)ijφ∈End (M[j])i.
A A
COMPLETION BY DERIVED DOUBLE CENTRALIZER 5
The homotopy category of DGMA is K˜(DGMA), and the derivedcategory(got-
ten by inverting the quasi-isomorphisms in the homotopy category) is D˜(DGMA).
All these categories are K-linear. If A happens to be a ring (i.e. Ai = 0 for i 6= 0)
then D˜(DGMA)=D(ModA), the usual derived category of left A-modules.
LetA♮ bethegradedalgebragottenfromAbyforgettingthedifferential;andthe
same for DG A-modules. Recall that a DG A-module P is called semi-free if there
is a subset X ⊂P consisting of nonzero homogeneouselements, and an exhaustive
non-negativeincreasingfiltration{Fi(X)}i∈Z ofX bysubsets(i.e.F−1(X)=∅and
X = F (X)), such that P♮ is a free graded A♮-module with basis X; and for
i
S
every i one has d(X ) ⊂ F (P), where F (P) := Ax ⊂ P. The set X
is called a semi-basisiof P.i−A1ny M ∈DGMiA admitPs ax∈qFuia(Xsi-)isomorphismP →M
with P semi-free. A DG A-module Q is K-projective if and only if it is homotopy
equivalent to a semi-free DG module P.
Example2.4. IfAisaring,thenanyboundedabovecomplexP offreeA-modules
is a semi-free DG A-module. Indeed, let j := sup{j ∈ Z | Pj 6= 0} (we assume
0
P 6=0). Choose a basis Y for the free module Pj, j ≤j . Define X := Y and
j 0 Sj j
F (X):= Y . Then X is a semi-basis for P.
i Sj≥j0−i j
LetK˜(DGMA) bethefullsubcategoryofK˜(DGMA)consistingofsemi-freeDG
sf
modules. This is a triangulated category. The canonical functor
(2.5) En:K˜(DGMA) →D˜(DGMA)
sf
is anequivalence oftriangulatedcategories. See [Sp,BN, Ke,YZ] fordetails. (The
name “En” stands for “enhancement”.)
Suppose B is another DG algebra, and f : A → B is a homomorphism of DG
algebras. There is an exact functor
rest =rest :DGMB →DGMA
B/A f
calledrestrictionofscalars(aforgetfulfunctor). Itpassestoatriangulatedfunctor
(2.6) rest =rest :D˜(DGMB)→D˜(DGMA).
B/A f
Incasef isaquasi-isomorphism,then(2.6)isanequivalence(see[YZ,Proposition
1.4]).
Definition 2.7. Let M be a DG A-module. Define
Exti (M):=Hom (M,M[i])
A D˜(DGMA)
for any i ∈ Z. Then Ext (M) := Exti (M) is a graded K-algebra with the
A Li∈Z A
Yoneda multiplication (i.e. composition of morphisms in D˜(DGMA)).
Suppose we are given a DG A-module P. Let B := End (P) be the algebra of
A
graded A-linear endomorphisms of P. This is a DG K-algebra,with differential as
in (2.1); and P is a left DG B-module.
Definition 2.8. Given a DG A-module M, choose any semi-free resolution P →
K. The derived endomorphism algebra of K is the DG K-algebra REnd (M) :=
A
End (P).
A
Here is the relation between Definitions 2.7 and 2.8.
6 MARCOPORTA,LIRANSHAULANDAMNONYEKUTIELI
Proposition 2.9. Let B := REnd (M) be the derived endomorphism algebra of
A
M, as in Definition 2.8, constructed using a semi-free resolution P → M. There
is a canonical isomorphism of graded K-algebras Ext (M)∼=H(B).
A
Proof. This is immediate from the equivalence (2.5), since
Hi(B)=Hom (P,P[i]).
K˜(DGMA)
(cid:3)
The dependence of the DG algebra REnd (M) on the resolution P → M is
A
explained in the next proposition.
Proposition 2.10. Let M be a DG A-module, and let P → M and P′ → M
be semi-free resolutions. Define B := End (P) and B′ := End (P′). Then
A A
there is a DG K-algebra B′′, and a DG B′′-module P′′, with DG K-algebra quasi-
isomorphisms B′′ →B and B′′ →B′, and with isomorphisms
restB′′/B(P)∼=P′′ ∼=restB′′/B′(P′)
in D˜(DGMB′′).
Proof. Choose a quasi-isomorphism φ : P′ → P in DGMA lifting the given quasi-
isomorphisms to M. Let L := cone(φ) ∈ DGMA, the mapping cone of φ. So as
graded A-module L=P ⊕P′[1]=hPP′[1]i; and the differential is dL =hd0P dPφ′[1]i,
where φ is viewed as a degree 1 homomorphism P′[1] → P. Of course L is an
acyclic DG module.
Take Q := Hom (P′[1],P), and let B′′ be the triangular matrix DG algebra
A
B′′ := B Q with the obvious matrix multiplication. This makes sense because
h0 B′i
B′ ∼=End (P′[1]) as DG algebras, using the DG algebra isomorphism (2.3). Note
A
that B′′ is a subalgebra of End (L). We make B′′ into a DG algebra with differ-
A
entialdB′′ :=dEndA(L)|B′′. The projections B′′ →B andB′′ →B′ onthe diagonal
entries are DG algebra quasi-isomorphisms, because their kernels are the acyclic
complexes Hom (P′[1],L) and Hom (L,P) respectively.
A A
Now restB′′/B(P) = [P0] and restB′′/B′(P′) = (cid:2)P0′(cid:3) as DG B′′-modules. Define
P′′ := [P0]. It remains to find an isomorphism χ : P′′ −≃→ restB′′/B′(P′) in D(B′′).
Consider the exact sequence
0→[P0]→L→hP′0[1]i→0
in DGMB′′. There is an induced distinguished triangle (cid:2)P0′(cid:3) −→χ [P0] → L −→△ in
D˜(DGMB′′). But L is acyclic, so χ is an isomorphism. (cid:3)
Corollary 2.11. Let B := REnd (M) be the derived endomorphism algebra of
A
M, as in Definition 2.8, constructed using a semi-free resolution P → M. Then
the graded K-algebra Ext (P) is independent, up to isomorphism, of the resolution
B
P →M.
Proof. Let’s go back to the proof of Proposition 2.10. Since B′′ → B is a quasi-
isomorphism of DG algebras,it follows that
restB′′/B :D˜(DGMB)→D˜(DGMB′′)
COMPLETION BY DERIVED DOUBLE CENTRALIZER 7
isanequivalenceoftriangulatedcategories. ThereforerestB′′/B inducesagradedK-
algebraisomorphismExtB(P)−≃→ExtB′′(P′′). SimilarlywegetagradedK-algebra
isomorphism ExtB′(P′)−≃→ExtB′′(P′′). (cid:3)
Definition 2.12. Let M be a DG A-module, and let B := REnd (M) be the
A
derivedendomorphismalgebraofM,asinDefinition2.8,constructedusingasemi-
freeresolutionP →M. ThederiveddoublecentralizerofM isthegradedK-algebra
Ext (M):=Ext (P).
B B
The corollary says that the graded algebra Ext (M) is well-defined up to iso-
B
morphism.
Remark 2.13. Corollary2.11is sufficientfor the purposesof this paper (see The-
orem 0.1). However it would be nice to know that Ext (M) is well-defined up to
B
a unique isomorphism.
Furthermore,wewouldliketohaveabettercharacterization(maybesomesortof
axiomatization)ofthe DG algebraB =REnd (M), andthe lift of M to anobject
A
ofD˜(DGMB). LikewisetheDGalgebraREnd (M)shouldbecharacterized
REndA(M)
uniquely in a suitable sense, together with a homomorphism of DG algebras
A→REnd (M).
REndA(M)
Presumably all that should be done in the context of Quillen homotopical alge-
bra.
3. Supplement on Derived Morita Theory
Derived Morita theory goes back to Rickard’swork [Ri], which dealt with rings.
Further generalizations can be found in [Ke, BV]. Theorem 3.5 and Corollary 3.8
are“folklore”results,andhere wegivecompleteproofs,sincewe needthese results
for Section 4.
Suppose A and B are DG K-algebras, and P is a DG module over A⊗KBop.
Given a left DG B-module N, there is a left DG A-module P ⊗ N. We get a
B
functor
P ⊗ −:DGMB →DGMA.
B
Thetensoroperationrespectshomotopyequivalences. Byrestrictingittosemi-free
DG modules we get a triangulated functor
P ⊗ −:K˜(DGMB) →K˜(DGMA).
B sf
This applies in particular to the case B := End (P)op, since P is automatically a
A
DG A⊗KEndA(P) - module. (Warning: even if P is a semi-free DG A-module, it
is usually not even K-flat over End (P).)
A
For a DG A-module K we have the derived endomorphism DG algebra
REnd (K), which is well-defined up to quasi-isomorphism. See Definition 2.8 and
A
Proposition 2.10.
Proposition3.1. LetEbeabeafulltriangulatedsubcategoryofD˜(DGMA),closed
under infinite direct sums, and let K be an object of E. Define B :=REnd (K)op.
A
Then there is a K-linear triangulated functor G:D˜(DGMB)→E with these prop-
erties:
(i) G commutes with infinite direct sums, and G(B)∼=K.
8 MARCOPORTA,LIRANSHAULANDAMNONYEKUTIELI
(ii) Let P → K be the semi-free resolution used to define B, namely B =
End (P)op. Then the functor
A
G◦En:K˜(DGMB) →D˜(DGMA)
sf
is isomorphic to P ⊗ −.
B
Moreover, such a functor G is unique up to isomorphism.
Proof. Choose a semi-free resolution P → K, and define G(Q) := P ⊗ Q for a
B
semi-free DG B-module Q. The semi-free enhancement (2.5) for the DG algebra
B says that G extends, uniquely up to isomorphism, to a triangulated functor
G:D˜(DGMB)→E. This functor G commutes with infinite direct sums, property
(ii) holds, and G(B)=P ⊗ B ∼=P ∼=K. (cid:3)
B
Definition3.2. LetEbeabeafulltriangulatedsubcategoryofD˜(DGMA),closed
under infinite direct sums. A DG A-module K is said to be compact relative to E
if for any collection {N } of objects of E, the canonical homomorphism
i i∈I
Hom (K,N )→Hom K, N
M D˜(DGMA) i D˜(DGMA)(cid:16) M i(cid:17)
i∈I i∈I
is bijective.
As usual, if K is itself in E, then one calls K a compact object of E.
Let P be a DG module over A⊗KBop, as above. For any N ∈DGMA, we have
a DG B-module Hom (P,N). Thus we get a functor
A
Hom (P,−):DGMA→DGMB.
A
The functor Hom (P,−) respects homotopies, and hence we get an induced trian-
A
gulated functor
Hom (P,−):K˜(DGMA)→K˜(DGMB).
A
Proposition 3.3. Let K be a DG A-module, and let B :=REnd (K)op. There is
A
a K-linear triangulated functor
F :D˜(DGMA)→D˜(DGMB)
with these properties:
(i) F(K)∼=B in D˜(DGMB).
(ii) Let E be a be a full triangulated subcategory of D˜(DGMA), closed under
infinite direct sums. The functor F|E : E → D˜(DGMB) commutes with
infinite direct sums if and only if K is a compact object relative to E.
(iii) Let P → K be the semi-free resolution used to define B, namely B =
End (P)op. Then the functor
A
F ◦En:K˜(DGMA) →D˜(DGMB)
sf
is isomorphic to Hom (P,−).
A
Moreover, the functor F is unique up to isomorphism.
Proof. ExistenceofF,andproperty(iii),areimmediatefromtheequivalence(2.5).
Since K ∼=P in D˜(DGMA) it follows that F(K)∼=F(P)∼=Hom (P,P)=B.
A
It remains to consider property (ii). We know that
HomD˜(DGMA)(K,N[j])∼=Hj(RHomA(K,N))∼=Hj(F(N)),
COMPLETION BY DERIVED DOUBLE CENTRALIZER 9
functorially for N ∈ D˜(DGMA). So K is compact relative to E if and only if the
functors Hj ◦F commute with direct sums in E. But that is the same as asking F
to commute with direct sums in E. (cid:3)
Lemma 3.4. Let E be a triangulated category with infinite direct sums, let F,G :
D˜(DGMA) → E be triangulated functors that commute with infinite direct sums,
and let η : F → G be a morphism of triangulated functors. Assume that η :
A
F(A)→G(A) is an isomorphism. Then η is an isomorphism.
△
Proof. Suppose we are given a distinguished triangle M′ → M → M′′ −→ in
D˜(DGMA), such that two of the three morphisms ηM′, ηM and ηM′′ are isomor-
phisms. Then the third is also an isomorphism.
Since both functors F,G commute with shifts and direct sums, and since η is
A
anisomorphism,itfollowsthatη isanisomorphismforanyfreeDGA-module P.
P
Nextconsiderasemi-freeDGmoduleP. Chooseanysemi-basisX = X ofP.
j
S
Thisgivesrisetoafiltration{Fj(P)}j∈Z ofP byDGsubmodules,withF−1(P)=0.
For every j we have a distinguished triangle
F (P)−θ→j F (P)→F (P)/F (P)−→△
j−1 j j j−1
in D˜(DGMA), where θ :F (P)→F (P) is the inclusion. Since F (P)/F (P)
j j−1 j j j−1
is free, by induction we conclude that η is an isomorphism for every j. The
Fj(P)
telescope construction (see [BN, Remark 2.2]) gives a distinguished triangle
Θ △
F (P)−−→ F (P)→P −→,
Mj∈N j Mj∈N j
with
Θ| :=(id,−θ ):F (P)→F (P)⊕F (P).
Fj−1(P) j j−1 j−1 j
This shows that η is an isomorphism.
P
Finally, any DG module M admits a quasi-isomorphism P → M with P semi-
free. Therefore η is an isomorphism. (cid:3)
M
Theorem 3.5. Let E be a be a full triangulated subcategory of D˜(DGMA), closed
under infinite direct sums, and let K be a compact object of E. Define B :=
REnd (K)op. Consider the K-linear triangulated functors G : D˜(DGMB) → E
A
and F :E→D˜(DGMB) from the previous propositions.
(1) Therearemorphisms oftriangulatedfunctorsη :1→F◦Gandζ :G◦F →
1, that make F into a right adjoint of G.
(2) The morphism η is an isomorphism. Hence the functor G is fully faithful.
(3) Let M ∈E. Then M is in the essential image of the functor G if and only
if the morphism ζ :(G◦F)(M)→M is an isomorphism.
M
Proof. Let P → K be the semi-free resolution used to construct B; namely B =
End (P)op.
A
Take any M ∈E and N ∈D˜(DGMB). We have to construct a bijection
HomD˜(DGMA)(G(N),M)∼=HomD˜(DGMB)(N,F(M)),
10 MARCOPORTA,LIRANSHAULANDAMNONYEKUTIELI
which is bifunctorial. Choose a semi-free resolution Q→N over B. Since the DG
A-moduleP⊗ Qissemi-free,wehaveasequenceofisomorphisms(ofK-modules)
B
HomD˜(DGMA)(G(N),M)∼=H0(RHomA(G(N),M))
∼=H0(Hom (P ⊗ Q,M))∼=H0 Hom (Q,Hom (P,M))
A B B A
(cid:0) (cid:1)
∼=H0(RHomB(N,F(M))∼=HomD˜(DGMB)(N,F(M)).
The only choice made was inthe semi-free resolutionQ→N, so allis bifunctorial.
The corresponding morphisms 1 → F ◦G and G◦F → 1 are denoted by η and ζ
respectively.
We have to prove that the morphism η :N →(F ◦G)(N) in D˜(DGMB) is an
N
isomorphism. Since the functors 1 andF ◦G commute with infinite directsums, it
suffices (by Lemma 3.4) to check for N = B. But in this case η is the canonical
B
homomorphismofDGB-modulesB →Hom (P,P⊗ B),whichisclearlybijective.
A B
It remains to prove part (3). If ζ is an isomorphism then trivially M is in the
M
essential image of G. Conversely, assume that M ∼=G(N) for some DG B-module
N. It is enough to prove that ζ is an isomorphism. But under the bijection
G(N)
HomD˜(DGMB)(N,N)∼=HomD˜(DGMA)(G(N),G(N))
induced by G, 1 goes to ζ . So ζ is invertible. (cid:3)
N G(N) G(N)
Definition 3.6. Let E be a triangulated category. An object K ∈ E is called a
generatorifforanynonzeroM ∈EthereissomeintegerisuchthatHomE(K,M[i])
is nonzero.
Remark 3.7. Thenotionof“generator”aboveistheweakestamongseveralfound
in the literature. See [BV] for discussion.
Corollary 3.8. In the situation of Theorem 3.5, suppose that K is a compact
generator of E. Then the K-linear triangulated functor G : D˜(DGMB) → E is an
equivalence, with quasi-inverse F.
Proof. In view of property (2) of Theorem 1.5, all we have to prove is that G is
essentially surjective on objects. Take any L ∈ E, and consider the distinguished
triangle(G◦F)(L)−ζ→L L→M −→△ inE,inwhichM isthemappingconeofζ . Ap-
L
plyingF andusingηwegetadistinguishedtriangleF(L)−1−F−(L→) F(L)→F(M)−→△ .
So F(M)= 0. But RHomA(K,M)∼= F(M), and therefore HomD(A)(K,M[i])= 0
foreveryi. SinceK isageneratorofEwegetM =0. Henceζ isanisomorphism,
L
and so L is in the essential image of G. (cid:3)
Remark3.9. TheproofsaboveworkalsoforthetriangulatedcategoryD(C),where
C is any abelian category with exact infinite direct sums and enough projectives.
The changes needed are minor – one needs the K-projective enhancement of D(C).
Remark 3.10. A result similar to Theorem 3.5 should be true for the derived
category D(C) of a Grothendieck abelian category C; for instance C := ModA,
where (X,A) is a ringed space. Here one needs the K-injective enhancement of
the triangulated category D(C). See [KS, Theorem 14.3.1]. The details are more
difficult.