COHOMOLOGICAL QUOTIENTS AND SMASHING
LOCALIZATIONS
HENNING KRAUSE
Abstract. The quotient of a triangulated category modulo a subcategory wa*
*s de-
fined by Verdier. Motivated by the failure of the telescope conjecture, w*
*e introduce a
new type of quotients for any triangulated category which generalizes Ver*
*dier's con-
struction. Slightly simplifying this concept, the cohomological quotients*
* are flat epi-
morphisms, whereas the Verdier quotients are Ore localizations. For any c*
*ompactly
generated triangulated category S, a bijective correspondence between the*
* smashing
localizations of S and the cohomological quotients of the category of com*
*pact objects
in S is established. We discuss some applications of this theory, for ins*
*tance the prob-
lem of lifting chain complexes along a ring homomorphism. This is motivat*
*ed by some
consequences in algebraic K-theory and demonstrates the relevance of the *
*telescope
conjecture for derived categories. Another application leads to a derive*
*d analogue
of an almost module category in the sense of Gabber-Ramero. It is shown t*
*hat the
derived category of an almost ring is of this form.
Contents
Introduction 2
Acknowledgements 5
1. Modules 5
2. Cohomological functors 6
3. Flat epimorphisms 7
4. Cohomological quotient functors 10
5. Flat epimorphic quotients 11
6. A criterion for exactness *
*15
7. Exact quotient functors 16
8. Exact ideals 18
9. Factorizations 20
10. Compactly generated triangulated categories and Brown representability *
* 22
11. Smashing localizations 23
12. Smashing subcategories 24
13. The telescope conjecture 28
14. Homological epimorphisms of rings 31
15. Homological localizations of rings *
* 35
16. Almost derived categories 38
Appendix A. Epimorphisms of additive categories 40
____________
Key words and phrases. Triangulated category, derived category, cohomological*
* quotient, smashing
localization, telescope conjecture, non-commutative localization, homological e*
*pimorphism, algebraic
K-theory, almost ring.
Version from August 29, 2003.
1
2 HENNING KRAUSE
Appendix B. The abelianization of a triangulated category *
* 42
References 44
Introduction
The telescope conjecure from stable homotopy theory is a fascinating challeng*
*e for
topologists and algebraists. It is a conjecture about smashing localizations, *
*saying
roughly that every smashing localization is a finite localization. The failure*
* of this
conjecture forces us to develop a general theory of smashing localizations whic*
*h covers
the ones which are not finite. This is precisely the subject of the first part *
*of this pa-
per. The second part discusses some applications of the general theory in the c*
*ontext
of derived categories of associative rings. In fact, we demonstrate the relevan*
*ce of the
telescope conjecture for derived categories, by studying some applications in a*
*lgebraic
K-theory and in almost ring theory.
Let us describe the main concepts and results from this paper. We fix a compa*
*ctly
generated triangulated category S, for example, the stable homotopy category of*
* CW-
spectra or the unbounded derived category of an associative ring. A smashing lo*
*caliza-
tion functor is by definition an exact functor F :S ! T between triangulated ca*
*tegories
having a right adjoint G which preserves all coproducts and satisfies F OG ~= I*
*dT.
Such a functor induces an exact functor Fc:Sc ! Tc between the full subcategori*
*es of
compact objects, and the telescope conjecture [5, 27] claims that the induced f*
*unctor
Sc=Ker Fc ! Tc is an equivalence up to direct factors. Here, Ker Fc denotes th*
*e full
triangulated subcategory of objects X in Sc such that FcX = 0, and Sc=Ker Fc is*
* the
quotient in the sense of Verdier [31]. The failure of the telescope conjecture *
*[15, 20] mo-
tivates the following generalization of Verdier's definition of a quotient of a*
* triangulated
category. To be precise, there are examples of proper smashing localization fun*
*ctors F
where Ker Fc = 0. Nonetheless, the functor Fc is a cohomological quotient funct*
*or in
the following sense.
Definition. Let F :C ! D be an exact functor between triangulated categories. W*
*e call
F a cohomological quotient functor if for every cohomological functor H :C ! A *
*satisfy-
ing Ann F Ann H, there exists, up to a unique isomorphism, a unique cohomolog*
*ical
functor H0:D ! A such that H = H0OF .
Here, Ann F denotes the ideal of all maps OE in C such that F OE = 0. The pro*
*perty of F
to be a cohomological quotient functor can be expressed in many ways, for insta*
*nce more
elementary as follows: every object in D is a direct factor of some object in t*
*he image
of F , and every map ff: F X ! F Y in D can be composed with a split epimorphism
F ß :F X0! F X such that ff OF ß belongs to the image of F .
Our main result shows a close relation between cohomological quotient functor*
*s and
smashing localizations.
Theorem 1. Let S be a compactly generated triangulated category, and let F :Sc *
*! T
be a cohomological quotient functor. Denote by R the full subcategory of object*
*s X in
S such that every map C ! X from a compact object C factors through some map in
Ann F .
COHOMOLOGICAL QUOTIENTS 3
(1)The category R is a triangulated subcategory of S and the quotient functo*
*r S !
S=R is a smashing localization functor which induces a fully faithful and*
* exact
functor T ! S=R making the following diagram commutative.
Sc _____F_____//_T
|inc| ||
fflffl|can |fflffl
S __________//_S=R
(2)The triangulated category S=R is compactly generated and the subcategory *
*of
compact objects is precisely the closure of the image of T ! S=R under fo*
*rming
direct factors.
(3)There exists a fully faithful and exact functor G: T ! S such that
S(X, GY ) ~=T (F X, Y )
for all X in Sc and Y in T .
One may think of this result as a generalization of the localization theorem *
*of Neeman-
Ravenel-Thomason-Trobaugh-Yao [24, 27, 30, 34]. To be precise, Neeman et al. co*
*nsid-
ered cohomological quotient functors of the form Sc ! Sc=R0 for some triangulat*
*ed
subcategory R0 of Sc and analyzed the smashing localization functor S ! S=R whe*
*re
R denotes the localizing subcategory generated by R0.
Our theorem provides a bijective correspondence between smashing localization*
*s of S
and cohomological quotients of Sc; it improves a similar correspondence [18] - *
*the new
ingredient in our proof being a recent variant [19] of Brown's Representability*
* Theorem
[6]. The essential invariant of a cohomological quotient functor F :Sc ! T is t*
*he ideal
Ann F . The ideals of Sc which are of this form are called exact and are precis*
*ely those
satisfying the following properties:
(1)I2 = I.
(2)I is saturated, that is, for every exact triangle X0!ffX fi!X00! X0 and *
*every
map OE: X ! Y in Sc, we have that OE Off, fi 2 I implies OE 2 I.
(3) I = I.
Let us rephrase the telescope conjecture in terms of exact ideals and cohomol*
*ogical
quotient functors. To this end, recall that a subcategory of S is smashing if i*
*t is of the
form Ker F for some smashing localization functor F :S ! T .
Corollary. The telescope conjecture for S is equivalent to each of the followin*
*g state-
ments.
(1)Every smashing subcategory of S is generated by compact objects.
(2)Every exact ideal is generated by idempotent elements.
(3)Every cohomological quotient functor F :Sc ! T induces up to direct facto*
*rs an
equivalence Sc=Ker F ! T .
(4)Every flat epimorphism F :Sc ! T satisfying (Ann F ) = Ann F is an Ore
localization.
This reformulation of the telescope conjecture is based on our approach to vi*
*ew a
triangulated category as a ring with several object. In this setting, the cohom*
*ological
quotient functors are the flat epimorphisms, whereas the Verdier quotient funct*
*ors are
4 HENNING KRAUSE
the Ore localizations. The reformulation in terms of exact ideals refers to the*
* classical
problem from ring theory of finding idempotent generators for an idempotent ide*
*al,
studied for instance by Kaplansky [13] and Auslander [1]. We note that the tele*
*scope
conjecture becomes a statement about the category of compact objects. Moreover,*
* we
see that the smashing subcategories of S form a complete lattice which is isomo*
*rphic to
the lattice of exact ideals in Sc.
The second part of this paper is devoted to studying non-commutative localiza*
*tions
of rings. We do this by using unbounded derived categories and demonstrate that*
* the
telescope conjecture is relevant in this context. This is inspired by recent wo*
*rk of Neeman
and Ranicki [26]. They study the problem of lifting chain complexes up to homot*
*opy
along a ring homomorphism R ! S. To make this precise, let us denote by Kb(R) t*
*he
homotopy category of bounded complexes of finitely generated projective R-modul*
*es.
(1)We say that the chain complex lifting problem has a positive solution, if*
* every
complex Y in Kb(S) such that for each i we have Y i= P i R S for some fin*
*itely
generated projective R-module P i, is isomorphic to X R S for some compl*
*ex
X in Kb(R).
(2)We say that the chain map lifting problem has a positive solution, if for*
* every
pair X, Y of complexes in Kb(R) and every map ff: X R S ! Y R S in Kb(S*
*),
there are maps OE: X0 ! X and ff0:X0 ! Y in Kb(R) such that OE R S is
invertible and ff = ff0 R S O(OE R S)-1 in Kb(S).
Note that complexes can be lifted whenever maps can be lifted. For example, ma*
*ps
and complexes can be lifted if R ! S is a commutative localization. However, th*
*ere are
obstructions in the non-commutative case, and this leads to the concept of a ho*
*mological
epimorphism. Recall from [12] that R ! S is a homological epimorphism if S R S*
* ~=S
and TorRi(S, S) = 0 for all i 1. For example, every commutative localization *
*is a flat
epimorphism and therefore a homological epimorphism. The following observation*
* is
crucial for both lifting problems.
Proposition. A ring homomorphism R ! S is a homological epimorphism if and only
if - R S :Kb(R) ! Kb(S) is a cohomological quotient functor.
This shows that we can apply our theory of cohomological quotient functors, a*
*nd we
see that the telescope conjecture for the unbounded derived category D(R) of a *
*ring R
becomes relevant. In particular, we obtain a non-commutative analogue of Thomas*
*on-
Trobaugh's localization theorem for algebraic K-theory [30].
Theorem 2. Let R be a ring such that the telescope conjecture holds true for D(*
*R).
Then the chain map lifting problem has a positive solution for a ring homomorph*
*ism
f :R ! S if and only if f is a homological epimorphism. Moreover, in this case*
* f
induces a sequence
K(R, f) -! K(R) -! K(S)
of K-theory spectra which is a homotopy fibre sequence, up to failure of surjec*
*tivity of
K0(R) ! K0(S). In particular, there is induced a long exact sequence
. .-.! K1(R) -! K1(S) -! K0(R, f) -! K0(R) -! K0(S)
of algebraic K-groups.
Unfortunately, not much seems to be known about the telescope conjecture for *
*derived
categories. Note that the telescope conjecture has been verified for D(R) prov*
*ided R
COHOMOLOGICAL QUOTIENTS 5
is commutative noetherian [23]. On the other hand, there are counter examples w*
*hich
arise from homological epimorphisms where not all chain maps can be be lifted [*
*15].
In the final part of this paper, we introduce the derived analogue of an almo*
*st module
category in the sense of [9]. In fact, there is a striking parallel between alm*
*ost rings and
smashing localizations: both concepts depend on an idempotent ideal. Given a ri*
*ng R
and an idempotent ideal a, the category of almost modules is by definition the *
*quotient
Mod (R, a) = Mod R=(a? ),
where Mod R denotes the category of right R-modules and a? denotes the Serre su*
*b-
category of R-modules annihilated by a. Given an idempotent ideal I of Kb(R) wh*
*ich
satifies I = I, the objects in D(R) which are annihilated by I form a triangul*
*ated
subcategory, and we call the quotient category
D(R, I) = D(R)=(I? )
an almost derived category. It turns out that the almost derived categories are*
*, up to
equivalence, precisely the smashing subcategories of D(R). Moreover, as one sh*
*ould
expect, the derived category of an almost ring is an almost derived category.
Theorem 3. Let R be a ring and a be an idempotent ideal such that a R a is flat*
* as left
R-module. Then the maps in Kb(R) which annihilate all suspensions of the mappi*
*ng
cone of the natural map a R a ! R form an idempotent ideal A, and D(R, A) is
equivalent to the unbounded derived category of Mod (R, a).
Acknowledgements. I would like to thank Ragnar Buchweitz, Bernhard Keller, and
Amnon Neeman for several stimulating discussions during a visit to the Mathemat*
*ical
Sciences Institute in Canberra in July 2003.
1. Modules
The homological properties of an additive category C are reflected by propert*
*ies of
functors from C to various abelian categories. In this context, the abelian cat*
*egory Ab
of abelian groups plays a special role, and this leads to the concept of a C-mo*
*dule. In
this section we give definitions and fix some terminology.
Let C and D be additive categories. We denote by Hom (C, D) the category of f*
*unctors
from C to D. The natural transformations between two functors form the morphism*
*s in
this category, but in general they do not form a set. A category will be called*
* large to
point out that the morphisms between fixed objects are not assumed to form a se*
*t.
A C-module is by definition an additive functor Cop ! Ab into the category Ab
of abelian groups, and we denote for C-modules M and N by Hom C(M, N) the class
of natural transformations M ! N. We write Mod C for the category of C-modules
which is large, unless C is small, that is, the isomorphism classes of objects *
*in C form
a set. A sequence L ! M ! N of maps between C-modules is exact if the sequence
LX ! MX ! NX is exact for all X in C. We denote for every X in C by HX = C(-, X)
the corresponding representable functor and recall that Hom C(HX , M) ~=MX for *
*every
module M by Yoneda's lemma. It follows that HX is a projective object in Mod C.
A C-module M is called finitely presented if it fits into an exact sequence
C(-, X) -! C(-, Y ) -! M ! 0
6 HENNING KRAUSE
with X and Y in C. Note that Hom C(M, N) is a set for every finitely presented *
*C-module
M by Yoneda's lemma. The finitely presented C-modules form an additive category*
* with
cokernels which we denote by mod C.
Now let F :C ! D be an additive functor. This induces the restriction functor
F*: Mod D -! Mod C, M 7-! M OF,
and its left adjoint
F *:Mod C -! Mod D
which sends a C-module M, written as a colimit M = colimff2MXC(-, X) of repre-
sentable functors, to
F *M = colimff2MXD(-, F X).
Note that every C-module can be written as a small colimit of representable fun*
*ctors
provided C is small. The finitely presented C-modules are precisely the finite *
*colimits of
representable functors. We denote the restriction of F *by
F ?:mod C -! mod D
and observe that F ?is the unique right exact functor mod C ! mod D sending C(-*
*, X)
to D(-, F X) for all X in C.
Finally, we define
Ann F = the ideal of all maps OE 2 C with F OE = 0, and
Ker F = the full subcategory of all objects X 2 C with F X = 0.
Recall that an ideal I in C consists of subgroups I(X, Y ) in C(X, Y ) for ever*
*y pair of
objects X, Y in C such that for all OE in I(X, Y ) and all maps ff: X0! X and f*
*i :Y ! Y 0
in C the composition fi OOE Off belongs to I(X0, Y 0). Note that all ideals in *
*C are of the
form Ann F for some additive functor F .
Given any class of maps in C, we say that an object X in C is annihilated b*
*y ,
if Ann C(-, X). We denote by ? the full subcategory of objects in C which *
*are
annihilated by .
2.Cohomological functors
Let C be an additive category and suppose mod C is abelian. Note that mod C *
*is
abelian if and only if every map Y ! Z in C has a weak kernel X ! Y , that is, *
*the
sequence C(-, X) ! C(-, Y ) ! C(-, Z) is eaxct. In particular, mod C is abelian*
* if C is
triangulated. A functor F :C ! A to an abelian category A is called cohomologic*
*al if it
sends every weak kernel sequence X ! Y ! Z in C to an exact sequence F X ! F Y !
F Z in A. If C is a triangulated category, then a functor F :C ! A is cohomolog*
*ical if
and only if F sends every exact triangle X ! Y ! Z ! X in C to an exact sequen*
*ce
F X ! F Y ! F Z ! F X in A. The Yoneda functor
HC: C -! mod C, X 7! HX = C(-, X)
is the universal cohomological functor for C. More precisely, for every abelian*
* category
A, the functor
Hom (HC, A): Hom (mod C, A) -! Hom (C, A)
induces an equivalence
Hom ex(mod C, A) -! Hom coh(C, A),
COHOMOLOGICAL QUOTIENTS 7
where the subscripts ex = exact and coh = cohomological refer to the appropriat*
*e full
subcategories; see [8, 31] and also [18, Lemma 2.1].
Following [18], we call an ideal I in C cohomological if there exists a cohom*
*ological
functor F :C ! A such that I = Ann F . For example, if F :C ! D is an exact
functor between triangulated categories, then Ann F is cohomological because An*
*n F =
Ann(HD OF ). Note that the cohomological ideals of C form a complete lattice, p*
*rovided
C is small. For instance, given a family (Ii)i2 of cohomological ideals, we ha*
*ve
"
infIi= Ii,
i
T
because iIi= Ann F for
Y
F :C -! Ai, X 7! (FiX)i2
i
where each Fi:C ! Ai is a cohomological functor satisfying Ii = Ann Fi. We obta*
*in
supIi by taking the infimum of all cohomological ideals J with Ii J for all i *
*2 .
3. Flat epimorphisms
The concept of a flat epimorphisms generalizes the classical notion of an Ore*
* local-
ization. We study flat epimorphisms of additive categories, following the idea *
*that an
additive category may be viewed as a ring with several objects. Given a flat ep*
*imorphism
C ! D, it is shown that the maps in D are obtained from those in C by a general*
*ized
calculus of fractions. There is a close link between flat epimorphisms and quo*
*tients
of abelian categories. It is the aim of this section to explain this connectio*
*n which is
summarized in Theorem 3.8. We start with a brief discussion of quotients of ab*
*elian
categories.
Let C be an abelian category. A full subcategory B of C is called a Serre sub*
*category
provided that for every exact sequence 0 ! X0 ! X ! X00! 0 in C, the object X
belongs to B if and only if X0 and X00belong to B. The quotient C=B with respec*
*t to a
Serre subcategory B is by definition the localization C[ -1], where denotes t*
*he class
of maps OE in C such that Ker OE and CokerOE belong to B; see [10, 11]. The loc*
*alization
functor Q: C ! C=B yields for every category E a functor
Hom (Q, E): Hom (C=B, E) -! Hom (C, E)
which induces an isomorphism onto the full subcategory of functors F :C ! E suc*
*h that
F OE is invertible for all OE 2 . Note that C=B is abelian and Q is exact with*
* KerQ = B.
Up to an equivalence, a localization functor can be characterized as follows.
Lemma 3.1. Let F :C ! D be an exact functor between abelian categories. Then the
following are equivalent.
(1)F induces an equivalence C=Ker F ! D.
(2)For every abelian category A, the functor
Hom (F, A): Hom ex(D, A) -! Hom ex(C, A)
induces an equivalence onto the full subcategory of functors G: C ! A sat*
*isfying
Ker F KerG.
Proof.See [10, III.1].
8 HENNING KRAUSE
An exact functor between abelian categories satisfying the equivalent conditi*
*ons of
Lemma 3.1 is called an exact quotient functor. There is a further characteriza*
*tion in
case the functor has a right adjoint.
Lemma 3.2. Let F :C ! D be an exact functor between abelian categories and supp*
*ose
there is a right adjoint G: D ! C. Then F is a quotient functor if and only if *
*G is fully
faithful. In this case, G identifies D with the full subcategory of objects X i*
*n C satisfying
C(Ker F, X) = 0 and Ext1C(Ker F, X) = 0.
Proof.See Proposition III.3 and Proposition III.5 in [10].
Next we analyze an additive functor F :C ! D in terms of the induced functor
F ?:mod C ! mod D.
Lemma 3.3. Let F :C ! D be an additive functor and suppose F ?:mod C ! mod D
is an exact quotient functor of abelian categories.
(1)Every object in D is a direct factor of some object in the image of F .
(2)For every map ff: F X ! F Y in D, there are maps ff0:X0! Y and ß :X0! X
in C such that F ff0= ff OF ß and F ß is a split epimorphism.
Proof.The functor F ?:mod C ! mod D is, up to an equivalence, a localization fu*
*nctor.
Therefore the objects in mod D coincide, up to isomorphism, with the objects in*
* mod C.
Moreover, the maps in mod D are obtained via a calculus of fractions from the m*
*aps in
mod C; see [11, I.2.5].
(1) Fix an object Y in D. Then D(-, Y ) ~=F ?M for some M in mod C. If M is a
quotient of C(-, X), then F ?M is a quotient of D(-, F X). Thus Y is a direct f*
*actor of
F X.
(2) Fix a map ff: F X ! F Y . The corresponding map D(-, ff) in mod D is a fr*
*action,
that is, of the form
(F?ff)-1 ? F?ffi ?
D(-, F X) = F ?C(-, X) -----! F M --! F C(-, Y ) = D(-, F Y )
for some M in mod C. Choose an epimorphism æ: C(-, X0) ! M for some X0 in C.
Now define ff0:X0 ! Y by C(-, ff0) = OE Oæ, and define ß :X0 ! X by C(-, ß) = o*
*e Oæ.
Clearly, F ß is a split epimorphism since D(-, F X) is a projective object in m*
*od D.
Remark 3.4. Conditions (1) and (2) in Lemma 3.3 imply that every map in D is a *
*direct
factor of some map in the image of F . To be precise, we say that a map ff: X !*
* X0 is
a direct factor of a map fi :Y ! Y 0if there is a commutative diagram
__"__// __i__//
X Y X
ff|| fi|| |ff|
fflffl|"fflffl|fflffl|0i0
X0 ____//_Y_0__//Y 0
such that ß O" = idX and ß0O"0= idX0.
Recall that an additive functor F :C ! D is an epimorphism of additive catego*
*ries,
or simply an epimorphism, if G OF = G0OF implies G = G0for any pair G, G0:D ! E
of additive functors.
Lemma 3.5. Let F :C ! D be an additive functor having the following properties:
(1)Every object in D belongs to the image of F .
COHOMOLOGICAL QUOTIENTS 9
(2)For every map ff: F X ! F Y in D, there are maps ff0:X0! Y and ß :X0! X
in C such that F ff0= ff OF ß and F ß is a split epimorphism.
Then F is an epimorphism.
Proof.Let G, G0:D ! E be a pair of additive functors satisfying G OF = G0OF . T*
*he
first condition implies that G and G0 coincide on objects, and the second condi*
*tion
implies that G and G0coincide on maps. Thus G = G0.
Next we explain the notion of a flat functor. A Cop-module M is called flat *
*if for
every map fi :Y ! Z in C and every y 2 Ker(Mfi), there exists a map ff: X ! Y i*
*n C
and some x 2 MX such that (Mff)x = y. We call an additive functor F :C ! D flat*
* if
the Cop-module D(X, F -) is flat for every X in D. We record without proof a nu*
*mber
of equivalent conditions which justify our terminology.
Lemma 3.6. Let F :C ! D be an additive functor. Suppose C is small and mod C is
abelian. Then the following are equivalent.
(1)D(X, F -) is a flat Cop-module for every X in D.
(2)F preserves weak kernels.
(3)F ?:mod C ! mod D is exact.
(4)F *:Mod C ! Mod D is exact.
Given an additive functor F :C ! D, we continue with a criterion on F such th*
*at
F ?:mod C ! mod D is an exact quotient functor.
Lemma 3.7. Let F :C ! D be an additive functor. Suppose C is small and mod C is
abelian. If F is a flat and F*: Mod D ! Mod C is fully faithful, then F ?:mod *
*C !
mod D is an exact quotient functor of abelian categories.
Proof.The functor F *:Mod C ! Mod D is exact because F is flat, and it is a qu*
*otient
functor because F* is fully faithful. This follows from Lemma 3.2 since F* is t*
*he right
adjoint of F *. We conclude that the restriction F ?= F *|mod Cto the category *
*of finitely
presented modules is an exact quotient functor, for instance by [17, Theorem 2.*
*6].
Let F :C ! D be an additive functor, and denote by D0 the full subcategory of*
* D
formed by the objects in the image of F . We say that F is an epimorphism up to*
* factors,
if the induced functor C ! D0 is an epimorphism, and if every object in D is a *
*direct
factor of some object in D0.
The following result summarizes our discussion and provides a characterizatio*
*n of flat
epimorphisms.
Theorem 3.8. Let F :C ! D be an additive functor between additive categories. S*
*up-
pose mod C is abelian and F is flat. Then the following are equivalent.
(1)The exact functor F ?:mod C ! mod D, sending C(-, X) to D(-, F X) for all
X in C, is a quotient functor of abelian categories.
(2)Every object in D is a direct factor of some object in the image of F . A*
*nd for
every map ff: F X ! F Y in D, there are maps ff0:X0 ! Y and ß :X0 ! X in
C such that F ff0= F ß Off and F ß is a split epimorphism.
(3)F is an epimorphism up to factors.
Proof.The implication (1) ) (2) follows from Lemma 3.3, and the implication (2)*
* ) (3)
from Lemma 3.5. To prove the implication (3) ) (1), assume that F is an epimorp*
*hism
10 HENNING KRAUSE
up to factors. We need to enlarge our universe so that C becomes a small categ*
*ory.
Note that this does not affect our assumption on F , by Lemma A.6. It follows *
*from
Proposition A.5 that F*: Mod D ! Mod C is fully faithful, and Lemma 3.7 implies*
* that
F ?:mod C ! mod D is a quotient functor.
4.Cohomological quotient functors
In this section we introduce the concept of a cohomological quotient functor *
*between
two triangulated categories. This concept generalizes the classical notion of a*
* quotient
functor C ! C=B which Verdier introduced for any triangulated subcategory B of *
*C; see
[31].
Definition 4.1. Let F :C ! D be an exact functor between triangulated categorie*
*s.
We call F a cohomological quotient functor if for every cohomological functor H*
* :C ! A
satisfying Ann F Ann H, there exists, up to a unique isomorphism, a unique co*
*homo-
logical functor H0:D ! A such that H = H0OF .
Let us explain why a quotient funtor C ! C=B in the sense of Verdier is a coh*
*omo-
logical quotient functor. To this end we need the following lemma.
Lemma 4.2. Let F :C ! D be an exact functor between triangulated categories and
suppose F induces an equivalence C=B ! D for some triangulated subcategory B of*
* C.
Then Ann F is the ideal of all maps in C which factor through some object in B.
Proof.The quotient C=B is by definition the localization C[ -1] where is the *
*class
of maps X ! Y in C which fit into an exact triangle X ! Y ! Z ! X with Z in
B. Now fix a map _ :Y ! Z in Ann F . The maps in C=B are described via a calcul*
*us
of fractions. Thus F _ = 0 implies the existence of a map OE: X ! Y in such *
*that
_ OOE = 0. Complete OE to an exact triangle X ! Y ! Z0 ! X. Clearly, _ facto*
*rs
through Z0 and Z0 belongs to B. Thus Ann F is the ideal of maps which factor th*
*rough
some object in B.
Example 4.3. A quotient functor F :C ! C=B is a cohomological quotient functor.
To see this, observe that a cohomological functor H :C ! A with Ker H containing
B factors uniquely through F via some cohomological functor H0:C=B ! A; see [31,
Corollaire II.2.2.11]. Now use that
B KerH () Ann F Ann H,
which follows from Lemma 4.2.
It turns out that cohomological quotients are closely related to quotients of*
* additive
and abelian categories. The following result makes this relation precise and pr*
*ovides a
number of characterizations for a functor to be a cohomological quotient functo*
*r.
Theorem 4.4. Let F :C ! D be an exact functor between triangulated categories. *
*Then
the following are equivalent.
(1)F is a cohomological quotient functor.
(2)The exact functor F ?:mod C ! mod D, sending C(-, X) to D(-, F X) for all
X in C, is a quotient functor of abelian categories.
(3)Every object in D is a direct factor of some object in the image of F . A*
*nd for
every map ff: F X ! F Y in D, there are maps ff0:X0 ! Y and ß :X0 ! X in
C such that F ff0= F ß Off and F ß is a split epimorphism.
COHOMOLOGICAL QUOTIENTS 11
(4)F is up to factors an epimorphism of additive categories.
Proof.All we need to show is the equivalence of (1) and (2). The rest then fol*
*lows
from Theorem 3.8. Fix an abelian category A and consider the following commutat*
*ive
diagram
Hom (F?,A)
Hom ex(mod D, A)__________//_Homex(mod C, A)
Hom|(HD,A)| Hom|(HC,A)|
fflffl| Hom(F,A) fflffl|
Hom coh(D, A)_____________//_Homcoh(C, A)
where the vertical functors are equivalences. Observe that Hom (HC, A) identi*
*fies the
exact functors G: mod C ! A satisfying Ker F ? KerG with the cohomological fun*
*c-
tors H :C ! A satisfying Ann F Ann H. This follows from the fact that each M
in mod C is of the form M = Im C(-, OE) for some map OE in C. We conclude that*
* the
property of F ?to be an exact quotient functor, is equivalent to the property o*
*f F to be
a cohomological quotient functor.
We complement the description of cohomological quotient functors by a charact*
*eriza-
tion of quotient functors in the sense of Verdier.
Proposition 4.5. Let F :C ! D be an exact functor between triangulated categori*
*es.
Then the following are equivalent.
(1)F induces an equivalence C=Ker F ! D.
(2)Every object in D is isomorphic to some object in the image of F . And fo*
*r every
map ff: F X ! F Y in D, there are maps ff0:X0! Y and ß :X0! X in C such
that F ff0= F ß Off and F ß is an isomorphism.
Proof.Let B = Ker F and denote by Q: C ! C=B the quotient functor, which is the
identity on objects. Given objects X and Y in C, the maps X ! Y in C=B are frac*
*tions
of the form
(Qi)-1 0 Qff
X ----! X --! Y
such that F ß is an isomorphism. This shows that (1) implies (2). To prove the *
*converse,
denote by G: C=B ! D the functor which is induced by F . The description of the*
* maps
in C=B implies that G is full. It remains to show that G is faithful. To this e*
*nd choose
a map _ :Y ! Z such that F _ = 0. We complete _ to an exact triangle
X -ffi!Y -_! Z -! X
and observe that F OE is a split epimorphism. Choose an inverse ff: F Y ! F X*
* and
write it as F ff0O(F ß)-1, using (2). Thus Q(OE Off0) is invertible, and _ OOE *
*Off0= 0 implies
Q_ = 0 in C=B. We conclude that G is faithful, and this completes the proof.
5. Flat epimorphic quotients
In this section we establish a triangulated structure for every additive cate*
*gory which
is a flat epimorphic quotient of some triangulated category.
Theorem 5.1. Let C be a triangulated category, and let D be an additive categor*
*y with
split idempotents. Suppose F :C ! D is a flat epimorphism up to factors satisf*
*ying
(Ann F ) = Ann F . Then there exists a unique triangulated structure on D such*
* that
12 HENNING KRAUSE
F is exact. Moreover, a triangle in D is exact if and only if there is an exa*
*ct triangle
in C such that is a direct factor of F .
Note that an interesting application arises if one takes for D the idempotent*
* comple-
tion of C. In this case, one obtains the main result of [2].
The proof of Theorem 5.1 is given in several steps and requires some preparat*
*ions.
Assuming the suspension : D ! D is already defined, let us define the exact tr*
*iangles
in D. We call a triangle in D exact, if there exists an exact triangle in C*
* such that
is a direct factor of F , that is, there are triangle maps OE: ! F and _*
* :F !
such that _ OOE = id .
From now on assume that F :C ! D is a flat epimorphism up to factors, satisfy*
*ing
(Ann F ) = Ann F . We simplify our notation and identify C with the image of *
*the
Yoneda functor C ! mod C. The same applies to the Yoneda functor D ! mod D.
Moreover, we identify F ?= F and ? = . Note that F :mod C ! mod D is an exact
quotient functor by Theorem 3.8. In particular, the maps in mod D are obtained *
*from
maps in mod C via a calculus of fractions. Let us construct the suspension for *
*D.
Lemma 5.2. There is an equivalence 0: mod D ! mod D making the following dia-
gram commutative.
__F__//
mod C mod D
|| ||0
fflffl|F fflffl|
mod C_____//modD
The equivalence 0is unique up to a unique isomorphism.
Proof.Every object in mod D is isomorphic to F M for some M in mod C. And every
map ff: F M ! F N is a fraction, that is, of the form
(Fff)-1
F M -Fffi-!F N0 ----! F N.
Now define 0(F M) = F ( M) and 0ff = F ( oe)-1 OF ( OE).
We shall abuse notation and identify 0 = . Now fix M, N in mod D. We may
assume that M = F M0 and N = F N0. We have a natural map
~M0,N0: Hom C(M0, N0) -! Ext3C( M0, N0)
which is induced from the triangulated structure on C; see Appendix B. This map
induces a natural map
~M,N : Hom D(M, N) -! Ext3D( M, N)
since every map F M0 ! F N0 is a fraction of maps in the image of F . Recall t*
*hat
~M = ~M,M (idM ). Let
: X -ff!Y -fi!Z -fl! X
be a triangle in D and put M = Kerff. We call pre-exact, if fl induces a map *
*Z ! M
such that the sequence
0 -! M -! X -ff!Y -fi!Z -! M -! 0
is exact in mod D and represents ~M 2 Ext3D( M, M). Note that every exact tria*
*ngle
in D is pre-exact, by Proposition B.2.
COHOMOLOGICAL QUOTIENTS 13
Lemma 5.3. Given a commutative diagram
fi fl
(5.1) X __ff_//Y____//Z_____// X
|ffi| |_| ||ffi
fflffl|ffflffl|f0fi0fl0 fflffl|
X0 ____//_Y_0__//_Z0___// X0
in D such that both rows are pre-exact triangles, there exists a map æ: Z ! Z0 *
*such that
the completed diagram commutes. Moreover, if OE2 = OE and _2 = _, then there ex*
*ists a
choice for æ such that æ2 = æ.
Proof.Let M = Kerff and M0 = Kerff0. The pair OE, _ induces a map ~: M ! M0 and
we obtain the following diagram in mod D.
(5.2) ~M : 0_____// -2M _____//Z____//_ M_____//0
||-2~ |~|
fflffl| fflffl|
~M0 : 0_____// -2M0_____//Z0___//_ M0____//_0
Here we use a dimension shift to represent ~M and ~M0 by short exact sequences*
*. The
map ~M,N is natural in M and N, and therefore ~M,~(~M ) = ~~,M0(~M0). This impl*
*ies
the existence of a map æ: Z ! Z0making the diagram (5.2) commutative. Note that*
* we
can choose æ to be idempotent if ~ is idempotent. It follows that the map æ com*
*pletes
the diagram (5.1) to a map of triangles.
Lemma 5.4. Every map X ! Y in D can be completed to an exact triangle X ! Y !
Z ! X.
Proof.A map in D is a direct factor of some map in the image of F by Theorem 3.*
*8;
see also Remark 3.4. Thus we have a commutative square
_Fff//_ 0
F X0 F Y
|ffi| _||
fflffl|Ffffflffl|
F X0 ____//_F Y 0
such that OE and _ are idempotent and the map X ! Y equals the map Im OE ! Im _
induced by F ff. We complete ff to an exact triangle in C and extend the pair*
* OE, _ to
an idempotent triangle map ": F ! F , which is possible by Lemma 5.3. The im*
*age
Im" is an exact triangle in D, which completes the map X ! Y .
We are now in the position to prove the octahedral axiom for D. Note that we
have already established that D is a pre-triangulated category. We say that a p*
*air of
composable maps ff: X ! Y and fi :Y ! Z can be completed to an octahedron if th*
*ere
14 HENNING KRAUSE
exists a commutative diagram of the form
__ff_//_______// _____//_
X Y U X
|| | | ||
|| fi| | ||
||fi Offfflffl| |fflffl ||
X _____//_Z_____//V_____//_ X
| |
| |
fflffl| |fflffl
W _______W
| |
| |
fflffl| |fflffl
Y ____//_ U
such that all triangles which occur are exact.
We shall use the following result due to Balmer and Schlichting.
Lemma 5.5. Let ff: X ! Y and fi :Y ! Z be maps in a pre-triangulated category.
Suppose there are objects X0, Y 0, Z0 such that
h i
[ff000] 0 0 fi000 0
X q X0- --! Y q Y and Y q Y ----! Z q Z
can be completed to an octahedron. Then ff and fi can be completed to an octahe*
*dron.
Proof.See the proof of Theorem 1.12 in [2].
Lemma 5.6. Every pair of composable maps in D can be completed to an octahedron.
Proof.Fix two maps ff: X ! Y and fi :Y ! Z in D. We proceed in two steps. First
assume that X = F A, Y = F B, and Z = F C. We use the description of the maps i*
*n D
which is given in Theorem 3.8. We consider the map fi :Y ! Z and obtain new maps
_ :B0! C and ß :B0! B in C such that F _ = fi OF ß and F ß is a split epimorphi*
*sm.
We get a decomposition F B0= Y q Y 0and an automorphism ": Y q Y 0! Y q Y 0such
that F _ O" = [fi0]. The same argument, applied to the composite
hid i
Y
X -ff!Y -0! Y q Y 0,
gives a map OE: A0! B0 in C, a decomposition F A0= X q X0, and an automorphism
ffi :X q X0! X q X0 such that F OE Offi = [ff000]. We know that the pair OE, _ *
*in C can be
completed to an octahedron. Thus F OE and F _ can be completed to an octahedron*
* in
D. It follows that [ff000]and [fi0]can be completed to an octahedron. Using Lem*
*ma 5.5,
we conclude that the pair ff, fi can be completed to an octahedron.
In the second step of the proof, we assume that the objects X, Y , and Z are *
*arbitrary.
Applying again the description of the maps in D, we find objects X0, Y 0, and Z*
*0 in D
such that X q X0, Y q Y 0, and Z q Z0 belong to the image of F . We know from t*
*he first
part of the proof that the maps
h i
[ff000] 0 0 fi000 0
X q X0- --! Y q Y and Y q Y ----! Z q Z
can be completed to an octahedron. From this it follows that ff and fi can be c*
*ompleted
to an octahedron, using again Lemma 5.5. This finishes the proof of the octahe*
*dral
axiom for D.
COHOMOLOGICAL QUOTIENTS 15
Let us complete the proof of Theorem 5.1.
Proof of Theorem 5.1.We have constructed an equivalence : D ! D, and the exact
triangles in D are defined as well. We need to verify the axioms (TR1) - (TR4) *
*from
[31]. Let us concentrate on the properties of D, which are not immediately clea*
*r from
our set-up. In Lemma 5.4, it is shown that every map in D can be completed to *
*an
exact triangle. In Lemma 5.3, it is shown that every partial map between exact *
*triangles
can be completed to a full map. Finally, the octahedral axiom (TR4) is establis*
*hed in
Lemma 5.6.
6. A criterion for exactness
Given an additive functor C ! D between triangulated categories, it is a natu*
*ral
question to ask when this functor is exact. We provide a criterion in terms of*
* the
induced functor mod C ! mod D and the extension ~M in Ext3( ?M, M) defined for
each M in mod C; see Appendix B. There is an interesting consequence. Given a*
*ny
factorization F = F2O F1 of an exact functor, the functor F2 is exact provided *
*that F1
is a cohomological quotient functor.
Proposition 6.1. Let F :C ! D be an additive functor between triangulated categ*
*ories.
Then F is exact if and only if the following holds:
(1)The right exact functor F ?:mod C ! mod D, sending C(-, X) to D(-, F X) f*
*or
all X in C, is exact.
(2)There is a natural isomorphism j :F O C ! D OF .
(3)F ?~M = Ext3D(j?M, F ?M)(~F?M ) for all M in mod C.
Here, we denote by j? the natural isomorphism F ?O ?C! ?DOF ?which extends j,
that is, j?C(-,X)= D(-, jX ) for all X in C.
Proof.Suppose first that (1) - (3) hold. Let
: X -ff!Y -! Z -! CX
be an exact triangle in C. We need to show that F sends this triangle to an exa*
*ct triangle
in D. To this end complete the map F ff to an exact triangle
F X Fff-!F Y -! Z0- ! D (F X).
Now let M = Ker C(-, ff). We use a dimension shift to represent the class ~M *
*by a
short exact sequence corresponding to an element in Ext1C( ?CM, -2M). Analogou*
*sly,
we represent ~F?M by a short exact sequence. Next we use the exactness of F ?t*
*o obtain
the following diagram in mod D.
F ?~M : 0 ____//_F ?( -2M)____//D(-, F Z)___//F ?( ?CM)___//_0
|| | ?
|| |''M
|| fflffl|
~F?M : 0 ____//_ -2(F ?M)____//D(-, Z0)____// ?D(F ?M)___//_0
The diagram can be completed by a map D(-, F Z) ! D(-, Z0) because F ?~M =
Ext3D(j?M, F ?M)(~F?M ). Let OE: F Z ! Z0 be the new map which is an isomorphi*
*sm,
16 HENNING KRAUSE
since j?M is an isomorphism. We obtain the following commutative diagram
_Fff//_ ____//_ _____//
F X F Y F Z F ( CX)
|| ||
|| || |ffi |''X
|| || | |
|| Fff || fflffl| fflffl|
F X ____//_F Y____//Z0____// D (F X)
and therefore F is an exact triangle. Thus F is an exact functor. It is not d*
*ifficult to
show that an exact functor F satisfies (1) - (3), and therefore the proof is co*
*mplete.
Corollary 6.2. Let F :C ! D and G: D ! E be additive functors between triangula*
*ted
categories. Suppose F and G OF are exact. Suppose in addition that F is a cohom*
*ological
quotient functor. Then G is exact.
Proof.We apply Proposition 6.1. First observe that G?: mod D ! mod E is exact
because the composite G?F ?= (GF )? is exact and F ?is an exact quotient functo*
*r,
by Theorem 4.4. Denote by jF :F ? ?C! ?DF ?and jGF :(G?F ?) ?C! ?E(G?F ?)
the natural isomorphisms which exists because F and GF are exact. In order to d*
*efine
jG :G? ?D! ?EG?, we use again the fact that F ?:mod C ! mod D is an exact quot*
*ient
functor. Thus every object in mod D is isomorphic to F ?M for some M in mod C.
Moreover, any morphism F ?M ! F ?N is a fraction, that is, of the form
?ffi (F?ff)-1
F ?M -F-! F ?N0 -----! F ?N.
Now define jGF?M as the composite
(G?''FM)-1? ?? ''GFM ? ? ?
jGF?M:(G? ?D)F ?M ------! (G F C)M --! ( EG )F M.
The map is natural, because jF and jGF are natural transformations, and maps F *
*?M !
F ?N come from maps in mod C. A straightforward calculation shows that G?~N =
Ext3D(jGN, G?N)(~G?N ) for all N = F ?M in mod D. Thus G is exact by Propositio*
*n 6.1.
7. Exact quotient functors
The definition of a cohomological quotient functor between two triangulated c*
*ate-
gories involves cohomological functors to an abelian category. It is natural to*
* study the
analogue where the cohomological functors are replaced by exact functors to a t*
*riangu-
lated category.
Definition 7.1. Let F :C ! D be an exact functor between triangulated categorie*
*s.
We call F an exact quotient functor if for every triangulated category E and ev*
*ery exact
functor G: C ! E satisfying Ann F Ann G, there exists, up to a unique isomorp*
*hism,
a unique exact functor G0:D ! E such that G = G0OF .
The motivating examples for this definition are the quotient functors in the *
*sense of
Verdier.
Example 7.2. A quotient functor F :C ! C=B is an exact quotient functor. To see
this, observe that an exact functor G: C ! D with KerG containing B factors uni*
*quely
through F via some exact functor G0:C=B ! D; see [31, Corollaire II.2.2.11]. No*
*w use
that
B KerG () Ann F Ann G,
COHOMOLOGICAL QUOTIENTS 17
which follows from Lemma 4.2.
We want to relate cohomological and exact quotient functors.
Lemma 7.3. Let F :C ! D be a cohomological quotient functor, and denote by D0 t*
*he
smallest full triangulated subcategory containing the image of F . Then the re*
*striction
F 0:C ! D0 of F has the following properties.
(1)F 0is a cohomological quotient functor.
(2)F 0is an exact quotient functor.
Proof.(1) Use the characterization of cohomological quotient functors in Theore*
*m 4.4.
(2) For simplicity we assume D0 = D. Let G: C ! E be an exact functor satisfy*
*ing
Ann F Ann G. Then the composite HE OG with the Yoneda functor factors through
F because F is a cohomological quotient functor. We have the following sequenc*
*e of
inclusions
E ~E mod E
where ~Edenotes the idempotent completion of E. We obtain a functor D ! mod E a*
*nd
its image lies in ~E, since every object in D is a direct factor of some object*
* in the image
of F . Thus we have a functor G0:D ! ~Ewhich is exact by Corollary 6.2. Our add*
*itional
assumption on F implies that Im G0 E. We conclude that G factors through F via*
* an
exact functor D ! E.
The following example has been suggested by B. Keller. It shows that there ar*
*e exact
quotient functors which are not cohomological quotient functors.
Example 7.4. Let A be the algebra of upper 2 x 2 matrices over a field k, and l*
*et B =
kxk. We consider the bounded derived categories C = Db(mod A) and D = Db(mod B).
Restriction along the algebra homomorphism f :B ! A, (x, y) 7! x00y, induces an
exact functor F :C ! D which is an exact quotient functor but not a cohomologic*
*al
quotient functor. In fact, f has a left inverse A ! B, [xz0y]7! (x, y), which i*
*nduces a
right inverse G: D ! C for F . Thus every exact functor F 0:C ! E satisfying An*
*n F
Ann F 0factors uniquely through F , by Lemma 7.5 below. However, the exact func*
*tor
F ?:mod C ! mod D extending F does not induce an equivalence mod C=Ker F ?!
mod D.
Let us describe mod C. To this end denote by 0 ! X1 ! X3 ! X2 ! 0 the unique
non-split exact sequence in mod A involving the simple A-modules X1 and X2. Th*
*is
sequence induces an exact triangle X1 ! X3 ! X2 ! X1 in C. Note that each
indecomposable object in C is determined by its cohomology and is therefore of *
*the form
nXifor some n 2 Z and some i 2 {1, 2, 3}. Thus the indecomposable objects in m*
*od C
are precisely the objects of the form
C(-, nXi)=radjC(-, nXi) with n 2 Z, i 2 {1, 2, 3}, j 2 {0,,1}
where rad0M = M and rad1M is the intersection of all maximal subobjects of M. T*
*he
restriction functor mod A ! mod B sends 0 ! X1 ! X3 ! X2 ! 0 to a split exact
sequence. Thus F kills the map X2 ! X1 in C, and we have that F ?M = 0 for some
indecomposable M in mod C if and only if M ~=C(-, nX2)=rad C(-, nX2) for some
n 2 Z. It follows that the canonical functor
a a
mod A -! mod C=Ker F ?, (Mn)n2Z 7-! C(-, nMn)
n2Z n2Z
18 HENNING KRAUSE
is an equivalence.
We have seen that mod C=Ker F ?is not a semi-simple category, whereas in mod D
every object is semi-simple. More specifically, the cohomological functor
H :C -! mod C -! mod C=Ker F ?
does not factor through F via some cohomological functor D ! mod C=Ker F ?, even
though Ann H = Ann F . We conclude that F is not a cohomological quotient funct*
*or.
Lemma 7.5. Let F :C ! D be an additive functor having a right inverse G: D ! C,
that is, F OG = IdD. Then every additive functor F 0:C ! E satisfying Ann F A*
*nn F 0
factors uniquely through F .
Proof.We have F 0= (F 0OG) OF .
8. Exact ideals
Given a cohomological quotient functor F :C ! D, the ideal Ann F is an import*
*ant
invariant. In this section we investigate the collection of all ideals which ar*
*e of this form.
Definition 8.1. Let C be a triangulated category. An ideal I of C is called exa*
*ct if there
exists a cohomological quotient functor F :C ! D such that I = Ann F .
The exact ideals are partially ordered by inclusion and we shall investigate *
*the struc-
ture of this poset.
Theorem 8.2. Let C be a small triangulated category. Then the exact ideals in C*
* form
a complete lattice, that is, given a family (Ii)i2 of exact ideals, the suprem*
*um supIi
and the infimum infIi exist. Moreover, the supremum coincides with the supremum*
* in
the lattice of cohomological ideals.
Our strategy for the proof is to use a bijection between the cohomological id*
*eals of C
and the Serre subcategories of mod C. We proceed in several steps and start wit*
*h a few
definitions. Given an ideal I of C, we define
ImI = {M 2 mod C | M ~=Im C(-, OE) for some OE}2.I
The next definition is taken from [3].
Definition 8.3. Let C be a triangulated category. An ideal I of C is called sat*
*urated
if for every exact triangle X0 ff!X !fiX00! X0 and every map OE: X ! Y in Sc, *
*we
have that OE Off, fi 2 I implies OE 2 I.
The following characterization combines [18, Lemma 3.2] and [3, Theorem 3.1].
Lemma 8.4. Let C be a triangulated category. Then the following are equivalent *
*for an
ideal I of C.
(1)I is cohomological.
(2)I is saturated.
(3)Im I is a Serre subcategory of mod C.
Moreover, the map J 7! Im J induces a bijection between the cohomological ideal*
*s of C
and the Serre subcategories of mod C.
COHOMOLOGICAL QUOTIENTS 19
Proof.(1) ) (2): Let I = Ann F for some cohomological functor F :C ! A. Fix an
exact triangle X0!ffX !fiX00! X0 and a map OE: X ! Y in C. Suppose OE Off, fi *
*2 I.
Then F ff is an epimorphism, and therefore F OE OF ff = 0 implies F OE = 0. Thu*
*s OE 2 I.
(2) ) (3): Let 0 ! F 0! F ! F 00! 0 be an exact sequence in mod C. Using
that I is an ideal, it is clear that F 2 Im I implies F 0, F 002 Im I. Now supp*
*ose that
F 0, F 002 Im I. Using that mod C is a Frobenius category, we find maps OE: X *
*! Y
and ff: X0 ! X such that F = Im C(-, OE) and F 0= Im C(-, OE Off). Now form ex*
*act
triangles
[ffffl] fi 00 0
W -ffl!X -ffi!Y -! W and X0q W - ! X -! X -! (X q W )
in C, and observe that F 00= Im C(-, fi). We have OE O[ffffl]and fi in I. Thus *
*OE 2 I since
I is saturated. It follows that F = Im C(-, OE) belongs to Im I.
(3) ) (1): Let F be the composite of the Yoneda functor C ! mod C with the
quotient functor mod C ! mod C=Im I. This functor is cohomological and we have
I = Ann F .
We need some more terminology. Fix an abelian category A. A Serre subcatego*
*ry
B of A is called localizing if the quotient functor A ! A=B has a right adjoint*
*. If A
is a Grothendieck category, then B is localizing if and only if B is closed und*
*er taking
coproducts [10, Proposition III.8]. We denote for any subcategory B by lim-!B t*
*he full
subcategory of filtered colimits lim-!Xi in A such that Xi belongs to B for all*
* i.
Now let C be a small additive category and suppose mod C is abelian. Given a *
*Serre
subcategory S of mod C, then lim-!S is a localizing subcategory of Mod C; see [*
*17, Theo-
rem 2.8]. This has the following consequence which we record for later referenc*
*e.
Lemma 8.5. Let C be a small triangulated category and I be a cohomological idea*
*l of
C. Then lim-!ImI is a localizing subcategory of Mod C.
Proof.Use Lemma 8.4.
We call a Serre subcategory S of mod C perfect if the right adjoint of the qu*
*otient
functor Mod C ! Mod C=lim-!Sis an exact functor. We have a correspondence betwe*
*en
perfect Serre subcategories of mod C and flat epimorphisms starting in C. To ma*
*ke this
precise, we call a pair F1: C ! D1 and F2: C ! D2 of flat epimorphisms equivale*
*nt if
KerF1?= KerF2?.
Lemma 8.6. Let C be a small additive category and suppose mod C is abelian. Th*
*en
the map
(F :C ! D) 7-! Ker F ?
induces a bijection between the equivalence classes of flat epimorphisms starti*
*ng in C,
and the perfect Serre subcategories of mod C.
Proof.We construct the inverse map as follows. Let S be a perfect Serre subcate*
*gory
of mod C and consider the quotient functor Q: Mod C ! Mod C=lim-!S. Observe th*
*at
Q preserves projectivity since the right adjoint of Q is exact. Now define D t*
*o be
the full subcategory formed by the objects QC(-, X) with X in C, and let F :C !*
* D
be the functor which sends X to QC(-, X). It follows that F *:Mod C ! Mod D
induces an equivalence Mod C=lim-!S! Mod D. Thus F is a flat epimorphism satisf*
*ying
S = KerF ?.
20 HENNING KRAUSE
Lemma 8.7. Let C be a small additive category and suppose mod C is abelian. If *
*(Si)i2
is a family of perfect Serre subcategories of mod C, then the smallest Serre su*
*bcategory
of mod C containing all Si is perfect.
Proof.For each i 2 , let Mi be the full subcategory of C-modules M satisfying
Hom C(lim-!Si, M) = 0 = Ext1C(lim-!Si, M). Note that the right adjoint adjoint *
*of the quo-
tient functor Q: Mod C ! Mod C=lim-!Siidentifies Mod C=lim-!Siwith Mi; see Lemm*
*a 3.2.
Let S = supSi. Then the full subcategory lim-!S is the smallest localizing subc*
*ategory
of Mod C containing all Si. Let M be the full subcategory ofTC-modules M satisf*
*ying
Hom C(lim-!S, M) = 0 = Ext1C(lim-!S, M). We claim that M = iMi. To see this, *
*let Ii
be the full subcategory of injective objects in Mi. Note that a C-module M belo*
*ngs to
lim-!Si if and only if Hom C(M, Ii) = 0, and M belongs to MiTif and only if the*
*re is an
exact sequence 0 ! M ! I0 ! I1 with I0, I1 in Ii. Let I = iIi. Then we have th*
*atTa
C-module M belongs to lim-!S if and only if Hom C(M, I) = 0, and M belongs to *
* iMi
if andTonly if there is an exact sequence 0 ! M ! I0 ! I1 with I0, I1 in I. Thi*
*s proves
M = iMi. It follows that the inclusion M ! Mod C is exact because each inclus*
*ion
Mi! Mod C is exact. Thus S is a perfect Serre subcateory.
Proposition 8.8. Let C be a small triangulated category, and let I be an ideal *
*in C
satisfying I = I. Then the following are equivalent.
(1)I is an exact ideal.
(2)Im I is a perfect Serre subcategory of mod C.
(3)There exists a flat epimorphism F :C ! D such that Ann F = I.
Proof.(1) ) (2): Suppose I is an exact ideal, that is, there is a cohomological*
* quotient
functor F :C ! D such that I = Ann F . Then F is a flat epimorphism by Theorem *
*4.4.
Now observe that Im I = KerF ?. Thus Im I is perfect by Lemma 8.6.
(2) ) (3): Apply again Lemma 8.6 to obtain a flat epimorphisms F :C ! D with
Ann F = I.
(3) ) (1): We may assume that idempotents in D split. It follows from Theorem*
* 5.1
that D is a triangulated category and that F is an exact functor. Moreover, The*
*orem 4.4
implies that F is a cohomological quotient functor. Thus I is an exact ideal.
We collect our findings to obtain the proof of the theorem from the beginning*
* of this
section.
Proof of Theorem 8.2.Let (Ii)i2 be a family of exact ideals in C and consider *
*the cor-
responding Serre subcategories Si= Im Iiof mod C which are perfect by Propositi*
*on 8.8.
It follows from Lemma 8.7 that S = supSi is perfect. There is a cohomological i*
*deal I
in C satisfying S = Im I and we have I = supIi in the lattice of cohomological *
*ideals
by Lemma 8.4. Applying again Proposition 8.8, we see that the ideal I is exact.*
* This
completes the proof.
9. Factorizations
Let C be a triangulated category. If B is a full triangulated subcategory of *
*C, then
the quotient functor Q: C ! C=B in the sense of Verdier is a cohomological and *
*exact
quotient functor (in the sense of Definition 4.1 and Definition 7.1). This fact*
* motivates
the following definition.
COHOMOLOGICAL QUOTIENTS 21
Definition 9.1. Let F :C ! D be an exact functor between triangulated categorie*
*s.
We call F a CE-quotient functor if F is a cohomological quotient functor and an*
* exact
quotient functor.1
In this section we study the collection of all CE-quotient functors starting *
*in a fixed
triangulated category. Given a pair F1: C ! D1 and F2: C ! D2 of CE-quotient
functors, we define
F1 ~ F2 () there exists an equivalence G: D1 ! D2 such that F2 = G OF1,
F1 F2 () there exists an exact functor G: D1 ! D2 such that F2 = G OF1.
We obtain a partial ordering on the equivalence classes of CE-quotient functors*
*, which
may be rephrased as follows.
F1 ~ F2 () Ann F1 = Ann F2,
F1 F2 () Ann F1 Ann F2.
The ideals of the form Ann F arising from CE-quotient functors F :C ! D form a
complete lattice. This has been established in Theorem 8.2, and we obtain the f*
*ollowing
immediate consequence.
Theorem 9.2. Let C be a small triangulated category.
(1)The equivalence classes of CE-quotient functors starting in C form a comp*
*lete
lattice.
(2)The assignment F 7! Ann F induces an anti-isomorphism between the lattice*
* of
CE-quotient functors starting in C and the lattice of exact ideals of C.
(3)Given a family (Fi)i2 of CE-quotient functors, we have
Ann (infFi) = sup(Ann Fi)
i2 i2
where the supremum is taken in the lattice of cohomological ideals of C.
Proof.The ideals of the form Ann F for some CE-quotient functor F :C ! D are pr*
*e-
cisely the exact ideals of C. This follows from Lemma 7.3. Now apply Theorem 8.*
*2.
The completeness of the CE-quotient functor lattice yields a canonical factor*
*ization
for every exact functor between two triangulated categories.
Corollary 9.3. Let C be a small triangulated category. Then every exact functor*
* F :C !
D to a triangulated category D has a factorization
Q F0
C _______//_C0______//D
having the following properties:
(1)Q is a CE-quotient functor and F 0is exact.
(2)Given a factorization
Q0 F00
C _______//_C00_____//D
____________
1The terminology refers to the properties `cohomological' and `exact'. In add*
*ition, we wish to honour
Cartan and Eilenberg.
22 HENNING KRAUSE
of F such that Q0 is a CE-quotient functor and F 00is exact, there exists*
*, up to
a unique isomorphism, a unique exact functor G: C00! C0
0 q8C00MM8
Qqqqqq | MMF00MMM
qqqq | MMM&&
C MMM G|| qqD88
MMQMMM | F0qqqq
M&&fflffl|Mqqqq
C0
such that Q = G OQ0and F 00~=F 0OG.
Proof.We obtain the CE-quotient functor Q: C ! C0 by taking the infimum over all
CE-quotient functors Q0:C ! C00admitting a factorization
Q0 F00
C _______//_C00____//_D.
Note that F factors through Q because Ann Q Ann F . This follows from the fac*
*t that
Ann F is cohomological and Ann Q0 Ann F for all Q0.
10.Compactly generated triangulated categories and Brown
representability
We recall the definition of a compactly generated triangulated category, and *
*we review
a variant of Brown's Representability Theorem which will be needed later on.
Let S be a triangulated category and suppose that arbitrary coproducts exist *
*in S.
An`object X in S is`called compact if for every family (Yi)i2Iin S, the canonic*
*al map
iS(X, Yi) ! S(X, iYi) is an isomorphism. We denote by Sc the full subcategory*
* of
compact objects in S and observe that Sc is a triangulated subcategory of S. Fo*
*llowing
[25], the category S is called compactly generated provided that the isomorphis*
*m classes
of objects in Sc form a set, and S(C, X) = 0 for all C in Sc implies X = 0 for *
*every
object X in S.
A basic tool for studying a compactly generated triangulated category S is th*
*e coho-
mological functor
HS :S -! Mod Sc, X 7! HX = S(-, X)|Sc
which we call restricted Yoneda functor. Our notation does not distinguish bet*
*ween
the Yoneda functor HS :S ! mod S and the restricted Yoneda functor. However, the
meaning of HS and HX for some X in S will be clear from the context.
Next we recall from [19] a variant of Brown's Representability Theorem [6]. *
*Let
S be a triangulated category with arbitrary products. An object U in S is call*
*ed a
perfect cogenerator if S(X, U) = 0 implies X = 0 for every object X in S, and f*
*or every
countable family of maps Xi! Yi in S, the induced map
Y Y
S( Yi, U) -! S( Xi, U)
i i
is surjective provided that the map S(Yi, U) ! S(Xi, U) is surjective for all i.
Proposition 10.1 (Brown representability). Let S be a triangulated category wit*
*h ar-
bitrary products and a perfect cogenerator U.
COHOMOLOGICAL QUOTIENTS 23
(1)A functor H :S ! Ab is cohomological and preserves all products if and on*
*ly if
H ~=S(X, -) for some object X in S.
(2)S coincides with its smallest full triangulated subcategory which contain*
*s U and
is closed under taking all products.
Proof.See Theorem A in [19].
There is an immediate consequence which we shall use.
Corollary 10.2. Let S be a triangulated category with arbitrary products and a *
*perfect
cogenerator U. Then a functor S ! T between triangulated categories is exact *
*and
preserves all products if and only if it has a left adjoint.
Proof.The left adjoint of a functor F :S ! T sends an object X in T to the obje*
*ct in
S representing T (X, F -).
Take as an example for S a compactly generated triangulated category. Then
a
U = C
C2Sc
is a perfect cogenerator for Sop. If I is an injective cogenerator for Mod Sc,*
* then the
object V satisfying
Hom Sc(HS-, I) ~=S(-, V )
is a perfect cogenerator for S. Note that V exists because Sop is perfectly cog*
*enerated.
11. Smashing localizations
We establish for any compactly generated triangulated category S a bijective *
*corre-
spondence between the smashing localizations of S and the cohomological quotien*
*ts of
Sc.
This result is divided into two parts. In this section we show that any smas*
*hing
localization induces a cohomological quotient. Let us recall the relevant defin*
*itions.
An exact functor F :S ! T between triangulated categories is a localization f*
*unctor
if it has a right adjoint G such that F OG ~=IdT. Note that the condition F OG *
*~=IdT
is equivalent to the fact that F induces an equivalence S=Ker F ! T , where S=K*
*er F
denotes quotient in the sense of Verdier [31]. It is often useful to identify a*
* localization
functor F :S ! T with the idempotent functor L: S ! S defined by L = G OF . The*
* L-
acyclic objects are those in KerF and the L-local object are those which are is*
*omorphic
to some object in the image of G. The localization F is called smashing if G pr*
*eserves
all coproducts which exist in T .2
Theorem 11.1. Let S be a compactly generated triangulated category and F :S ! T
be an exact functor between triangulated categories. Then F is a smashing local*
*ization
if and only if the following holds:
(1)T is a compactly generated triangulated category.
(2)F preserves coproducts.
(3)F induces a functor Fc:Sc ! Tc which is a cohomological quotient functor.
We need a few preparations before we can give the proof of this result.
____________
2If S carries a smash product ^:S xS ! S with unit S, then LX = X ^LS provide*
*d F is smashing.
24 HENNING KRAUSE
Lemma 11.2. Let S be a compactly generated triangulated category and F :S ! T *
*be
an exact functor between triangulated categories. Suppose F preserves coproduct*
*s. Then
the right adjoint of F preserves coproducts if and only if F preserves compactn*
*ess.
Proof.Combine the definition of compactness and the adjointness isomorphism; se*
*e [25,
Theorem 5.1].
Lemma 11.3. Let F :S ! T be an exact functor between compactly generated triang*
*u-
lated categories. Suppose F has a right adjoint G: T ! S which preserves coprod*
*ucts.
Then the following diagram commutes.
____F_____//____G_____//
S T S
|HS| |HT| HS||
fflffl|(Fc)*fflffl|(Fc)*fflffl|
Mod Sc ____//_ModTc____//_ModSc
Proof.See Proposition 2.6 in [18].
We are now in the position that we can prove the main result of this section.
Proof of Theorem 11.1.Suppose first that F is a smashing localization. Thus F *
*has
a right adjoint G which preserves coproducts. This implies that F induces a fu*
*nctor
Fc:Sc ! Tc, by Lemma 11.2. Using the adjointness formula T (F X, Y ) ~=S(X, GY *
*), one
sees that T is generated by the image of Fc. Thus T is compactly generated. It *
*remains
to show that Fc is a cohomological quotient functor. To this end denote by M th*
*e class
of Tc-modules M such that the natural map ((Fc)*O(Fc)*)M ! M is an isomorphism.
Observe that (Fc)*O(Fc)* composed with the Yoneda embedding Tc ! Mod Tc equals
the composite HT OF OG|Tc, by Lemma 11.3. Our assumption implies F OG ~=IdT, and
therefore M contains all representable functors. The composite (Fc)*O(Fc)* pres*
*erves
all colimits and therefore M is closed under taking colimits. We conclude that
(Fc)*O(Fc)* ~=IdModTc
since every module is a colimit of representable functors. Thus Fc is up to fac*
*tors an
epimorphism by Proposition A.5, and therefore a cohomological quotient functor *
*by
Theorem 4.4.
Now suppose that F satisfies (1) - (3). An application of Brown's Representab*
*ility
Theorem shows that F has a right adjoint G, since F preserves coproducts. Moreo*
*ver,
G preserves coproducts by Lemma 11.2, since F preserves compactness. It remains*
* to
show that F OG ~=IdT. To this end denote by T 0the class of objects X in T such*
* that
the natural map (F OG)X ! X is an isomorphism. Our assumption on Fc implies
(Fc)*O(Fc)* ~=IdModTc.
Using again Lemma 11.3, we see that Tc T 0. The objects in T 0form a triangul*
*ated
subcategory which is closed under taking coproducts. It follows that T 0= T sin*
*ce T is
compactly generated. This finishes the proof.
12. Smashing subcategories
Let S be a compactly generated triangulated category. In this section we comp*
*lete
the correspondence between smashing localizations of S and cohomological quotie*
*nts of
COHOMOLOGICAL QUOTIENTS 25
Sc. In order to formulate this, let us define the following full subcategories *
*of S for any
ideal I in Sc:
FiltI = {X 2 S | every map C ! X, C 2 Sc, factors through some map in}I,
I? = {X 2 S | S(OE, X) = 0 for all OE}2.I
Theorem 12.1. Let S be a compactly generated triangulated category, and let I be
an exact ideal in Sc. Then there exists a smashing localization F :S ! T havi*
*ng the
following properties:
(1)The right adjoint of F identifies T with I? .
(2)Ker F = FiltI.
(3)Sc\ Ann F = I.
The proof of this result requires some preparations. We start with descripti*
*ons of
FiltI and I? which we take from [18].
Lemma 12.2. Let I be an ideal in Sc and X be an object in S.
(1)X 2 FiltI if and only if HX 2 lim-!ImI.
(2)X 2 I? if and only if Hom Sc(Im I, HX ) = 0.
Proof.For (1), see Lemma 3.9 in [18]. (2) follows from the fact that Hom Sc(-, *
*HX ) is
exact when restricted to mod Sc.
Now suppose that I is a cohomological ideal in Sc and observe that L = lim-!I*
*mI is
a localizing subcategory of Mod Sc, by Lemma 8.5. Thus we obtain a quotient fun*
*ctor
Q: Mod Sc ! Mod Sc=L which has a right adjoint R; see [10, Proposition III.8]. *
*Note
that R identifies Mod Sc=L with the full subcategory M of Sc-modules M satisfyi*
*ng
Hom Sc(L, M) = 0 = Ext1Sc(L, M); see Lemma 3.2. Moreover, every Sc-module M fits
into an exact sequence
0 -! M0 -! M -! (R OQ)M -! M00-! 0
with M0, M00in L.
Lemma 12.3. An object X in S belongs to I? if and only if Hom Sc(L, HX ) = 0 =
Ext1Sc(L, HX ).
Proof.Suppose X 2 I? . Then we have Hom Sc(L, HX ) = 0 because Hom Sc(Im I, HX *
*) =
0 by Lemma 12.2. Thus we have an exact sequence
0 -! HX - ! (R OQ)HX - ! M -! 0.
We claim that M = 0. For this it is sufficient to show that every map OE: M0 !*
* M
from a finitely presented module M0 is zero. The map OE factors through some M0*
*0in
ImI because M 2 L. Now we use that Ext1Sc(-, HX ) vanishes on finitely present*
*ed
modules; see [18, Lemma 1.6]. Thus M00! M factors through (R OQ)HX . However,
Hom Sc(L, M) = 0, and this implies OE = 0. Therefore M = 0, and HX belongs to M.
Thus the proof is complete because the other implication is trivial.
Lemma 12.4. Let I be an ideal in Sc satisfying I = I. If I is exact or I2 = I,
then I? is a triangulated subcategory of S which is closed under taking product*
*s and
coproducts.
26 HENNING KRAUSE
Proof.Clearly, I? is closed under taking products and coproducts. Also, (I? ) *
*= I?
is clear. It remains to show that I? is closed under forming extensions. Let
X -ff!Y -fi!Z -fl! X
be a triangle in S with X and Y in I? , which induces an exact sequence
0 -! CokerHff-! HZ -! KerH ff-! 0
in Mod Sc. If I2 = I, then HY is annihilated by I because CokerHffand Ker H ffa*
*re
annihilated by I. Thus Z belongs to I? . Now suppose that I is exact. We apply *
*Propo-
sition 8.8 to see that the category M of Sc-modules M satisfying Hom Sc(L, M) =*
* 0 =
Ext1Sc(L, M) is closed under taking kernels, cokernels, and extensions. Thus Co*
*kerHff
and KerH ffbelong to M, and therefore HZ as well. We conclude again that Z belo*
*ngs
to I? . This finishes the proof.
We are now in the position that we can prove Theorem 12.1.
Proof of Theorem 12.1.We know from Lemma 8.5 that L = lim-!ImI is a localizing
subcategory of Mod Sc. Denote by R the right adjoint of the quotient functor Mo*
*d Sc !
Mod Sc=L. Recall that R identifies Mod Sc=L with the full subcategory M of Sc-m*
*odules
M satisfying Hom Sc(L, M) = 0 = Ext1Sc(L, M). The quotient Mod Sc=L is an abeli*
*an
Grothendieck category. Thus there is an injective cogenerator, say I, and we de*
*note by
U the object in S satisfying
Hom Sc(HS-, RI) ~=S(-, U).
Let T = I? , which is a triangulated subcategory of S and closed under taking p*
*roducts
and coproducts by Lemma 12.4. We claim that U is a perfect cogenerator for T .*
* To
see this, let X be an object in T satisfying S(X, U) = 0. We have HX 2 M by
Lemma 12.3, and therefore HX = 0 since RI is a cogenerator for M. Thus X = 0. N*
*ow
let Xi! Yi be a set of maps in T such that the map S(Yi, U) ! S(Xi, U) is surje*
*ctive
for all i. Using the fact that RI is an injective cogenerator for M, we see th*
*at each
map HXi ! HYi is a monomorphism. Thus their product is a monomorphism, and
thereforeQthe mapQHQ iXi! HQ iYiis a monomorphism. We conclude that the map
S( iYi, U) ! S( iXi, U) is surjective, since RI is an injective object. Now *
*we can
apply Corollary 10.2. It follows that the inclusion T ! S has a left adjoint F *
*which is
a smashing localization since T is closed under taking coproducts.
It remains to describe Ker F and Ann F . To this end let X be an object in S.*
* We
have F X = 0 iff S(X, U) = 0 iff Hom Sc(HX , RI) = 0 iff HX 2 L iff X 2 FiltI,*
* by
Lemma 12.2. Now let OE: X ! Y be a map in Sc. If F OE = 0, then S(OE, U) = 0.
Now use that S(OE, U) = 0 iff Hom Sc(Hffi, RI) = 0 iff Im Hffi2 L iff Im Hffi2 *
*Im I iff
OE 2 I. Conversely, suppose OE 2 I. Then F Y 2 I? implies F OE = 0 since T (F O*
*E, F Y ) ~=
S(OE, F Y ). Thus Sc\ Ann F = I, and the proof is complete.
Combining Theorem 11.1 and Theorem 12.1, one obtains a bijection between smas*
*hing
localizations of S and exact ideals of Sc. It is convenient to formulate this i*
*n terms of
smashing subcategories. Recall that a subcategory of S is smashing if it of th*
*e form
KerF for some smashing localization functor F :S ! T . Note that the kernel Ker*
*F of
any localization functor F is a localizing subcategory, that is, KerF is a full*
* triangulated
subcategory which is closed under taking coproducts. Thus a subcategory R of S*
* is
COHOMOLOGICAL QUOTIENTS 27
smashing if and only if R is a localizing subcategory admitting a right adjoint*
* for the
inclusion R ! S which preserves coproducts.
Corollary 12.5. Let S be a compactly generated triangulated category. Then the *
*maps
I 7! FiltI and R 7! {OE 2 Sc | OE factors through some object in}R
induce mutually inverse bijections between the set of exact ideals of Sc and th*
*e set of
smashing subcategories of S.
A similar result appears as Theorem 4.9 in [18]. However, the proof given the*
*re is not
correct for two reasons: it uses an unnecessary assumption and relies on an err*
*oneous
definition of an exact ideal.3
Let us formulate further consequences of Theorem 12.1 about cohomological quo-
tient functors. I am grateful to B. Keller for pointing out to me the followin*
*g simple
description of the exact ideals of Sc.
Corollary 12.6. Let S be a compactly generated triangulated category. Then an i*
*deal
I of Sc is exact if and only if the following conditions hold.
(1) I = I.
(2)I is saturated.
(3)I is idempotent, that is, I2 = I.
Proof.Suppose first that I is exact. Applying Corollary 12.5, the ideal I is th*
*e collection
of maps in Sc which factor through an object in FiltI. Now fix a map OE: X ! Y *
*in
I. Then OE factors through an object Y 0in FiltI via a map OE0:X ! Y 0, and OE0*
*factors
through a map OE1: X ! Y 00in I since Y 0belongs to FiltI. Thus OE = OE2O OE1,*
* and
OE2: Y 00! Y belongs to I because it factors through an object in FiltI.
Now suppose that I satisfies (1) - (3). The proof of Theorem 12.1 works with *
*these
assumptions, thanks to Lemma 12.4. The conclusion of Theorem 12.1 shows that I =
Ann Fc for some smashing localization F :S ! T . Thus I is exact because Fc is*
* a
cohomological quotient functor by Theorem 11.1.
One may think of the following result as a generalization of the localization*
* theorem
of Neeman-Ravenel-Thomason-Trobaugh-Yao [24, 27, 30, 34]. To be precise, Neeman
et al. considered cohomological quotient functors of the form Sc ! Sc=R0 for so*
*me
triangulated subcategory R0 of Sc and analyzed the smashing localization functo*
*r S !
S=R where R denotes the localizing subcategory generated by R0.
Corollary 12.7. Let S be a compactly generated triangulated category, and let F*
* :Sc !
T be a cohomological quotient functor.
(1)The category R = Filt(Ann F ) is a smashing localizing subcategory of S a*
*nd the
quotient functor S ! S=R induces a fully faithful and exact functor T ! S*
*=R
making the following diagram commutative.
Sc _____F_____//_T
|inc| ||
fflffl|can |fflffl
S __________//_S=R
____________
3The error occurs in Lemma 4.10. The claim that f*(~) is an isomorphism is on*
*ly correct if f is a
cohomological quotient functor.
28 HENNING KRAUSE
(2)The triangulated category S=R is compactly generated and (S=R)c is the cl*
*osure
of the image of T ! S=R under forming direct factors.
(3)There exists a fully faithful and exact functor G: T ! S such that
S(X, GY ) ~=T (F X, Y )
for all X in Sc and Y in T .
Proof.The ideal Ann F is exact and we obtain from Theorem 12.1 a smashing local*
*iza-
tion functor Q: S ! S=R. The induced functor Qc: Sc ! (S=R)c is a cohomological
quotient functor with Ann Qc = Ann F , by Theorem 11.1. The proof of Lemma 7.3
shows that Qc factors through F , since idempotents in (S=R)c split. Moreover,*
* the
functor T ! S=R is fully faithful since it induces an equivalence mod T ! mod (*
*S=R)c.
Note that every compact object in S=R is a direct factor of some object in the *
*image
of Qc by Theorem 4.4. To obtain the functor G: T ! S, take the fully faithful *
*right
adjoint S=R ! S of Q, and compose this with the functor T ! S=R.
13.The telescope conjecture
The telescope conjecture due to Bousfield and Ravenel is originally formulate*
*d for
the stable homotopy category of CW-spectra; see [5, 3.4], [27, 1.33] (and [20] *
*for an
unsuccessful attempt to disprove the conjecture). The stable homotopy category*
* is a
compactly generated triangulated category. This fact suggests the following for*
*mulation
of the telescope conjecture for a specific triangulated category S which is com*
*pactly
generated.
Telescope Conjecture. Every smashing subcategory of S is generated as a localiz*
*ing
subcategory by objects which are compact in S.
Recall that a subcategory of S is smashing if it of the form KerF for some sm*
*ashing
localization functor F :S ! T . Note that Ker F is a localizing subcategory of *
*S, that
is, Ker F is a full triangulated subcategory which is closed under taking copro*
*ducts.
A localizing subcategory of S is generated by a class X of objects if it is the*
* smallest
localizing subcategory of S which contains X .
The telescope conjecture in this general form is known to be false. In fact,*
* Keller
gives an example of a smashing subcategory which contains no non-zero compact o*
*bject
[15]; see also Section 15. However, there are classes of compactly generated tr*
*iangulated
categories where the conjecture has been verified.
We have seen that smashing subcategories of S are closely related to cohomolo*
*gical
quotients of Sc. It is therefore natural to translate the telescope conjecture*
* into a
statement about cohomological quotients. Roughly speaking, the telescope conjec*
*ture
for S is equivalent to the assertion that every flat epimorphism Sc ! T is an *
*Ore
localization. We need some preparations in order to make this precise.
Lemma 13.1. Let S be a compactly generated triangulated category and F :S ! T *
*be
a smashing localization functor. Then the following are equivalent.
(1)The localizing subcategory KerF is generated by objects which are compact*
* in S.
(2)The ideal Sc\ Ann F of Sc is generated by identity maps.
Proof.Let R = KerF and I = Sc\ Ann F .
(1) ) (2): Suppose R is generated by compact objects. Then every object in R *
*is
a homotopy colimit of objects in R \ Sc; see [25]. Let OE: X ! Y be a map in *
*I. It
COHOMOLOGICAL QUOTIENTS 29
follows from Lemma 4.2 that OE factors through a homotopy colimit of objects in*
* R \ Sc.
Thus OE factors through some object in R \ Sc since X is compact. We conclude t*
*hat I
is generated by the identity maps of all objects in R \ Sc.
(2) ) (1): Let R0 be a class of compact objects and suppose I is generated by
the identity maps of all objects in R0. Let R0 be the localizing subcategory w*
*hich
is generated by R0. This category is smashing and we have a localization funct*
*or
F 0:S ! T 0with R0= Ker F 0. Clearly, Ann Fc Ann Fc0since idX belongs to Ann *
*Fc0
for all X in R0. On the other hand, R0 R since R0 R. Thus Ann Fc0 Ann Fc
by Lemma 4.2, and Ann Fc0= Ann Fc follows. We conclude that R0 = R because
Theorem 12.1 states that a smashing subcategory is determined by the correspond*
*ing
exact ideal in Sc. Thus R is generated by compact objects.
Proposition 13.2. Let F :C ! D be an exact functor between triangulated categor*
*ies.
Then the following are equivalent.
(1)F induces an equivalence C=Ker F ! D.
(2)F induces an equivalence C[ -1] ! D where = {OE 2 C | F OE is an iso}.
(3)F is a CE-quotient functor and the ideal Ann F is generated by identity m*
*aps.
Proof.We put B = KerF and denote by Q: C ! C=B the quotient functor.
(1) , (2): The quotient C=B is by definition C[ -1] where is the class of m*
*aps
X ! Y in C which fit into an exact triangle X ! Y ! Z ! X with Z in B. The
exactness of F implies that is precisely the class of maps OE in C such that *
*F OE is
invertible.
(1) ) (3): We have seen in Example 4.3 and Example 7.2 that Q is a CE-quotient
functor. Lemma 4.2 implies that the ideal Ann Q is generated by the identity ma*
*ps of
all objects in B.
(3) ) (1): The functor F induces an exact functor C=B ! D. Now suppose that
Ann F is generated by identity maps. Then Ann F is generated by the identity ma*
*ps
of all objects in B, and therefore Ann Q = Ann F by Lemma 4.2. Thus Q factors
through F by an exact functor D ! C=B since F is an exact quotient functor. The
uniqueness of D ! C=B and C=B ! D implies that both functors are mutually inver*
*se
equivalences.
Lemma 13.3. Let F :C ! D be a cohomological quotient functor. Then the following
are equivalent.
(1)F induces a fully faithful functor C=Ker F ! D.
(2)The ideal Ann F is generated by identity maps.
Proof.Let D0 be the smallest full triangulated subcategory of D containing the *
*image
of F . It follows from Lemma 7.3 that the induced functor F 0:C ! D0is a CE-quo*
*tient
functor. Now the assertion follows from Proposition 13.2 since Ann F = Ann F 0.
We obtain the following reformulation of the telescope conjecture. Note in pa*
*rticu-
lar, that the telescope conjecture becomes a statement about the category of co*
*mpact
objects.
Theorem 13.4. Let S be a compactly generated triangulated category. Then the fo*
*llow-
ing are equivalent.
(1)Every smashing subcategory of S is generated by objects which are compact*
* in S.
(2)Every smashing subcategory of S is a compactly generated triangulated cat*
*egory.
30 HENNING KRAUSE
(3)Every exact ideal in Sc is generated by idempotent elements.
(4)Every CE-quotient functor F :Sc ! T induces an equivalence Sc=Ker F ! T .
(5)Every cohomological quotient functor F :Sc ! T induces a fully faithful f*
*unctor
Sc=Ker F ! T .
(6)Every flat epimorphism F :Sc ! T satisfying (Ann F ) = Ann F induces an
equivalence Sc[ -1] ! T where = {OE 2 Sc | F OE is an iso}.
Proof.We use the bijection between smashing subcategories of S and exact ideals*
* of Sc;
see Corollary 12.5. Recall that an ideal is by definition exact if it is of the*
* form Ann F
for some cohomological functor F :Sc ! T .
(1) , (2): The inclusion R ! S of a smashing subcategory preserves compactnes*
*s.
(1) , (3): Apply Lemma 13.1. Note that any ideal in Sc which is generated by
idempotent maps is also generated by identity maps. This follows from the fact*
* that
idempotents in Sc split.
(3) , (4): Apply Proposition 13.2.
(3) , (5): Apply Lemma 13.3.
(5) ) (6): Let F :Sc ! T be a flat epimorphism satisfying (Ann F ) = Ann F*
* .
Composing it with the idempotent completion T ! ~Tgives a cohomological quotient
functor Sc ! ~T, by Theorem 4.4 and Theorem 5.1. Now use that Sc[ -1] = Sc=Ker *
*F.
Thus Sc[ -1] ! T is fully faithful, and it is an equivalence since F is surjec*
*tive on
objects by Lemma A.2.
(6) ) (5): Let F :Sc ! T be a cohomological quotient functor, and denote by T*
* 0the
full subcategory of T whose objects are those in the image of F . The induced f*
*unctor
Sc ! T 0is a flat epimorphism. Now use again that Sc[ -1] = Sc=Ker F. Thus the
induced functor Sc=Ker F ! T is fully faithful.
Remark 13.5. Let C be a triangulated category and F :C ! D be a flat functor sa*
*tisfying
(Ann F ) = Ann F . Then = {OE 2 C | F OE is an iso} is a multiplicative syst*
*em, that
is, admits a calculus of left and right fractions in the sense of [11].
We say that an additive functor C ! D is an Ore localization if it induces an*
* equiv-
alence C[ -1] ! D for some multiplicative system in C. Using this terminolog*
*y,
Theorem 13.4 suggests the following reformulation of the telescope conjecture.
Corollary 13.6. The telescope conjecture holds true for a compactly generated t*
*riangu-
lated category S if and only if every flat epimorphism F :Sc ! T satisfying (A*
*nn F ) =
Ann F is an Ore localization.
The reformulation of the telescope conjecture in terms of exact ideals raises*
* the ques-
tion when an idempotent ideal is generated by idempotent elements. This follows*
* from
Corollary 12.6 where it is shown that the exact ideals are precisely the idempo*
*tent ideals
which satisfy some natural extra conditions.
Corollary 13.7. The telescope conjecture holds true for a compactly generated t*
*rian-
gulated category S if and only if every idempotent and saturated ideal I of Sc *
*satisfying
I = I is generated by idempotent elements.
The problem of finding idempotent generators for an idempotent ideal is a ver*
*y clas-
sical one from ring theory. For instance, Kaplansky introduced the class of SBI*
* rings,
where SBI stands for `suitable for building idempotent elements' [13, III.8]. A*
*lso, Aus-
lander asked the question for which rings every idempotent ideal is generated b*
*y an
COHOMOLOGICAL QUOTIENTS 31
idempotent element [1, p. 241]. One can show for an additive category C, that *
*every
idempotent ideal is generated by idempotent elements provided that C is perfect*
* in the
sense of Bass [21]. Recall that C is perfect if every object in C decomposes i*
*nto a fi-
nite coproduct of indecomposable objects with local endomorphism rings, and, fo*
*r every
sequence
X1 ffi1-!X2 ffi2-!X3 ffi3-!. . .
of non-isomorphisms between indecomposable objects, the composition OEn O. .O.O*
*E2O OE1
is zero for n sufficiently large. Note that the category Sc of compact objects *
*is perfect
if and only if every object in S is a coproduct of indecomposable objects with *
*local
endomorphism rings [18, Theorem 2.10].
14.Homological epimorphisms of rings
A commutative localization R ! S of an associative ring R is always a flat ep*
*imor-
phism. For non-commutative localizations, there is a weaker condition which is*
* often
satisfied. Recall from [12] that a ring homomorphism R ! S is a homological epi*
*mor-
phism if S R S ~= S and TorRi(S, S) = 0 for all i 1. Homological epimorphis*
*ms
frequently arise in representation theory of finite dimensional algebras, in pa*
*rticular via
universal localizations [7, 28].
In recent work of Neeman and Ranicki [26], homological epimorphisms appear wh*
*en
they study the following chain complex lifting problem for a ring homomorphism *
*R ! S.
We denote by Kb(R) the homotopy category of bounded complexes of finitely gener*
*ated
projective R-modules.
Definition 14.1. Fix a ring homomorphism R ! S.
(1)We say that the chain complex lifting problem has a positive solution, if*
* every
complex Y in Kb(S) such that for each i we have Y i= P i R S for some fin*
*itely
generated projective R-module P i, is isomorphic to X R S for some compl*
*ex
X in Kb(R).
(2)We say that the chain map lifting problem has a positive solution, if for*
* every
pair X, Y of complexes in Kb(R) and every map ff: X R S ! Y R S in Kb(S*
*),
there are maps OE: X0 ! X and ff0:X0 ! Y in Kb(R) such that OE R S is
invertible and ff = ff0 R S O(OE R S)-1 in Kb(S).
The following observation shows that both lifting problems are closely relate*
*d. In
fact, it seems that the more general problem of lifting maps is the more natura*
*l one.
Lemma 14.2. Given a ring homomorphism R ! S, the chain complex lifting problem
has a positive solution whenever the chain map lifting problem has a positive s*
*olution.
Proof.Fix a complex Y in Kb(S). We proceed by induction on its length `(Y ) =*
* n.
If n = 0, then Y is concentrated in one degree, say i, and therefore Y = X R S*
* for
X = P i. If n > 0, choose an exact triangle Y1 ! Y2 ! Y ! Y1 with `(Yi) < n for
i = 1, 2. By our assumption, we have Yi ~=Xi R S for some complexes X1 and X2 in
Kb(R). Moreover, using the positive solution of the chain map lifting problem, *
*the map
X1 R S ! X2 R S is of the form ff R S O(OE R S)-1 for some maps OE: X01! X1 a*
*nd
ff: X01! X2 in Kb(R). We complete ff to an exact triangle X01! X2 ! X ! X01and
conclude that Y ~=X R S.
32 HENNING KRAUSE
The example of a proper field extension k ! K shows that both lifting problem*
*s are
not equivalent. In fact, the chain complex lifting problem for k ! K has a pos*
*itive
solution, but the chain map lifting problem does not.
The proof of Lemma 14.2 suggests the following reformulation of the chain com*
*plex
lifting problem.
Lemma 14.3. Given a ring homomorphism R ! S, the chain complex lifting problem
has a positive solution if and only if the full subcategory of Kb(S) formed by *
*the objects
in the image of - R S is a triangulated subcategory.
Proof.Clear.
We continue with a reformulation of the chain map lifting problem.
Proposition 14.4. Let R ! S be a ring homomorphism and denote by K the full
subcategory of Kb(R) formed by the complexes X such that X R S = 0. Then the
following are equivalent.
(1)The chain map lifting problem has a positive solution.
(2)The functor - R S induces a fully faithful functor Kb(R)=K ! Kb(S).
Proof.(1) ) (2): Denote by D the full subcategory of Kb(S) which is formed by a*
*ll
objects in the image of - R S. Using the description of the maps in D, we obs*
*erve
that D is a triangulated subcategory of Kb(S). It follows from Proposition 4.5 *
*that F
induces an equivalence Kb(R)=K ! D.
(2) ) (1): The maps in Kb(R)=K can be described as fractions; see for instan*
*ce
Proposition 4.5. The functor Kb(R)=K ! Kb(S) being full implies the positive so*
*lution
of the chain map lifting problem.
Given a ring homomorphism R ! S, we shall see that the problem of lifting com*
*plexes
and their maps is closely related to the question, when the derived functor - *
*LRS is a
smashing localization.
Let us denote by D(R) the unbounded derived category of R. Note that the incl*
*usion
Kb(R) ! D(R) induces an equivalence Kb(R) ! D(R)c.
Theorem 14.5. For a ring homomorphism R ! S the following are equivalent.
(1)The derived functor - LRS :D(R) ! D(S) is a smashing localization.
(2)The functor - R S :Kb(R) ! Kb(S) is a cohomological quotient functor.
(3)The map R ! S is a homological epimorphism.
Proof.The functor F = - LRS :D(R) ! D(S) has a right adjoint G: D(S) ! D(R)
which is simply restriction of scalars, that is, G = R Hom S(S, -). Clearly, G *
*preserves
coproducts. Thus F is a smashing localization if and only if F is a localizatio*
*n functor.
Note that F is a localization functor if and only if F OG ~=IdD(S). Moreover, F*
* OG is
exact and preserves coproducts. Using infinite devissage, one sees that F OG ~=*
*IdD(S)if
and only if the canonical map X LRS ! X is an isomorphism for the complex X = S
which is concentrated in degree 0. Clearly, this condition is equivalent to S *
*R S ~=S
and TorRi(S, S) = 0 for all i 1. This proves the equivalence of (1) and (3).*
* The
equivalence of (1) and (2) follows from Theorem 11.1 since F |Kb(R)= - R S.
We obtain the following conditions for solving the chain map lifting problem.
COHOMOLOGICAL QUOTIENTS 33
Theorem 14.6. Given a ring homomorphism R ! S, the chain map lifting problem
has a positive solution if and only if
(1)R ! S is a homological epimorphism, and
(2)every map OE in Kb(R) satisfying OE R S = 0 factors through some X in Kb*
*(R)
such that X R S = 0.
Proof.Suppose first that (1) and (2) hold. Condition (1) says that F = - R S :K*
*b(R) !
Kb(S) is a cohomological quotient functor. This follows from Theorem 14.5. Appl*
*ying
Lemma 13.3, we conclude from (2) that F induces a fully faithful functor Kb(R)=*
*Ker F !
Kb(S). The positive solution of the chain map lifting problem follows from Pro*
*posi-
tion 14.4.
Now suppose that the chain map lifting problem has a positive solution. The d*
*escrip-
tion of the maps in the image Im F of F implies that the full subcategory forme*
*d by
the objects in Im F is a triangulated subcategory of Kb(S). It contains a gener*
*ator of
Kb(S) and therefore every object in Kb(S) is a direct factor of some object in *
*Im F .
Now we apply Theorem 4.4 and see that F is a cohomological quotient functor. Th*
*us (1)
holds by Theorem 14.5. The induced functor Kb(R)=Ker F ! Kb(S) is fully faithfu*
*l by
Proposition 14.4. It follows from Lemma 13.3 that (2) holds. This finishes the *
*proof.
Corollary 14.7. Let R be a ring such that the telescope conjecture holds true f*
*or D(R).
Then the chain map lifting problem has a positive solution for a ring homomorph*
*ism
f :R ! S if and only if f is a homological epimorphism.
Note that the telescope conjecture has been verified for D(R) provided R is c*
*om-
mutative noetherian [23]. On the other hand, Keller has given an example of a r*
*ing R
such that the telescope conjecture for D(R) does not hold [15]. Let us mention *
*that the
validity of the telescope conjecture is preserved under homological epimorphims.
Proposition 14.8. Let R ! S be a homological epimorphism. If the telescope conj*
*ecture
holds for D(R), then the telescope conjecture holds for D(S).
Proof.The derived functor F = - LRS :D(R) ! D(S) is a smashing localization by
Theorem 14.5. Now suppose that G: D(S) ! T is a smashing localization. A compos*
*ite
of smashing localizations is a smashing localizations. Thus KerF is generated b*
*y a class
X of compact objects since the telescope conjecture holds for D(R). It follows*
* that
KerG is generated by F X .
The work of Neeman and Ranicki [26] on the problem of lifting chain complexes
is motivated by some applications in algebraic K-theory. In fact, they general*
*ize the
classical long exact sequence which is induced by an injective Ore localization*
*. More
precisely, they show that every universal localization f :R ! S which is a homo*
*logical
epimorphism induces a long exact sequence
. .-.! K1(R) -! K1(S) -! K0(R, f) -! K0(R) -! K0(S)
in algebraic K-theory [26, Theorem 10.11]. Our analysis of the chain map liftin*
*g problem
suggests that being a homological epimorphism and satisfying the additional hyp*
*othesis
(2) in Theorem 14.6 is the crucial property for such a sequence. We sketch the *
*construc-
tion of this sequence which uses the machinery developed by Waldhausen in [32].*
* Our
exposition follows closely the ideas of Thomason-Trobaugh [30] and Neeman-Ranic*
*ki
[26].
34 HENNING KRAUSE
We fix a ring homomorphism f :R ! S. Denote by W(R) the complicial biWald-
hausen category of bounded chain complexes of finitely generated projective R-m*
*odules
[30, 1.2.11]. We denote by K(R) the corresponding Waldhausen K-theory spectrum
K(W(R)); see [30, 1.5.2]. Note that K(R) is homotopy equivalent to the Quillen*
* K-
theory spectrum of the exact category projR of finitely generated projective R-*
*modules
[30, 1.11.2]. The algebraic K-groups Kn(R) = ßnK(R) are by definition the homot*
*opy
groups of the spectrum K(R). Now let W(R, f) be the complicial biWaldhausen sub-
category of W(R) consisting of those complexes X in W(R) such that X R S is ac*
*yclic,
and put K(R, f) = K(W(R, f)).
Theorem 14.9. Let f :R ! S be a homological epimorphism and suppose f satisfies
condition (2) in Theorem 14.6. Then f induces a sequence
W(R, f) -! W(R) -! W(S)
of exact functors such that
K(R, f) -! K(R) -! K(S)
is a homotopy fibre sequence, up to failure of surjectivity of K0(R) ! K0(S). I*
*n partic-
ular, there is induced a long exact sequence
. .-.! K1(R) -! K1(S) -! K0(R, f) -! K0(R) -! K0(S)
of algebraic K-groups.
Proof.The proof is modeled after that of Thomason-Trobaugh's localization theor*
*em
[30, Theorem 5.1]. We recall that a complicial biWaldhausen category comes equi*
*pped
with cofibrations and weak equivalences [30, 1.2.11]. The cofibrations of W(R) *
*are by
definition the chain maps which are split monomorphism in each degree, and the *
*weak
equivalences are the quasi-isomorphisms. We define a new complicial biWaldhaus*
*en
category W(R=f) as follows. The underlying category is that of W(R), the cofibr*
*ations
are those of W(R), and the weak equivalences are the chain maps whose mapping c*
*one
lies in W(R, f). We denote by K(R=f) the K-theory spectrum of W(R=f) and obtain
an induced sequence
W(R, f) -! W(R) -! W(R=f)
of exact functors such that
K(R, f) -! K(R) -! K(R=f)
is a homotopy fibre sequence by Waldhausen's localization theorem [30, 1.8.2]. *
*The func-
tor W(R) ! W(S) factors through W(R) ! W(R=f) and induces an exact functor
W(R=f) ! W(S). Note that any exact functor A ! B between complicial biWald-
hausen categories induces a homotopy equivalence of K-theory spectra K(A) ! K(B)
provided the functor induces an equivalence Ho(A) ! Ho(B) of the derived homoto*
*py
categories [30, 1.9.8]. Observe that Ho(W(R)) = Kb(R). Moreover, W(R) ! W(R=f)
induces an equivalence
Ho (W(R))=Ho (W(R, f)) -~!Ho (W(R=f)).
COHOMOLOGICAL QUOTIENTS 35
Thus we have the following commutative diagram
Ho (W(R, f)) ____//_Ho(W(R))____//_Ho(W(R=f))_____//Ho(W(S))
|o| |o| o|| |o|
fflffl| fflffl| fflffl| fflffl|
K ____________//Kb(R)________//_Kb(R)=K________//Kb(S)
where K denotes the full subcategory of Kb(R) consisting of all complexes X suc*
*h that
X R S = 0. Next we use our assumption about the ring homomorphism f and apply
Proposition 14.4 and Theorem 14.6. It follows that W(R=f) ! W(S) induces a func*
*tor
Ho (W(R=f)) -! Ho(W(S))
which is an equivalence up to direct factors. We conclude from the cofinality t*
*heorem
[30, 1.10.1] that
K(R, f) -! K(R) -! K(S)
is a homotopy fibre sequence, up to failure of surjectivity of K0(R) ! K0(S).
15.Homological localizations of rings
Let R be an associative ring and let be a class of maps between finitely ge*
*nerated
projective R-modules. The universal localization of R with respect to is the *
*universal
ring homomorphism R ! S such that OE R S is an isomorphism of S-modules for al*
*l OE
in ; see [7, 28]. To construct S, one formally inverts all maps from in the *
*category
C = projR of finitely generated projective R-modules and puts S = C[ -1](R, R).*
* The
following concept generalizes universal localizations.
Definition 15.1. We call a ring homomorphism f :R ! S a homological localization
with respect to a class of complexes in Kb(R) if
(1)X R S = 0 in Kb(S) for all X in , and
(2)given any ring homomorphism f0:R ! S0 such that X R S0= 0 in Kb(S0) for
all X in , there exists a unique homomorphism g :R ! R0such that f0 = g *
*Of.
Any universal localization is a homological localization. In fact, any map OE*
*: P ! Q
between finitely generated projective R-modules may be viewed as a complex of l*
*ength
one by taking its mapping cone Cone OE. If R ! S is a ring homomorphism, then O*
*E R S
is an isomorphism if and only if (Cone OE) R S = 0 in Kb(S).
The following example, which I learned from A. Neeman, shows that a homologic*
*al
localization need not to exist.
Example 15.2. Let k be a field and R = k[x, y]. Let P be the complex
[xy] [y-x]
. .-.! 0 -! R -! R q R -! R -! 0 -! . . .
which is a projective resolution of R=(x, y). Then we have P R R[x-1] = 0 and
P R R[y-1] = 0. Now suppose there is a homological localization R ! S with res*
*pect
to P . Viewing R[x-1] and R[y-1] as subrings of k(x, y), we have R[x-1] \ R[y-1*
*] = R.
Therefore the identity R ! R factors through R ! S. This is a contradiction and*
* shows
that the homological localization with respect to P cannot exist.
36 HENNING KRAUSE
Next we consider an example of a homological epimorphism which is not a homol*
*ogical
localization. Keller used this example in order to disprove the telescope conje*
*cture for
the derived category of a ring [15]. Let us explain the idea of Keller's exampl*
*e. He uses
the following observation.
Lemma 15.3. Let R be a ring and a be a two-sided ideal which is contained in the
Jacobson radical of R. Then X R R=a = 0 implies X = 0 for every X in Kb(R).
Proof.Using induction on the length of the complex X, the assertion follows from
Nakayama's lemma.
In [33], Wodzicki has constructed an example of a ring R such that the Jacobs*
*on
radical r is non-zero and satisfies
TorRi(R=r, R=r) = 0 for alli 1.
Thus R ! R=r is a homological epimorphism which induces a cohomological quotient
functor
F = - R S :Kb(R) -! Kb(R=r)
satisfying Ker F = 0 and Ann F 6= 0. It follows that R ! R=r is not a homologic*
*al lo-
calization since KerF = 0. Moreover, Theorem 13.4 shows that the telescope conj*
*ecture
does not hold for D(R).
One can find more examples along these lines, as R. Buchweitz kindly pointed *
*out
to me. Take any B'ezout domain R, that is, an integral domain such that every f*
*initely
generated ideal is principal. We have for every ideal a
TorR1(R=a, R=a) ~=a=a2 and TorRi(R=a, -) = 0 for alli > 1.
Thus for any idempotent ideal a, the natural map R ! R=a is a homological epimo*
*r-
phism. Specific examples arise from valuation domains, which are precisely the*
* local
B'ezout domains.
Our interest in homological localizations is motivated by the following obser*
*vation,
which shows that the positive solution of the chain map lifting problem forces *
*a ring
homomorphism to be a homological localization.
Proposition 15.4. Let f :R ! S be a ring homomorphism and suppose the chain map
lifting problem has a positive solution. Then f is a homological localization.
Proof.Denote by the set of complexes X in Kb(R) such that X R S = 0, and
denote by K the corresponding full subcategory. We have seen in Proposition 14.*
*4 that
- R S :Kb(R) ! Kb(S) induces a fully faithful functor Kb(R)=K ! Kb(S). Now
suppose that f0:R ! S0 is a ring homomorphism satisfying X R S0 = 0 for all X *
*in
. Then - R S0 factors through the quotient functor Kb(R) ! Kb(R)=K via some
functor G: Kb(R)=K ! Kb(S0). Clearly, G induces a homomorphism g :S ! S0 such
that f0 = g Of. The uniqueness of g follows from the uniqueness of G.
In [26], Neeman and Ranicki show that the chain complex lifting problem has a
positive solution for every universal localization which is a homological epimo*
*rphism.
We give an alternative proof of this result which is based on the criterion for*
* lifting
chain maps in Theorem 14.6.
Theorem 15.5. The chain map lifting problem has a positive solution for every h*
*omo-
logical epimorphism R ! S which is a universal localization.
COHOMOLOGICAL QUOTIENTS 37
We need some preparations for the proof of this result. Fix a homological epi*
*morphism
R ! S, and suppose it is the universal localization with respect to a class o*
*f maps
in the category C = projR of finitely generated projective R-modules. Thus we *
*have
projS = C[ -1]. We denote by Cone = {Cone OE | OE 2 } the corresponding cla*
*ss
of complexes of length one in Kb(R) = Kb(C), and we write for the thick
subcategory generated by Cone . Finally, denote by T the idempotent completio*
*n of
the quotient Kb(C)=, which one obtains for instance from Corollary 12.7*
* by
embedding Kb(C)= into the derived category D(R).
Lemma 15.6. The composite Q: Kb(C) ! Kb(C)= ! T has the following
properties.
(1)T ( n(QX), QY ) = 0 for all X, Y in C and n > 0.
(2)The functor Q|C: C ! T factors through the localization C ! C[ -1].
(3)The functor C[ -1] ! T extends to an exact functor Kb(C[ -1]) ! T .
(4)The functor - R S :Kb(C) ! Kb(C[ -1]) factors through Q: Kb(C) ! T
Proof.(1) The functor Q: Kb(C) ! T is a cohomological quotient functor. Thus we*
* can
apply Corollary 12.7 and obtain a fully faithful and exact `right adjoint' Q0:T*
* ! D(R)
such that
T (QA, B) ~=D(R)(A, Q0B) for all A 2 Kb(R) and B 2 T.
To compute T ( n(QX), QY ) for X, Y in C, it is sufficient to consider the case*
* X = R =
Y . We have
T ( n(QR), QR) ~=D(R)( nR, (Q0OQ)R) ~=H-n ((Q0OQ)R).
Now we apply Corollary 3.31 from [26] which says that TorRn(S, S) = 0 for all n*
* > 0
implies H-n ((Q0OQ)R) = 0 for all n > 0.
(2) The functor Kb(C) ! Kb(C)= makes the maps in invertible by send*
*ing
the objects in Cone to zero. Therefore C ! T factors through the localizatio*
*n C !
C[ -1].
(3) This follows from the `universal property' of the homotopy category Kb(C[*
* -1])
which is the main result in [16]. More precisely, any additive functor F :D ! A*
*~from
an additive category D to the stable category of a Frobenius category A extends*
* to an
exact functor Kb(D) ! A~ provided that A~( n(F X), F Y ) = 0 for all X, Y in D*
* and
n > 0. Note that we are using (1) and the fact that T can be embedded into the *
*stable
category of a Frobenius category.
(4) We have Kb(S) = Kb(C[ -1]) and X R S = 0 for all X 2 Cone since R !
S is the universal localization with respect to . Thus - R S factors through*
* the
quotient functor Kb(C) ! Kb(C)=. Moreover, - R S factors through T sin*
*ce
idempotents in Kb(C[ -1]) split.
The following commutative diagram summarizes our findings from Lemma 15.6.
38 HENNING KRAUSE
(15.1) C ________//_C[ -1]___//Kb(C[ -1])
| | rrrrr
| | rrr
fflffl|Q fflffl|Fxxrrrr
Kb(C) _________//Trr
rrr
|T| rrrrr
fflffl|Gxxrr
Kb(C[ -1])
Proof of Theorem 15.5.We want to apply Theorem 14.6 and use the diagram (15.1).
More precisely, we need to show that Ann T = Ann Q, because this implies condit*
*ion
(2) in Theorem 14.6 since T = - R S and Ann Q is generated by identity maps. We
claim that Q = F OT . This follows from the `universal property' of the homoto*
*py
category Kb(C) since Q|C = F OT |C; see [14]. We obtain that Q = F OG OQ, and t*
*his
implies F OG ~=IdT since both functors agree on Kb(C)=. Thus G is faith*
*ful
and we conclude that Ann T = Ann Q. This completes the proof.
Remark 15.7. I conjecture that Theorem 15.5 remains true if one replaces `unive*
*rsal
localization' by `homological localization'.
16. Almost derived categories
Almost rings and modules have been introduced by Gabber and Ramero [9]. Here,
we analyze their formal properties and introduce their analogue for derived cat*
*egories.
Let us start with a piece of notation. Given a class of maps in some additive*
* category
C, we denote by
? = {X 2 C | C(OE, X) = 0 for all OE}2
the full subcategory of objects which are annihilated by .
Throughout this section we fix an associative ring R. We view elements of R a*
*s maps
R ! R. Thus a? for any ideal a of R denotes the category of R modules which are
annihilated by R.
The formal essence of an almost module category can be formulated as follows.
Proposition 16.1. Let A be a full subcategory of a module category Mod R. Then *
*the
following are equivalent.
(1)A is a Serre subcategory, and the inclusion has a left and a right adjoin*
*t.
(2)There exists an idempotent ideal a of R such that A = a? .
In this case, the quotient category Mod R=A is the category of almost modules *
*with
respect to a, which is denoted by Mod (R, a).
Proof.The proof of the first part is straightforward; see for instance [1, Prop*
*osition 7.1].
The second part is just the definition of an almost module category from [9].
The following result is the analogue of Proposition 16.1 for triangulated cat*
*egories.
Theorem 16.2. Let R be a full subcategory of a compactly generated triangulated*
* cat-
egory S. Then the following are equivalent.
(1)R is a triangulated subcategory, and the inclusion has a left and a right*
* adjoint.
(2)There exists an idempotent ideal I of Sc satisfying I = I, such that R =*
* I? .
COHOMOLOGICAL QUOTIENTS 39
In this case, the left adjoint of the quotient functor S ! S=R identifies S=R w*
*ith a
smashing subcategory of S. Moreover, every smashing subcategory of S arises in*
* this
way.
Proof.Let us denote by F :R ! S the inclusion functor.
(1) ) (2): The left adjoint E :S ! R of the inclusion R ! S is a smashing loc*
*al-
ization functor since E OF ~=IdR . It follows from Theorem 12.1 and its Corolla*
*ry 12.6
that I = Sc\ Ann E is an idempotent ideal satisfying I = I and I? = R.
(2) ) (1): Lemma 12.4 implies that I? = R is a triangulated subcategory. Let *
*us
replace I by the ideal J of all maps in Sc annihilating R. Thus J is a cohomolo*
*gical ideal
satisfying J = J and J? = R. The proof of Theorem 12.1 shows that R is perfect*
*ly
cogenerated. Thus the inclusion F :R ! S has a left adjoint by Corollary 10.2, *
*since F
preserves all products. Theorem 11.1 implies that R is compactly generated. Thu*
*s F
has a right adjoint by the dual of Corollary 10.2, since F preserves all coprod*
*ucts.
Now let us prove the second part. Suppose first that (1) - (2) hold. Then t*
*he left
adjoint E :S ! R of the inclusion R ! S is a smashing localization functor. It *
*follows
that the left adjoint of the quotient functor S ! S=R identifies S=R with KerE,*
* which
is by definition a smashing subcategory.
Finally, suppose that T is a smashing subcategory of S. Let I be the idempote*
*nt ideal
of all maps in Sc which factor through some object in T . Then R = I? is a tria*
*ngulated
subcategory of S, and the inclusion R ! S has a left and a right adjoint. It fo*
*llows that
the left adjoint of the quotient functor S ! S=R identifies S=R with T . This f*
*inishes
the proof.
Let us complete the parallel between module categories and derived categories*
*. Thus
we consider the unbounded derived category D(R) of the module category Mod R. C*
*om-
paring the statements of Proposition 16.1 and Theorem 16.2, we see that the for*
*mal
analogue of an almost module category is a triangulated category of the form
D(R, I) = D(R)=(I? )
for some idempotent ideal I of Kb(R) satisfying I = I. We call such a category*
* an
almost derived category.
Next we show that the derived category of an almost module category is an alm*
*ost
derived category.
Corollary 16.3. Let R be a ring and a be an idempotent ideal such that a R a i*
*s flat
as left R-module. Denote by A the maps in Kb(R) which annihilate all suspension*
*s of
the mapping cone of the natural map a R a ! R. Then we have
A2 = A and D(Mod (R, a)) = D(R, A).
Proof.The quotient functor F :Mod R ! Mod (R, a) has a left adjoint E, and we *
*have
(E OF )M = M R (a R a);
see for instance [29, p. 200]. The extra assumption on a implies that E is exac*
*t. Taking
derived functors, we obtain an adjoint pair of exact functors
RF :D(Mod R) -! D(Mod (R, a)) and LE :D(Mod (R, a)) -! D(Mod R)
such that RF OLE ~=IdD(Mod(R,a)). It follows that LE identifies D(Mod (R, a)) w*
*ith a
smashing subcategory R of D(R). Now observe that the mapping cone Cone(a R a !
40 HENNING KRAUSE
R) generates D(R)=R. In fact, the canonical map
R -! Cone(a R a ! R)
is an isomorphism in D(R)=R since (LE ORF )R = a R a. Therefore A is the exact *
*ideal
corresponding to R which is idempotent by Corollary 12.6. Moreover, the localiz*
*ation
functor RF identifies D(R)=(A? ) with D(Mod (R, a)).
We know that the derived category of a ring is compactly generated. This is n*
*o longer
true for almost derived categories [15]. In deed, the telescope conjecture expr*
*esses the
fact that all almost derived categories are compactly generated.
Corollary 16.4. The telescope conjecture holds for the derived category D(R) of*
* a
ring R if and only if every almost derived category D(R, I) is a compactly gene*
*rated
triangulated category.
Appendix A. Epimorphisms of additive categories
An additive functor F :C ! D between additive categories is called an epimorp*
*hism of
additive categories, or simply an epimorphism, if G OF = G0OF implies G = G0for*
* any
pair G, G0:D ! E of additive functors. In this section we characterize epimorph*
*isms of
additive categories in terms of functors between their module categories. This *
*material
is classical [22], but we need it in a form which slightly generalizes the usua*
*l approach.
Lemma A.1. Let F :C ! D be an additive functor and suppose F is surjective on
objects. Then the following are equivalent.
(1)The restriction F*: Mod D ! Mod C is fully faithful.
(2)F is an epimorphism.
Proof.(1) ) (2): Let G, G0:D ! E be a pair of additive functors satisfying G OF*
* =
G0OF . Clearly, G and G0 coincide on objects since F is surjective on objects.*
* Now
choose a map ff: X ! Y in D. We need to show that Gff = G0ff. The functor G0
induces a C-linear map
fl :F*D(-, Y ) -! (F*O G*)E(-, GY ),
which is defined by
flC :D(F C, Y ) -! E(G(F C), GY ), OE 7! G0OE
for each C in C. The fact that F* is full implies that fl = F*ffi for some D-li*
*near map
ffi :D(-, Y ) ! G*E(-, GY ). In particular, ffiX = flC for some C in C satisfyi*
*ng F C = X.
Thus we obtain the following commutative diagram
ffiY
D(Y, Y )__________//_E(GY, GY )
D(ff,Y|)| E(Gff,GY|)|
fflffl| ffiX fflffl|
D(X, Y )__________//_E(GX, GY )
which shows Gff = G0ff if we apply it to idY. We conclude that G = G0.
(2) ) (1): Let M, N be a pair of D-modules. We need to show that the canonical
map
(F*)M,N : Hom D(M, N) -! Hom C(F*M, F*N)
COHOMOLOGICAL QUOTIENTS 41
is bijective. Given a family OE = (OEX )X2D of maps OEX :MX ! NX, we define a
D-module Hffiby
~ ~
HffiX = MX q NX and Hffiff = Nff OOEMff 0
Y - OEX OMffNff
for each object X and each map ff: X ! Y in D. Note that
(OEX )X2D :M -! N
is D-linear if and only if Hffi= M q N. Now define _X = OEFX for each X in C. *
*Then
(_X )X2C :F*M -! F*N
is C-linear if and only if HffiOF = (M q N) OF . Thus F*OE = _ because HffiOF*
* =
(M q N) OF implies Hffi= M q N. We conclude that the map (F*)M,N is surjective.
To prove that (F*)M,N is injective, choose a non-zero map OE: M ! N. Thus Im OE*
* 6= 0.
We have Im(F*OE) = F*(Im OE) 6= 0 and therefore F*OE 6= 0. It follows that (F**
*)M,N is
injective.
Lemma A.2. Let F :C ! D be an additive functor. If F is an epimorphism, then F
is surjective on objects.
Proof.Suppose there is an object D in D which does not belong to the image of F*
* . We
construct a new additive category E which contains D as a full subcategory and *
*has one
additional object, denoted by D0. Let E(X, D0) = D(X, D) and E(D0, X) = D(D, X)
for all X in D, and let E(D0, D0) = D(D, D). Now define G: D ! E to be the incl*
*usion,
and define G0:D ! E by G0X = GX for all X in D, except for X = D, where we put
G0D = D0. Clearly, G OF = G0OF but G 6= G0. Thus an epimorphism is surjective on
objects.
Lemma A.3. Let F :C ! D be a functor having a left adjoint E :D ! C. Then F is
fully faithful if and only if E OF ~=IdC.
Proof.Fix an object X in C. The map
C(X, Y ) -! C(EF X, Y ) -~! D(F X, F Y )
is bijective for all Y in D if and only if EF X ~=X.
Lemma A.4. Let F :C ! D be an additive functor and suppose C is small. If the
restriction F*: Mod D ! Mod C is fully faithful, then every object in D is a d*
*irect
factor of some object in the image of F .
Proof.The restriction F* has a left adjoint F *:Mod C ! Mod D, and we have F **
*OF* ~=
IdModD by Lemma A.3. Now fix an object Y in D. Every module is a quotient of*
* a
coproduct of representable functors. Thus we have an epimorphism
a
C(-, Xi) -! F*D(-, Y ),
i2
and applying F *induces an epimorphism
a
D(-, F Xi) -! D(-, Y ).
i2
`
Using Yoneda's lemma, we see that Y is a direct factor of F ( i2 Xi) for some *
*finite
subset .
42 HENNING KRAUSE
Let F :C ! D be an additive functor, and denote by D0 the full subcategory of*
* D
formed by the objects in the image of F . We say that F is an epimorphism up to*
* factors,
if the induced functor C ! D0 is an epimorphism, and if every object in D is a *
*direct
factor of some object in D0.
Proposition A.5. Let F :C ! D be an additive functor and suppose C is small. Th*
*en
the following are equivalent.
(1)F is an epimorphism up to factors.
(2)F*: Mod D ! Mod C is fully faithful.
(3)F *OF* ~=IdModD.
Proof.We write F as composite
C -F1!D0-F2!D
where D0denotes the full subcategory of D formed by the objects in the image of*
* F .
(1) ) (2): The functor (F2)* is an equivalence, and (F1)* is fully faithful b*
*y Lemma A.1.
Thus F* is fully faithful.
(2) ) (1): Lemma A.4 implies that every object in D is a direct factor of som*
*e object
in D0. Thus (F1)* is fully faithful, and Lemma A.1 implies that F1 is an epimor*
*phism.
(2) , (3): Use Lemma A.3.
The property of being an epimorphism is invariant under enlarging the univers*
*e.
Lemma A.6. Let U and V be universes in the sense of Grothendieck [10, I.1], and
suppose U V. If F :C ! D is an epimorphism of additive U-categories, then F i*
*s an
epimorphism of additive V-categories.
Proof.Let G, G0:D ! E be a pair of additive functors into a V-category E satisf*
*ying
G OF = G0OF . We denote by F the smallest additive subcategory of E containing *
*the
image of G and G0. Observe that F is a U-category since D is a U-category. Th*
*us
the restrictions D ! F of G and G0 agree by our assumption on F . It follows t*
*hat
G = G0.
Appendix B. The abelianization of a triangulated category
Let C be a triangulated category. In this section we discuss some properties*
* of the
abelianization mod C of C.
Lemma B.1. The category mod C is an abelian Frobenius category, that is, there *
*are
enough projectives and enough injectives, and both coincide.
Proof.The representable functors are projective objects in mod C by Yoneda's le*
*mma.
Thus mod C has enough projectives. Using the fact that the Yoneda functors C ! *
*mod C
and Cop ! mod (Cop) are universal cohomological functors, we obtain an equivale*
*nce
(mod C)op ! mod (Cop) which sends C(-, X) to C(X, -) for all X in C. Thus the
representable functors are injective objects, and mod C has enough injectives.
The triangulated structure of C induces some additional structure on mod C. *
*This
involves the equivalence ?: mod C ! mod C which extends : C ! . By abuse
of notation, we identify ? = . Using this internal grading, the category mod*
* C is
(3, -1)-periodic [8]. Thus we obtain a canonical extension ~M in Ext3C( M, M) *
*for every
COHOMOLOGICAL QUOTIENTS 43
module M in mod C. Under some additional assumptions, this extension is induced*
* by
a Hochschild cocycle of degree (3, -1); it plays a crucial role in [4].
Proposition B.2. Let C be a triangulated category.
(1)Given a pair M, N of objects in mod C, there is a natural map
~M,N : Hom C(M, N) -! Ext3C( M, N)
and we write ~N = ~N,N(idN).
(2)Let
: X -ff!Y -fi!Z -fl! X
be a triangle in C and let N = Ker C(-, ff). Then is exact if and only *
*if the
map fl induces a map C(-, Z) -! N such that the sequence
C(-,ff) C(-,fi)
0 -! N -! C(-, X) ----! C(-, Y ) ----! C(-, Z) -! N -! 0
is exact in mod C and represents ~N .
Proof.(1) Let M = CokerC(-, fi) be a an object in mod C and complete fi :Y ! Z *
*to
an exact triangle X ! Y ! Z ! X to obtain a projective resolution
. .!.C(-, Y ) ! C(-, Z) ! C(-, X) ! C(-, Y ) ! C(-, Z) ! M ! 0
of M. The map ~M,N takes by definition a map OE: M ! N to the element in
Ext3C( M, N) which is represented the composition of OE with the projection C(-*
*, Z) !
M.
(2) Fix a triangle
: X -ff!Y -fi!Z -fl! X
in C and let N = KerC(-, ff). The definition of ~N implies that the induced seq*
*uence
C(-,ff) C(-,fi)
" :0 -! N -! C(-, X) ----! C(-, Y ) ----! C(-, Z) -! N -! 0
is exact in mod C and represents ~N . Conversely, suppose that " = ~N . Comple*
*te ff
to an exact triangle
0 fl0
0:X -ff!Y -fi!Z0- ! X
in C. We use dimension shift and replace both sequences " and " 0 by short ex*
*act
sequences
0 ! -2N ! C(-, Z) ! N ! 0 and 0 ! -2N ! C(-, Z0) ! N ! 0.
These represent the same element in Ext1C( N, -2N), and we obtain therefore an
isomorphism OE: Z ! Z0 which induces an isomorphism of triangles ! 0. Thus *
* is
exact.
Let us explain a more conceptual way to understand the natural map ~M,N . To *
*this
end denote by mod_C the stable category of mod C, that is, the objects are thos*
*e of mod C
and
Hom__C(M, N) = Hom C(M, N)=P(M, N)
where P denotes the ideal of all maps in mod C which factor through some projec*
*tive
object. Taking syzygies in mod C induces an equivalence
: mod_C -! mod_C
44 HENNING KRAUSE
since mod C is a Frobenius category. Moreover,
Hom__C( nM, N) ~=ExtnC(M, N) ~=Hom__C(M, -n N) for all M, N and n >.0
The map ~M,N induces a natural isomorphism
Hom__C(M, N) -! Ext3C( M, N),
and composing this with the natural isomorphism
Ext 3C( M, N) -! Hom__C( M, -3N)
induces a natural isomorphism between
~: mod_C -! mod_C and -3 :mod_C -! mod_C.
Note that the natural map ~M,N can be reconstructed from the natural isomorphi*
*sm
~ ~= -3.
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Henning Krause, Department of Pure Mathematics, University of Leeds, Leeds LS*
*2 9JT,
United Kingdom.
E-mail address: henning@maths.leeds.ac.uk