Triangulated Categories.

451 Pages·2001·1.78 MB·English

by  Amnon Neeman

Contents
0. Acknowledgements 3
1. Introduction 3
Chapter 1. Definition and elementary properties of triangulated
categories 29
1.1. Pre–triangulated categories 29
1.2. Corollariesof Proposition 1.1.20 37
1.3. Mapping cones, and the definition of triangulated categories 45
1.4. Elementary properties of triangulated categories 52
1.5. Triangulated subcategories 60
1.6. Direct sums and products, and homotopy limits and colimits 63
1.7. Some weak “functoriality” for homotopy limits and colimits 68
1.8. History of the results in Chapter 1 70
Chapter 2. Triangulated functors and localizations of triangulated
categories 73
2.1. Verdier localization and thick subcategories 73
2.2. Sets and classes 99
2.3. History of the results in Chapter 2 100
Chapter 3. Perfection of classes 103
3.1. Cardinals 103
3.2. Generated subcategories 103
3.3. Perfect classes 110
3.4. History of the results in Chapter 3 122
Chapter 4. Small objects, and Thomason’s localisation theorem 123
4.1. Small objects 123
4.2. Compact objects 128
4.3. Maps factor through (cid:104)S(cid:105)β 130
4.4. Maps in the quotient 135
4.5. A refinement in the countable case 144
4.6. History of the results in Chapter 4 150
Chapter 5. The category A(S) 153
v
5.1. The abelian category A(S) 153
5.2. Subobjects and quotient objects in A(S) 172
5.3. The functoriality of A(S) 177
5.4. History of the results in Chapter 5 182
Chapter 6. The category Ex Sop,Ab 183
6.1. Ex Sop,Ab isanabeliancategorysatisfying[AB3]and[AB3∗]183
(cid:0) (cid:1)
6.2. The case of S=Tα 201
6.3. Ex(cid:0)Sop,Ab(cid:1) satisfies [AB4] and [AB4∗], but not [AB5] or
[AB5∗] 206
6.4. Pro(cid:0)jectives(cid:1)and injectives in the categoryEx Sop,Ab 211
6.5. The relation between A(T) and Ex {Tα}op,A(cid:0)b (cid:1) 214
6.6. History of the results of Chapter 6(cid:16) (cid:17) 220
Chapter 7. Homological properties of Ex Sop,Ab 221
7.1. Ex Sop,Ab as a locally presentable category 221
(cid:0) (cid:1)
7.2. Homological objects in Ex Sop,Ab 224
(cid:0) (cid:1)
7.3. A technical lemma and some consequences 230
7.4. The derived functors of co(cid:0)limits in(cid:1)Ex Sop,Ab 253
7.5. The adjoint to the inclusion of Ex Sop,Ab 266
(cid:0) (cid:1)
7.6. History of the results in Chapter 7 271
(cid:0) (cid:1)
Chapter 8. Brown representability 273
8.1. Preliminaries 273
8.2. Brown representability 275
8.3. The first representability theorem 280
8.4. Corollariesof Brown representability 285
8.5. Applications in the presence of injectives 288
8.6. The second representability theorem: Brown representability
for the dual 303
8.7. History of the results in Chapter 8 306
Chapter 9. Bousfield localisation 309
9.1. Basic properties 309
9.2. The six gluing functors 318
9.3. History of the results in Chapter 9 319
Appendix A. Abelian categories 321
A.1. Locally presentable categories 321
A.2. Formal properties of quotients 327
A.3. Derived functors of limits 345
A.4. Derived functors of limits via injectives 354
A.5. A Mittag–Leffler sequence with non–vanishing limn 361
A.6. History of the results of Appendix A 366
vi
Appendix B. Homological functors into [AB5α] categories 369
B.1. A filtration 369
B.2. Abelian categoriessatisfying [AB5α] 378
B.3. History of the results in Appendix B 385
Appendix C. Counterexamples concerning the abelian category A(T)387
C.1. The submodules piM 387
C.2. A large R–module 392
C.3. The categoryA(S) is not well–powered 393
C.4. A categoryEx Sop,Ab without a cogenerator 395
C.5. History of the results of Appendix C 405
(cid:0) (cid:1)
Appendix D. Where T is the homotopy categoryof spectra 407
D.1. Localisation with respect to homology 407
D.2. The lack of injectives 420
D.3. History of the results in Appendix D 426
Appendix E. Examples of non–perfectly–generatedcategories 427
E.1. IfTisℵ –compactlygenerated,Top isnotevenwell–generated427
0
E.2. An example of a non ℵ –perfectly generated T 432
1
E.3. For T =K(Z), neither T nor Top is well–generated. 437
E.4. History of the results in Appendix E 442
Bibliography 443
Index 445
vii

0. Acknowledgements
The author would like to thank Andrew Brooke–Taylor,Daniel Chris-
tensen, Pierre Deligne, Jens Franke, Bernhard Keller, Shun–ichi Kimura,
Henning Krause, Jack Morava, Saharon Shelah and Vladimir Voevodsky
for helpful discussions, and for their valuable contributions to the book.
1. Introduction
Before describing the contents of this book, let me explain its origins.
ThebookbeganasajointprojectbetweentheauthorandVoevodsky. The
idea was to assemble coherently the facts about triangulated categories,
that might be relevant in the applications to motives. Since the presumed
reader would be interested in applications, Voevodsky suggested that we
keep the theory part of the book free of examples. The interested reader
should have an example in mind, and read the book to find out what the
general theory might have to say about the example. The theory should
be presented cleanly, and the examples kept separate.
The division of labor was that I should write the theory, Voevodsky
theapplicationstomotives. Whatthen happenedwasthatmypartofthis
jointprojectmushroomedoutofproportion. Thisbookconsistsjustofthe
formal theory of triangulated categories. In a sequel, we hope to discuss
the motivic applications.
The project was initially intended to be purely expository. We meant
tocovermanytopics,buthadnonewresults. Thiswastobeanexposition
of the known facts about Brown representability, Bousfield localisation, t–
structures and triangulated categories with tensor products. The results
should be presented in a unified, clear way, with the exposition accessible
to a graduatestudent wishing to learn the theory. The catch was that the
theory should be developed in the generality one would need motivically.
The motivic examples, unlike the classical ones, are not compactly gener-
atedtriangulatedcategories(whateverthismeans). Theclassicalliterature
basically does not treat the situation in the generality required.
Myjobamountedtomodifyingthe classicalarguments,toworkinthe
greatergenerality. AsIstarteddoingthis,Iquicklycametotheconclusion
that both the statements and the proofs given classically are very unsat-
isfactory. The proofs in the literature frequently rely on lifting problems
about triangulatedcategoriesto problemsabout morerigidmodels. Right
at the outset I decided that in this book, I will do everything to avoid
models. Part of the challenge was to see how much of the theory can be
developedwithouttheusualcrutch. Buttherewasafarmoreseriousprob-
lem. Many of the statements were known only in somewhat special cases,
decidedly not including the sort that come up in motives. Thomason once
told me that “compact objects are as necessary to this theory as air to
3
4 1. INTRODUCTION
breathe”. In his words, I was trying to develop the theory in the absence
of oxygen.
The book isthe result of my workon the subject. It treatsa narrower
scope of topics than initially planned; we deal basically only with Brown
representability and Bousfield localisation. But in some sense we make
great progress on the problems. In the process of setting up the theory in
the right generality and without lifting to models, we end up with some
new and surprising theorems. The book was meant to be an exposition of
knownresults. Thewayitturnedout,itdevelopsacompletelynewtheory.
And this theory gives interesting, new applications to very old problems.
Now it is time to summarise the mathematical content of the book.
The first two chapters of the book are nothing more than a self–
contained exposition of known results. Chapter 1 is the definitions and
elementary properties of triangulated categories, while Chapter 2 gives
Verdier’s construction of the quotient of a triangulated category by a tri-
angulated subcategory. This book was after all intended as a graduate
textbook, and therefore assumes little prior knowledge. We assume that
the reader is familiar with the language of categories and functors. The
readershouldknowYoneda’sLemma,thegeneralfactsaboutadjointfunc-
tors between categories, units and counits of adjunction, products and co-
products. It is also assumed that the reader has had the equivalent of
an elementary course on homological algebra. We assume familiarity with
abelian categories, exact sequences, the snake lemma and the 5–lemma.
But this is all we assume. In particular,the readeris not assumed to have
ever seen the definition of a triangulated category. In practice, since we
give no examples, the reader might wish to find one elsewhere, to be able
to keep it in mind as an application of the general theory. One place to
find a relatively simple, concrete exposition of one example, is the first
chapter of Hartshorne’s book [19]. This first chapter develops the derived
category. Note that, since we wishto study mostly triangulatedcategories
closed under all small coproducts, the derived category of most interest is
the unbounded derived category. In [19], this is the derived category that
receives probably the least attention. There is another short account of
the derivedcategoryin Chapter10, pages369–415of Weibel’s [37]. There
are, of course, many other excellent accounts. But they tend to be longer.
Anyway, if the reader is willing to forget the examples, begin with the ax-
ioms, and see what can be proved using them, then this book is relatively
self–contained.
Chapters1and2areanaccountoftheveryclassicaltheory. Thereare
someexpositoryinnovationsinthesetwochapters,butotherwiselittlenew.
Ifthereaderwantstobeabletocomparethetreatmentgivenherewiththe
oldertreatments, at the end of each chapter there is a historicalsummary.
In the body of the chapters, I rarely give references to older works. The
1. INTRODUCTION 5
historical surveys at the end of each chapter contain references to other
expositions. Theyalsotrytopointoutwhat,ifanything,distinguishesthe
exposition given here from older ones.
Starting with Chapter 3, little of what is in the book may be found in
the literature. For the reader who has some familiarity with triangulated
categories, it seems only fair that the introduction summarise what, if
anything, he or she can expect to find in this book which they did not
already know. It is inevitable, however, that such an explanation will
demand from the reader some prior knowledge of triangulated categories.
Thegraduatestudent,whohasneverbeforemettriangulatedcategories,is
advisedtoskiptheremainderoftheintroductionandproceedtoChapter1.
After reading Chapters 1 and 2, the rest of the introduction will make a
lot more sense.
Let me begin with the concrete. Few people have a stomach strong
enough for great generalities. Sweeping, general theorems about arbitrary
2–categories tend to leave us cold. We become impressed only when we
learn that these theorems teach us something new. Preferably something
newabout an old, concreteexamplethat weknowand love. BeforeI state
the results in the book in great generality, let me tell the reader what we
may conclude from them about a special case. Let us look at the special
case, where T is the homotopy category of spectra.
Let T be the homotopy category of spectra. Let E be a spectrum (ie.
an object of T). Following Bousfield, the full subcategory T ⊂ T, whose
E
objects are called the E–acyclic spectra, is defined by
Ob(T ) = {x∈Ob(T)|x∧E =0}.
E
The full subcategory ⊥T ⊂T, whose objects are called the E–local spec-
E
tra, is defined by
Ob ⊥T = {y∈Ob(T)|∀x∈Ob(T ), T(x,y)=0}.
E E
An old theo(cid:0)rem (cid:1)of Bousfield (see [6]) asserts that one can localise spectra
with respecttoanyhomologytheoryE. Inthe notationabove,Bousfield’s
theorem asserts
Theorem (Bousfield, 1979). Let E be a spectrum, that is E is an
object of T. Let T ⊂ T and ⊥T ⊂ T be defined as above. Suppose x is
E E
an object of T. Then there is a triangle in T
x −−−−→ x −−−−→ ⊥x −−−−→ Σx ,
E E E
with x ∈T , and ⊥x ∈⊥T .
E E E E
Bousfield’stheoremhasbeenknownforalongtime. Whatthisbookhasto
add, are surprising structure theorems about the categories T and ⊥T .
E E
We prove the following representability theorems
6 1. INTRODUCTION
Theorem (New, this book). Let E be a spectrum. Let T ⊂ T
E
and ⊥T ⊂T be defined as above. The representable functors
E
T (−,h) and ⊥T (−,h)
E E
can be characterised as the homological functors H : Top −→ Ab (respec-
E
tively H : ⊥Top −→ Ab) taking coproducts in T (respectively ⊥T ) to
E E E
products in Ab. The representable functors
T (h,−)
E
can be characterised as the homological functors H : T −→ Ab, taking
E
products to products.
Proof: The characterisationof the functors T (−,h) and ⊥T (−,h) may
E E
be found in Theorem D.1.12. More precisely, for T (−,h) see D.1.12.1,
E
whilethestatementfor⊥T (−,h)iscontainedinD.1.12.5. Thecharacter-
E
isation of the functors T (h,−) may be found in Lemma D.1.14. (cid:50)
E
Representability theorems are central to this subject. What we have
achieved here, is to extend Brown’s old representability theorem of [7].
Brown proved that the functors T(−,h) can be characterised as the ho-
mological functors Top −→ Ab taking coproducts to products. We have
generalised this to T , Top and ⊥T , but unfortunately not to ⊥Top. We
E E E E
do not know, whether the the functors ⊥T (h,−) can be characterised as
E
the homologicalfunctors taking products to products.
Another amusing fact we learn in this book, is that the categories T
E
and⊥T arenotequivalenttoTop. Thereare,infact,manymoreamusing
E
facts we prove. Let us give one more. We begin with a definition.
Definition. Let α be a regular cardinal. A morphism f : x −→ y
in T is called an α–phantommap if, for any spectrum s with fewer than α
cells, any composite
f
s −−−−→ x −−−−→ y
vanishes.
With this definition, we are ready to state another fun fact that we
learn in this book.
Theorem (New, this book). Let α > ℵ be a regular cardinal.
0
There is an object z ∈T, which admits no maximal α–phantom map y −→
z. That is, given any α–phantom map y −→ z, there is at least one α–
phantom map x−→z not factoring as
x −−−−→ y −−−−→ z.
1. INTRODUCTION 7
Proof: TheproofofthisfactfollowsfromPropositionD.2.5,coupledwith
Lemma 8.5.20. (cid:50)
Remark 1.1. It should be noted that the above is surprising. If α =
ℵ , the α–phantom maps are the maps vanishing on all finite spectra.
0
Theseareveryclassical,andhavebeenextensivelystudiedintheliterature.
Usually, they go by the name phantom maps; the reference to α = ℵ is
0
newtothisbook,wherewestudythenaturallarge–cardinalgeneralisation.
FromtheworkofChristensenandStrickland[9],weknowthateveryobject
z ∈ T admits a maximal ℵ –phantom map y −→ z. There is an ℵ –
0 0
phantom map y −→ z, so that all other ℵ –phantom maps x −→z factor
0
as
x −−−−→ y −−−−→ z.
What is quite surprising is that this is very special to α=ℵ .
0
Sofar,wehavegiventhereaderasamplingoffactsaboutthehomotopy
categoryofspectra,whichfollowfromthemoregeneralresultsofthisbook.
I could give more; but it is perhaps more instructive to indicate the broad
approach.
Theideaofthisbookistostudyacertainclassoftriangulatedcatego-
ries, the well–generated triangulated categories. And the thrust is to prove
great facts about them. We will show, among many other things
Theorem 1.2. The following facts are true:
1.2.1. Let T be the homotopy category of spectra. Let E be an
object of T. Then both the category T and the category ⊥T are
E E
well–generated triangulated categories.
1.2.2. Suppose T is a well–generated triangulated category. The
representable functors T(−,h) can be characterised as the homolog-
ical functors H :Top −→Ab, taking coproducts in T to products in
Ab.
Inotherwords,wewillproveavastgeneralisationofBrown’srepresentabil-
itytheorem. Not onlydoesit generalisetoT and ⊥T , but to verymany
E E
othercategoriesaswell. Thecategoriesthattypicallycomeupinthestudy
of motives are examples. And now it is probably time to tell the reader
what a well–generated triangulated category is. It turns out to be quite a
deep fact that this structure even makes sense.
LetT beatriangulatedcategory. Weremindthereader: ahomological
functor T −→ A is a functor from T to an abelian category A, taking
triangles to long exact sequences. We can consider the collection of all
homologicalfunctorsT −→A. AnoldtheoremofFreyd’s(see[13])asserts
that
8 1. INTRODUCTION
Theorem (Freyd, 1966). Among all the homological functors
T −→ A there is a universal one. There is an abelian category A(T) and
a homological functor T −→ A(T), so that any other homological functor
T −→A factors as
T −−−−→ A(T) −−−∃−!→ A
where the exact functor A(T) −→ A is unique up to canonical equiva-
lence. Any natural tranformation of homological functors T −→ A factors
uniquely through a natural transformation of the (unique) exact functors
A(T)−→A.
Thistheoremtellsusthat, associatednaturallytoeverytriangulatedcate-
goryT, thereisanabeliancategoryA(T). Theassociationiseasilyseento
be functorial. It takes the 2–categoryof triangulated categoriesand trian-
gulatedfunctorstothe2–categoryofabeliancategoriesandexactfunctors,
and is a lax functor.
Onecanwonderaboutthehomologicalalgebraoftheabeliancategory
A(T). Freyd provesalso
Proposition (Freyd, 1966). Let T be a triangulated category. The
abelian category A(T) of the previous theorem has enough projectives and
enough injectives. In fact, the projectives and injectives in A(T) are the
same. An object a∈A(T) is projective (equivalently, injective) if and only
if there exists an object b∈A(T), so that
a⊕b∈T ⊂A(T).
Thatis,aisadirectsummandofanobjecta⊕b,anda⊕bisintheimageof
the universal homological functor T −→ A(T). This universal homological
functor happens to be a fully faithful embedding; hence I allow myself to
write T ⊂A(T).
It turns out to be easy to deduce the following corollary:
Corollary 1.3. Let F :S−→T be a triangulated functor. If F has a
right adjoint G:T −→S, thenG is also triangulated, and A(G):A(T)−→
A(S) is right adjoint to A(F) : A(S) −→ A(T). But more interesting is
the following. If every idempotent in S splits, then F : S −→ T has a
right adjoint if and only if A(F) : A(S) −→ A(T) does. That is, if A(F) :
A(S)−→A(T) has a rightadjoint G˜ :A(T)−→A(S), thenF :S−→T has
a right adjoint G:T −→S, and of course A(G) is naturally isomorphic to
G˜.
Proof: Lemma 5.3.6 shows that the adjoint of a triangulated functor is
triangulated,Lemma5.3.8provesthatifGisrightadjointtoF thenA(G)
is right adjoint to A(F), while Proposition 5.3.9 establishes that if A(F)
has a right adjoint G˜, then F has a right adjoint G. (cid:50)