Table Of ContentHomological Properties of Differential Graded Algebras
DavidPauksztello
Submittedinaccordancewiththerequirements
forthedegreeofPhD.
TheUniversityofLeeds
DepartmentofPureMathematics
July2008
Thecandidateconfirmsthattheworksubmittedishisownworkandthatappropriate
credithasbeengivenwherereferencehasbeenmadetotheworkofothers.
Thiscopyhasbeensuppliedontheunderstandingthatitiscopyrightmaterialand
thatnoquotationfromthethesismaybepublishedwithoutproperacknowledgement.
i
Abstract
In this thesis we consider various homological properties of differential graded algebras,
andmoregenerally,propertiesofarbitrarytriangulatedcategorieswhichhavesetindexed
coproducts. A major example of such a triangulated category is the derived category
of a differential graded algebra. We present the background material to the theory of
triangulated categories, derived categories and differential graded algebras as well as a
briefresume´ ofclassicalhomologicalalgebrainChapters2and3.
InChapter4weconsiderthefollowingsituation: letRandS bedifferentialgradedal-
gebrasandM beadifferentialgradedR-S-bimodulesatisfyingsomefinitenesscondition.
InthischapterwegiveacharacterisationofwhenadifferentialgradedR-S-bimoduleM
inducesafullembeddingofderivedcategories
L
M ⊗ − : D(S) → D(R).
R S S
In particular, this characterisation generalises the theory of Geigle and Lenzing’s homo-
logicalepimorphismsofrings,describedin[24]. Furthermore,thereisanexampleofthe
main result in relation to Dwyer and Greenlees’s Morita theory and we are able to apply
the main result to give a characterisation of when the co-induction functor yields a full
embeddingofderivedcategories.
InChapter5weconsiderthenewnotionofaco-t-structureonatriangulatedcategory.
In the work of Hoshino, Kato and Miyachi, [31], the authors look at t-structures induced
by a compact object, C, of a triangulated category, T , which is rigid in the sense of
Iyama and Yoshino, [32]. Hoshino, Kato and Miyachi show that such an object yields
a non-degenerate t-structure on T whose heart is equivalent to Mod(End(C)op). Rigid
objects in a triangulated category can the thought of as behaving like chain differential
gradedalgebras(DGAs).
Analogously, looking at objects which behave like cochain DGAs naturally gives the
ii
dual notion of a corigid object. Here, we see that a compact corigid object, S, of a
triangulated category, T , induces a structure similar to a t-structure which we shall call
a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is
equivalenttoMod(End(S)op),andhenceanabeliansubcategoryofT .
At the end of this chapter we present a generalisation of the main result to the case
when we have more than one corigid object and make some conjectures about further
generalisations. We also highlight the role which co-t-structures may play in the theory
oftorsiontheoriesandtheirpositioncomplementingthetheoryoft-structures.
In Chapter 6 we consider generalised Moore spectra in a triangulated catgeory. This
was first addressed at this level of generality by Jørgensen in [35]. Let R be a principal
idealdomainandT beanR-lineartriangulatedcategory,thenundercertainniceassump-
tionsonR,givenanR-moduleA,Jørgensenlookedforthe“bestapproximation”ofAin
T ,M(A),calledthe“Moorespectrum”ofA.
InChapter6weobtainthefollowinggeneralisedconstruction. LetT beatriangulated
category and let C be a compact object of T satisfying certain nice “tilting” hypotheses.
Let S = End(C). Then we obtain a full embedding M : Mod(Sop) → M which is
left adjoint to the functor Hom (C,−) : M → Mod(Sop), where M is some auxilliary
T
full subcategory of T . We show that the functor M is well-behaved and relate the new
constructionwithJørgensen’sconstructionin[35]. Asanexample,werecoverthecanon-
icaltriangleembeddingofthecategoryoffinitelypresentedrightmodulesofahereditary
algebraintoitsu-clustercategoryofu > 2from[40].
iii
Contents
Abstract i
Acknowledgements vi
“TheLogicalExpressionist”byBethJervis vii
1 Introduction 1
1.1 Outlineofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 ClassicalHomologicalAlgebra 10
2.1 Abeliancategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Homologyandcohomology . . . . . . . . . . . . . . . . . . . . 14
2.3 Classicalderivedfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Projectiveandinjectiveobjects . . . . . . . . . . . . . . . . . . . 15
2.3.2 Projectiveandinjectivepresentationsandresolutions . . . . . . . 16
2.3.3 Projectiveandglobaldimensions . . . . . . . . . . . . . . . . . 17
2.3.4 Derivedfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.5 ThefunctorsExti (−,−)andTorR(−,−) . . . . . . . . . . . . . 19
R i
3 TriangulatedandDerivedCategoriesandDifferentialGradedAlgebras 22
3.1 Triangulatedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Derivedcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Thehomotopycategory . . . . . . . . . . . . . . . . . . . . . . 26
iv
3.2.2 Quasi-isomorphisms,localisationandthederivedcategory . . . . 28
3.3 Derivedfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Rightderivedfunctors . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.2 ThefunctorsRHom (−,−)andExti (−,−) . . . . . . . . . . . 32
R R
3.4 Differentialgradedalgebrasandmodules . . . . . . . . . . . . . . . . . 34
3.4.1 Differentialgradedalgebras . . . . . . . . . . . . . . . . . . . . 35
3.4.2 Differentialgradedmodules . . . . . . . . . . . . . . . . . . . . 36
3.5 ThecategoriesK (R)andD(R) . . . . . . . . . . . . . . . . . . . . . . 38
3.5.1 ThefunctorHom (−,−) . . . . . . . . . . . . . . . . . . . . . . 39
R
L
3.5.2 Thefunctors−⊗ −,− ⊗ −andTorR(−,−) . . . . . . . . . 40
R R i
3.6 K-projectivity,K-injectivityandK-flatness . . . . . . . . . . . . . . . . 42
3.6.1 K-projective,K-injectiveandK-flatobjects . . . . . . . . . . . 42
3.6.2 K-projectiveresolutionsforDGmodules . . . . . . . . . . . . . 45
L
3.6.3 ComputingRHom (−,−)and− ⊗ − . . . . . . . . . . . . . . 46
R R
4 HomologicalEpimorphismsofDifferentialGradedAlgebras 48
4.1 Compactobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Canonicalmaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Definingcanonicalmaps . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Anexamplecomputation . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3 Anotherexamplecomputation . . . . . . . . . . . . . . . . . . . 57
4.3 Characterisationfortherestrictionofscalarsfunctor . . . . . . . . . . . . 59
4.3.1 HomologicalepimorphismsofDGAs . . . . . . . . . . . . . . . 67
4.3.2 Versionsfortheboundedandfinitederivedcategories . . . . . . 70
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.1 DwyerandGreenlees’Moritatheory . . . . . . . . . . . . . . . 73
4.4.2 Homologicalepimorphismsofrings . . . . . . . . . . . . . . . . 74
4.4.3 Characterisationfortheco-inductionfunctor . . . . . . . . . . . 80
5 CorigidObjectsinTriangulatedCategoriesandCo-t-structures 82
5.1 Thedefinitionofaco-t-structure . . . . . . . . . . . . . . . . . . . . . . 83
5.1.1 Preenvelopesandperpendicularcategories . . . . . . . . . . . . 83
v
5.1.2 t-structuresandco-t-structures . . . . . . . . . . . . . . . . . . . 85
5.2 Someexamplesoft-structuresandco-t-structures . . . . . . . . . . . . . 88
5.2.1 Acanonicalexample . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2 ChainandcochainDGAs . . . . . . . . . . . . . . . . . . . . . . 89
5.2.3 At-structureobtainedfromarigidobject . . . . . . . . . . . . . 90
5.3 Aco-t-structureobtainedfromacorigidobject . . . . . . . . . . . . . . 91
5.3.1 ExistenceofanS⊥n-preenvelope . . . . . . . . . . . . . . . . . 93
5.3.2 Homotopycolimits . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.3 ExistenceofanS⊥∞-preenvelope . . . . . . . . . . . . . . . . . 97
5.3.4 Theinducedco-t-structure . . . . . . . . . . . . . . . . . . . . . 99
5.4 Thecoheartoftheco-t-structureofTheorem5.3.12 . . . . . . . . . . . . 102
5.5 Aversionformorethanoneobject . . . . . . . . . . . . . . . . . . . . . 106
5.6 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.1 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.2 ThefullsubcategoryB . . . . . . . . . . . . . . . . . . . . . . . 112
6 GeneralisedMooreSpectrainaTriangulatedCategory 114
6.1 Jørgensen’sconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Thegeneralconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.1 Theinductionhypothesesandbasestep . . . . . . . . . . . . . . 118
6.2.2 TheconstructionofthemapM . . . . . . . . . . . . . . . . . 120
k+1
6.2.3 FunctorialityofM . . . . . . . . . . . . . . . . . . . . . . . 123
k+1
6.2.4 Thescaffolding . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Someconsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3.1 Jørgensen’sconstructionrevisited . . . . . . . . . . . . . . . . . 132
6.3.2 ThefunctorM iswellbehaved . . . . . . . . . . . . . . . . . . . 133
6.3.3 AnalternativeproofofTheorem6.3.3 . . . . . . . . . . . . . . . 138
6.4 Anexamplefromu-clustercategories . . . . . . . . . . . . . . . . . . . 143
6.4.1 Aversionofthemainresultforfinitelypresentedmodules . . . . 144
6.4.2 u-clustercategories . . . . . . . . . . . . . . . . . . . . . . . . . 145
Bibliography 150
vi
Acknowledgements
Firstly, I would like to thank my supervisor, Prof Peter Jørgensen, for all his helpful
advice, encouragement and support during the preparation of this thesis. Thanks are also
due to the members of the Leeds Algebra Group for many useful discussions, seminars
and post-seminar pub trips, in particular, my fellow PhD students Graham and Marcel. I
am also grateful to an anonymous referee for useful comments on the material presented
inChapter5.
IwouldliketoacknowledgetheEPSRCandUniversityofLeedsforfinancialsupport
during the first three and a half years of my PhD and the Bank of Mum and Dad for the
remainingsixmonths!
In general, I am grateful to my family, cousins, aunts and uncles and all; in particular,
myMum,Dadandbrotherwhohavebeenagreatsupportthroughoutmystudies. Iwould
also like to make a special mention to my Grandma and Grandad, the latter for all those
years spent in the corner and for instilling in me from a young age the importance of
proof: “I’lltellyouandI’lltellyouforwhy”.
Thanks also to the inhabitants of Level 11, past and present, for making my time in
Leedssoenjoyable. Iamalsoindebtedtovariousfriendsaroundthecountryforallowing
metoescapeLeedsfromtimetotime,inparticular,OliverBridleandGarethBarnett.
Muchas gracias tambie´n to los Bollo for their hospitality on numerous hugely enjoy-
abletripstoMadrid,ZaragozaandAndorra. SpecialthankstoAliciaBolloFustero,firstly
for being so Spanish, then for her wicked sense of humour (in both senses of the word)
andforalwaysbeingright(especiallywhenshewaswrong!).
Thanks also to Beth Jervis whose wonderfully cheesy poem graces the beginning of
this thesis. And a final thank you to Matt Daws for kindly providing the template with
whichthisthesiswaswritten.
vii
The Logical Expressionist
by Beth Jervis
Everymathematicianis
Apoettrulyatheart,
Whohaschosennumericalreasoning
Asthemediumforhisart.
Everymathematicianhas
Apassionforrhymeandrhythm,
Flowingbeautifullyinalgebra
Andeccentricandinspiringinalgorithm.
Everymathematiciangets
Excitedaboutaproof,
Eruptswithpleasureateurekatime,
Andcriesout,exclaims,woohoo!
Everymathematiciandoes
Hishardandtiresometrials,
Heknowshissoulfromthedeepestpits
Ofpainstakingunworkablehours.
Butmostofall,
Themathematicianrecognisesart.
Hisbeautifulmaths
viii
Iscomposedassuch,
Onthecanvasofreasonandlogic.
Hebendsthemaslight,
Therulesofreality,
Butonlywithfaithfultools,
Andarrivesatsingularity,
Withcontingenceinmind,
Thatcaninfinitelyandrandomlydisperse.
Hetakeshismodel,
Hisartist’sgift,
Andappliesittotheworld,
Andsometimesitisappliedmechanically,
Andsometimesitturnsoutpure.
Butstatistically,
Themathematicianknowsthisisart.
Foreverymathematician,
Istrulyapoetatheart.
Andeverymathematician,
Hasastorythatspringsfromthisheart.
Hegrappleswithabstraction,
Andinvestshisbelief,
Andreducestheworldtonumber.
Hecountsonthisnumberlike
Sleeplesscountingsheep,
Andpocketstheworldtokeep.
Chapter1.Introduction 1
Chapter 1
Introduction
TheareaofmathematicsnowknownashomologicalalgebrafirstaroseintheearlyTwen-
tieth Century from algebraic topology, in which the homology of chain complexes was
used to provide invariants for topological spaces. In particular, topological spaces whose
homologies are different cannot be homeomorphic to one another. In the introduction
to [57], Weibel describes homological algebra as a “tool used to prove nonconstructive
existence theorems in algebra and in algebraic topology”. Weibel also tells us that ho-
mological algebra can be used to provide us with obstructions to whether certain con-
structions can be carried out. Since the beginning of the second half of the Twentieth
Century, homological algebra has blossomed, starting with Cartan and Eilenberg’s semi-
naltext[17]whichintroducedthehomologicalfunctorsExt(−,−)andTor(−,−)which
are the mainstay of classical homological algebra. It has become ubiquitous in many ar-
eas of mathematics, for example in the theory of representations of algebras, algebraic
geometry,algebraictopologyandinthetheoryofBanachalgebras.
SinceGrothendieckinthe1960s,thelanguageofhomologicalalgebrahasbeenhighly
categorical; the notions of abelian categories, projective and injective objects and resolu-
tionsbysuchobjects,andderivedfunctorshavebecomecentral,providingpowerfultools
in diverse branches of mathematics. During the second half of the Twentieth Century the
notions of triangulated categories and derived categories have become increasingly im-
portant tools in homological algebra. Derived categories of abelian categories (or, more
generally, of additive categories) are mathematical objects which encode the homologi-
cal information about the given mathematical object or property one is studying, and as
