Rotman-An Introduction to Homological Algebra, 2nd

721 Pages·2008·9.53 MB·English

Universitext
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http://www.springer.com/series/223
Joseph J. Rotman
An Introduction to
Homological Algebra
Second Edition
123
JosephJ.Rotman
DepartmentofMathematics
UniversityofIllinoisatUrbana-Champaign
UrbanaIL61801
USA
[email protected]
Editorialboard:
SheldonAxler,SanFranciscoStateUniversity
VincenzoCapasso,Universita`degliStudidiMilano
CarlesCasacuberta,UniversitatdeBarcelona
AngusMacIntyre,QueenMary,UniversityofLondon
KennethRibet,UniversityofCalifornia,Berkeley
ClaudeSabbah,CNRS,E´colePolytechnique
EndreSu¨li,UniversityofOxford
WojborWoyczynski,CaseWesternReserveUniversity
ISBN:978-0-387-24527-0 e-ISBN:978-0-387-68324-9
DOI10.1007/978-0-387-68324-9
LibraryofCongressControlNumber:2008936123
MathematicsSubjectClassification(2000):18-01
(cid:2)c SpringerScience+BusinessMedia,LLC2009
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To the memory of my mother
Rose Wolf Rotman
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Contents
PrefacetotheSecondEdition . . . . . . . . . . . . . . . . . . . x
HowtoReadThisBook . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter1 Introduction
1.1 SimplicialHomology . . . . . . . . . . . . . . . . . . . . . . 1
1.2 CategoriesandFunctors . . . . . . . . . . . . . . . . . . . . . 7
1.3 SingularHomology . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter2 HomandTensor
2.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 TensorProducts . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.2.1 AdjointIsomorphisms. . . . . . . . . . . . . . . . . . . . . . 91
Chapter3 SpecialModules
3.1 ProjectiveModules . . . . . . . . . . . . . . . . . . . . . . . 98
3.2 InjectiveModules . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3 FlatModules . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3.1 Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Chapter4 SpecificRings
4.1 SemisimpleRings . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2 vonNeumannRegularRings . . . . . . . . . . . . . . . . . . 159
4.3 HereditaryandDedekindRings . . . . . . . . . . . . . . . . . 160
vii
viii Contents
4.4 SemihereditaryandPru¨ferRings . . . . . . . . . . . . . . . . 169
4.5 Quasi-FrobeniusRings . . . . . . . . . . . . . . . . . . . . . 173
4.6 SemiperfectRings . . . . . . . . . . . . . . . . . . . . . . . . 179
4.7 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.8 PolynomialRings . . . . . . . . . . . . . . . . . . . . . . . . 203
Chapter5 SettingtheStage
5.1 CategoricalConstructions . . . . . . . . . . . . . . . . . . . . 213
5.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
5.3 AdjointFunctorTheoremforModules . . . . . . . . . . . . . 256
5.4 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.4.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.4.2 SheafConstructions . . . . . . . . . . . . . . . . . . . . . . . 294
5.5 AbelianCategories . . . . . . . . . . . . . . . . . . . . . . . 303
5.5.1 Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Chapter6 Homology
6.1 HomologyFunctors . . . . . . . . . . . . . . . . . . . . . . . 323
6.2 DerivedFunctors . . . . . . . . . . . . . . . . . . . . . . . . 340
6.2.1 LeftDerivedFunctors . . . . . . . . . . . . . . . . . . . . . . 343
6.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
6.2.3 CovariantRightDerivedFunctors . . . . . . . . . . . . . . . 364
6.2.4 ContravariantRightDerivedFunctors . . . . . . . . . . . . . 369
6.3 SheafCohomology . . . . . . . . . . . . . . . . . . . . . . . 377
6.3.1 CˇechCohomology . . . . . . . . . . . . . . . . . . . . . . . 384
6.3.2 Riemann–RochTheorem . . . . . . . . . . . . . . . . . . . . 392
Chapter7 TorandExt
7.1 Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
7.1.1 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
7.1.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
7.2 Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
7.2.1 BaerSum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
7.3 CotorsionGroups . . . . . . . . . . . . . . . . . . . . . . . . 438
7.4 UniversalCoefficients . . . . . . . . . . . . . . . . . . . . . . 448
Chapter8 HomologyandRings
8.1 DimensionsofRings . . . . . . . . . . . . . . . . . . . . . . 453
8.2 Hilbert’sSyzygyTheorem . . . . . . . . . . . . . . . . . . . 467
8.3 StablyFreeModules . . . . . . . . . . . . . . . . . . . . . . 476
8.4 CommutativeNoetherianLocalRings . . . . . . . . . . . . . 484
Contents ix
Chapter9 HomologyandGroups
9.1 GroupExtensions . . . . . . . . . . . . . . . . . . . . . . . . 495
9.1.1 SemidirectProducts . . . . . . . . . . . . . . . . . . . . . . . 500
9.1.2 GeneralExtensionsandCohomology. . . . . . . . . . . . . . 504
9.1.3 StabilizingAutomorphisms . . . . . . . . . . . . . . . . . . . 514
9.2 GroupCohomology . . . . . . . . . . . . . . . . . . . . . . . 519
9.3 BarResolutions . . . . . . . . . . . . . . . . . . . . . . . . . 525
9.4 GroupHomology . . . . . . . . . . . . . . . . . . . . . . . . 535
9.4.1 SchurMultiplier . . . . . . . . . . . . . . . . . . . . . . . . . 541
9.5 ChangeofGroups . . . . . . . . . . . . . . . . . . . . . . . . 559
9.5.1 RestrictionandInflation. . . . . . . . . . . . . . . . . . . . . 564
9.6 Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
9.7 TateGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
9.8 OuterAutomorphismsof p-Groups. . . . . . . . . . . . . . . 587
9.9 CohomologicalDimension . . . . . . . . . . . . . . . . . . . 591
9.10 DivisionRingsandBrauerGroups . . . . . . . . . . . . . . . 595
Chapter10 SpectralSequences
10.1 Bicomplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
10.2 FiltrationsandExactCouples . . . . . . . . . . . . . . . . . . 616
10.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
10.4 HomologyoftheTotalComplex . . . . . . . . . . . . . . . . 628
10.5 Cartan–EilenbergResolutions . . . . . . . . . . . . . . . . . 647
10.6 GrothendieckSpectralSequences . . . . . . . . . . . . . . . . 656
10.7 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660
10.8 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
10.9 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
10.10 Ku¨nnethTheorems . . . . . . . . . . . . . . . . . . . . . . . 678
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
SpecialNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
Preface to the Second Edition
HomologicalAlgebrahasgrowninthenearlythreedecadessincethefirstedi-
tionofthisbookappearedin1979. Twobooksdiscussingmorerecentresults
are Weibel, An Introduction to Homological Algebra, 1994, and Gelfand–
Manin, Methods of Homological Algebra, 2003. In their Foreword, Gelfand
andManindividethehistoryofHomologicalAlgebraintothreeperiods: the
first period ended in the early 1960s, culminating in applications of Homo-
logical Algebra to regular local rings. The second period, greatly influenced
bytheworkofA.GrothendieckandJ.-P.Serre,continuedthroughthe1980s;
it involves abelian categories and sheaf cohomology. The third period, in-
volvingderivedcategoriesandtriangulatedcategories, isstillongoing. Both
ofthesenewerbooksdiscussallthreeperiods(seealsoKashiwara–Schapira,
CategoriesandSheaves). Theoriginalversionofthisbookdiscussedthefirst
period only; this new edition remains at the same introductory level, but it
now introduces the second period as well. This change makes sense peda-
gogically, for there has been a change in the mathematics population since
1979; today, virtually all mathematics graduate students have learned some-
thing about functors and categories, and so I can now take the categorical
viewpointmoreseriously.
When I was a graduate student, Homological Algebra was an unpopular
subject. The general attitude was that it was a grotesque formalism, boring
to learn, and not very useful once one had learned it. Perhaps an algebraic
topologistwasforcedtoknowthisstuff, butsurelynooneelseshouldwaste
time on it. The few true believers were viewed as workers at the fringe of
mathematicswhokepttinkeringwiththeirelaboratemachine,smoothingout
roughpatcheshereandthere.
x
Prefaceto theSecond Edition xi
ThisattitudechangeddramaticallywhenJ.-P.Serrecharacterizedregular
localringsusingHomologicalAlgebra(theyarethecommutativenoetherian
local rings of “finite global dimension”), for this enabled him to prove that
any localization of a regular local ring is itself regular (until then, only spe-
cial cases of this were known). At the same time, M. Auslander and D. A.
Buchsbaum also characterized regular local rings, and they went on to com-
pleteworkofM.Nagatabyusingglobaldimensiontoprovethateveryregular
localringisauniquefactorizationdomain. AsGrothendieckandSerrerevolu-
tionizedAlgebraicGeometrybyintroducingschemesandsheaves,resistance
to Homological Algebra waned. Today, it is just another standard tool in a
mathematician’skit. Formoredetails,werecommendC.A.Weibel’schapter,
“HistoryofHomologicalAlgebra,”inthebookofJames,HistoryofTopology.
Homological Algebra presents a great pedagogical challenge for authors
and for readers. At first glance, its flood of elementary definitions (which
oftenoriginateinotherdisciplines)anditsspace-fillingdiagramsappearfor-
bidding. To counter this first impression, S. Lang set the following exercise
onpage105ofhisbook,Algebra:
Takeanybookonhomologicalalgebraandproveallthetheorems
withoutlookingattheproofsgiveninthatbook.
Taken literally, the statement of the exercise is absurd. But its spirit is ab-
solutely accurate; the subject only appears difficult. However, having rec-
ognized the elementary character of much of the early material, one is often
temptedto“waveone’shands”: topretendthatminutiaealwaysbehavewell.
It should come as no surprise that danger lurks in this attitude. For this rea-
son,Iincludemanydetailsinthebeginning,attheriskofboringsomereaders
by so doing (of course, such readers are free to turn the page). My intent is
twofold: to allow readers to see that complete proofs can, in fact, be written
compactly; to give readers the confidence to believe that they, too, can write
suchproofswhen,later,thelazyauthorasksthemto. However,wemustcau-
tion the reader; some “obvious” statements are not only false, they may not
evenmakesense. Forexample,if Risaringand Aand B areleft R-modules,
thenHom (A,B)maynotbean R-moduleatall;and,ifitisamodule,itis
R
sometimesaleftmoduleandsometimesarightmodule. Isanallegedfunction
withdomainatensorproductwell-defined? Isanisomorphismreallynatural?
Does a diagram really commute? After reading the first three chapters, the
readershouldbeabletodealwithsuchmattersefficiently.
This book is my attempt to make Homological Algebra lovable, and I
believethatthisrequiresthesubjectbepresentedinthecontextofothermath-
ematics. Forexample,Chapters2,3,and4formashortcourseinmodulethe-
ory,investigatingtherelationbetweenaringanditsprojective,injective,and
flatmodules. Makingthesubjectlovableismyreasonfordelayingtheformal
introduction of homology functors until Chapter 6 (although simplicial and
xii Prefaceto theSecond Edition
singular homology do appear in Chapter 1). Many readers wanting to learn
HomologicalAlgebraarefamiliarwiththefirstpropertiesofHomandtensor;
evenso,theyshouldglanceatthefirstchapters,fortheremaybesomeunfa-
miliaritemstherein. Somecategorytheoryappearsthroughout,butitmakesa
morebrazenappearanceinChapter5,wherewediscusslimits,adjointfunc-
tors,andsheaves. AlthoughpresheavesareintroducedinChapter1,wedonot
introducesheavesuntilwecanobservethattheyusuallyformanabeliancat-
egory. Chapter6constructshomologyfunctors,givingtheusualfundamental
resultsaboutlongexactsequences, naturalconnectinghomomorphisms, and
independence of choices of projective, injective, and flat resolutions used to
construct them. Applications of sheaves are most dramatic in the context of
SeveralComplexVariablesandinAlgebraicGeometry;alas,Isayonlyafew
words pointing the reader to appropriate texts, but there is a brief discussion
oftheRiemann–RochTheoremovercompactRiemannsurfaces. Chapters7,
8,and9considerthederivedfunctorsofHomandtensor,withapplicationsto
ringtheory(viaglobaldimension),cohomologyofgroups,anddivisionrings.
LearningHomologicalAlgebraisatwo-stageaffair. First,onemustlearn
the language of Ext and Tor and what it describes. Second, one must be
able to compute these things and, often, this involves yet another language,
thatofspectralsequences. Chapter10developsspectralsequencesviaexact
couples, always taking care that bicomplexes and their multiple indices are
visiblebecausealmostallapplicationsoccurinthismilieu.
Awordaboutnotation. Iamusuallyagainstspellingreform;ifeveryone
iscomfortablewithasymboloranabbreviation,whoamItosayotherwise?
However,Idouseanewsymboltodenotetheintegersmodmbecause,nowa-
days, two different symbols are used: Z/mZ and Z . My quarrel with the
m
firstsymbolisthatitistoocomplicatedtowritemanytimesinanargument;
myquarrelwiththesimplersecondsymbolisthatitisambiguous: when pis
aprime,thesymbolZ oftendenotesthe p-adicintegersandnottheintegers
p
mod p. SincecapitalIremindsusofintegersandsinceblackboardfontisin
commonuse,asinZ,Q,R,C,andF ,Idenotetheintegersmodm byI .
q m
It is a pleasure to thank again those who helped with the first edition.
I also thank the mathematicians who helped with this revision: Matthew
Ando, Michael Barr, Steven Bradlow, Kenneth S. Brown, Daniel Grayson,
Phillip Griffith, William Haboush, Aimo Hinkkanen, Ilya Kapovich, Randy
McCarthy, Igor Mineyev, Thomas A. Nevins, Keith Ramsay, Derek Robin-
son, and Lou van den Dries. I give special thanks to Mirroslav Yotov who
notonlymademanyvaluablesuggestionsimprovingtheentiretextbutwho,
having seen my original flawed subsection on the Riemann–Roch Theorem,
patientlyguidedmyrewritingofit.
JosephJ.Rotman
May2008
UrbanaIL