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EQUIVALENCE AND DUALITY FOR
MODULE CATEGORIES
(withTiltingandCotiltingforRings)
This book provides a unified approach to many of the theories of equiva-
lenceanddualitybetweencategoriesofmodulesthathavetranspiredoverthe
last 45 years. In particular, during the past dozen or so years many authors
(includingtheauthorsofthisbook)haveinvestigatedrelationshipsbetween
categoriesofmodulesoverapairofringsthatareinducedbybothcovariant
andcontravariantrepresentablefunctors,inparticularbytiltingandcotilting
theories.
Byherecollectingandunifyingthebasicresultsoftheseinvestigationswith
innovativeandeasilyunderstandableproofs,theauthors’aimistoprovidean
aidtofurtherresearchinthiscentraltopicinabstractalgebraandareference
forallwhoseresearchliesinthisfield.
RobertR.ColbyisProfessorEmeritusattheUniversityofHawaiiandInde-
pendentScholarattheUniversityofIowa.
KentR.FullerisaprofessorofmathematicsattheUniversityofIowa.
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CAMBRIDGETRACTSINMATHEMATICS
GeneralEditors
B.BOLLOBA`S,W.FULTON,A.KATOK,F.KIRWAN,
P.SARNAK,B.SIMON
161 Equivalence and Duality for
Module Categories
(with Tilting and Cotilting for Rings)
ii
CB678-BOOK-DVR CB678-Colby January23,2004 19:4 CharCount=0
EQUIVALENCE AND DUALITY
FOR MODULE CATEGORIES
(with Tilting and Cotilting for Rings)
ROBERT R. COLBY
UniversityofHawaiiandUniversityofIowa
KENT R. FULLER
UniversityofIowa
iii
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521838214
© Robert R. Colby and Kent R. Fuller 2004
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format
isbn-13 978-0-511-18531-1 eBook (NetLibrary)
isbn-10 0-511-18531-6
eBook (NetLibrary)
isbn-13 978-0-521-83821-4 hardback
isbn-10 0-521-83821-5 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
CB678-BOOK-DVR CB678-Colby January23,2004 19:4 CharCount=0
Contents
Preface page vii
Acknowledgment ix
1 Some Module Theoretic Observations 1
1.1 TheKernelofExt1(V, )1
R
1.2 Gen(V)andFiniteness 2
1.3 Add(V ) and Prod(V ) 8
R R
1.4 TorsionTheory 9
2 RepresentableEquivalences 12
2.1 AdjointnessofHom (V, )and ⊗ V 12
R S
2.2 Weak∗-modules 14
2.3 ∗-modules 18
2.4 ThreeSpecialKindsof∗-modules 22
3 TiltingModules 28
3.1 GeneralizedTiltingModules 28
3.2 TiltingModules 31
3.3 TiltingTorsionTheories 34
3.4 PartialTiltingModules 36
3.5 TheTiltingTheorem 40
3.6 GlobalDimensionandSplitting 45
3.7 GrothendieckGroups 49
3.8 TorsionTheoryCounterEquivalence 57
4 RepresentableDualities 66
4.1 TheU-dual 67
4.2 CostarModules 69
4.3 Quasi-DualityModules 75
4.4 MoritaDuality 84
5 Cotilting 86
5.1 CotiltingTheorem 86
5.2 CotiltingModules 91
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vi Contents
5.3 CotiltingBimodules 97
5.4 CotiltingviaTiltingandMoritaDuality 103
5.5 WeakMoritaDuality 108
5.6 FinitisticCotiltingModulesandBimodules 115
5.7 U-torsionlessLinearCompactness 119
5.8 ExamplesandQuestions 126
A AdjointsandCategoryEquivalence 131
A.1 TheYoneda–GrothendieckLemma 131
A.2 AdjointCovariantFunctors 132
A.3 EquivalenceofCategories 135
B NoetherianSerialRings 139
B.1 FinitelyGeneratedModules 139
B.2 InjectiveModules 143
Bibliography 147
Index 151
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Preface
Approximatelyforty-fiveyearsagoK.Moritapresentedthefirstmajorresults
on equivalences and dualities between categories of modules over a pair of
rings. These results, which characterized an equivalence between the entire
categories of right (or left) modules over two rings as being represented by
thecovariantHomandtensorfunctorsinducedbyabalancedbimodulethat
isaso-calledprogeneratoroneitherside,andwhichcharacterizedaduality
between reasonably large subcategories of right and left modules over two
rings as being represented by the contravariant Hom functors induced by a
balancedbimodulethatisaninjectivecogeneratoronbothsides,havecome
tobeknownastheMoritatheorems.
Morita’s theorems on equivalence are exemplified by the equivalence of
thecategoriesofrightmodulesoverasimpleartinianringandtherightvector
spacesoveritsunderlyingdivisionring.Morethanadozenyearslaterthesec-
ondauthor,expandingontherelationshipbetweenthecategoryofmodules
generatedbyasimplemoduleandthevectorspacesoveritsendomorphism
ring,introducedtheconceptofaquasi-progeneratortocharacterizetheequiv-
alencesbetweenasubcategoryofrightmodulesoveroneringthatisclosed
undersubmodules,epimorphicimages,anddirectsumsandthecategoryof
allrightmodulesoveranotherring.Intheinterim,employingthenotionof
linearcompactness,B.J.Mu¨llerhadcharacterizedthereflexivemodulesun-
der Morita duality and given a one-sided characterization of the bimodules
inducingthesedualities.Theseresultsledtoseveralinvestigationsofthedual
notionofaquasi-progeneratorthatwerefertoasaquasi-dualitymodule.
Intheearly1980sthenotionoftiltingmodulesandthetiltingtheoremfor
finitelygeneratedmodulesoverartinalgebraswasintroducedandpolishedin
papersbyS.BrennerandM.C.R.Butler,D.HappelandC.M.Ringel,and
K.Bongartztoprovidenewinsightandexamplesintherepresentationtheory
ofartinalgebras.Thishasprovedparticularlyeffectiveinthestudyofalgebras
ofvariousrepresentationtypes.Thedefinitionofatiltingmodulealsoapplies
tomodulesoverarbitraryringsand,inthissetting,inducestorsiontheoriesin
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viii Preface
thecategoriesofmodulesovertworingsandapairofequivalencesbetween
thetorsionandtorsion-freepartsofthetorsiontheories.Thesetiltingmodules
andequivalencesareamongtheprincipaltopicsofthisbook.
Neartheendofthe1980sC.MeniniandA.Orsattiintroducedatypeof
module, which has come to be known as a ∗-module, that generalizes both
quasi-progeneratorsandtiltingmodulesbyinducinganequivalencebetween
asubcategoryofmodulesoveroneringthatisclosedunderdirectsumsand
epimorphicimagesandasubcategoryofmodulesoverasecondringthatis
closedunderdirectproductsandsubmodules.
At about the same time, the first author introduced a generalization of
Morita duality and the notions of cotilting modules and a cotilting theorem
fornoetherianrings.Thisinspiredseveralversionsofcotiltingmodulesand
cotiltingtheoremsthatarefurtherprincipaltopicsofthisbook.
Just as progenerators, quasi-progenerators, and tilting modules are all
∗-modules, so are Morita duality modules, quasi-duality modules, and the
variousflavorsofcotiltingmodules,alldualversionsof∗-modules,akindof
modulethatwecallcostarmodules.Asissooftenthecaseinmathematics,
generalizationsofaconceptledtoabetterunderstandingoftheconcept.Here
weprovideunifiedproofsregardingthevarioustypesofequivalenceanddu-
ality,andthemodulesthatinducethem,byapproachingthemviathegeneral
notionsof∗-modulesandcostarmodules.Wefeelthatthisapproachshould
yieldimprovedaccessibilityto,andbetterunderstandingof,theseconcepts.
En route we present much of the relatively little that is known about how
propertiesof(themodulesover)oneoftheringsinquestionaretransformed
totheotherringunderthesevariousequivalencesanddualities.Wehopethat
thisexpositionwillinspirefurtherresearchinthisandrelateddirections.
This book contains some of the work of many authors that has inspired
and, to a small extent been inspired by, our own research. We have made
an attempt to give credit where it is due, but surely we have inadvertently
omitted references to some works that should have been mentioned. To the
authors of these works, we offer our sincere apologies, and note that they
haveundoubtedlybeenreferredtoinoneormoreofthepapersand/orbooks
inourbibliography.
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Acknowledgment
ColbywishestoexpresshisgratitudeforthehospitalityoftheUniversityof
Iowa,whereheisanindependentscholar.
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