Table Of ContentLONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
http://www.cambridge.org/mathematics
312 Foundationsofcomputationalmathematics,Minneapolis2002, F.CUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles, S.MÜLLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs, D.CVETKOVIC,P.ROWLINSON&S.SIMIC
315 Structuredringspectra, A.BAKER&B.RICHTER(eds)
316 Linearlogicincomputerscience, T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds)
317 Advancesinellipticcurvecryptography, I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds)
318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations, D.HENRY
319 DoubleaffineHeckealgebras, I.CHEREDNIK
320 L-functionsandGaloisrepresentations, D.BURNS,K.BUZZARD&J.NEKOVÁRˇ(eds)
321 Surveysinmodernmathematics, V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory, F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations, S.GUTTetal(eds)
324 Singularitiesandcomputeralgebra, C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciflow, P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype, J.E.HUMPHREYS
327 Surveysincombinatorics2005, B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds, R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups, Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology, A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005, L.MPARDO,A.PINKUS,E.SÜLI&
M.J.TODD(eds)
332 Handbookoftiltingtheory, L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition), A.KOCK
334 TheNavier–Stokesequations, N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability, A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules, I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory, J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory, N.YOUNG(ed)
339 GroupsStAndrews2005I, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory, J.B.CONREY,D.W.FARMER,F.MEZZADRI&
N.C.SNAITH(eds)
342 Ellipticcohomology, H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI, J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII, J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry, A.NEEMAN
346 Surveysincombinatorics2007, A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics, N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis, P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI, Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY
&A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII, Z.CHATZIDAKIS,D.MACPHERSON,
A.PILLAY&A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas, A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials, J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis, J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis, K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence, J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations, D.DELBOURGO
357 Algebraictheoryofdifferentialequations, M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory, M.R.BRIDSON,P.H.KROPHOLLER&
I.J.LEARY(eds)
359 Modulispacesandvectorbundles, L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA&
S.RAMANAN(eds)
360 Zariskigeometries, B.ZILBER
361 Words:Notesonverbalwidthingroups, D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories, R.BAUTISTA,L.SALMERÓN&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008, F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics, J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009, S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems, B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena, G.BLOWER
368 GeometryofRiemannsurfaces, F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIÉ
370 Theoryofp-adicdistributions, S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals, F.PRZYTYCKI&M.URBANSKI
372 Moonshine:Thefirstquartercenturyandbeyond, J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularityandcompleteintersection, J.MAJADAS&A.G.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction, S.ALINHAC
375 Triangulatedcategories, T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns, S.LINTON,N.RUŠKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications, G.BERHUY
378 Probabilityandmathematicalgenetics, N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory, J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities, M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
381 Symmetriesandintegrabilityofdifferenceequations, D.LEVI,P.OLVER,Z.THOMOVA&
P.WINTERNITZ(eds)
382 Forcingwithrandomvariablesandproofcomplexity, J.KRAJÍCˇEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI, R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII, R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems, B.MARCUS,K.PETERSEN&
T.WEISSMAN(eds)
386 Independence-friendlylogic, A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI, C.M.CAMPBELLetal(eds)
388 GroupsStAndrews2009inBathII, C.M.CAMPBELLetal(eds)
389 Randomfieldsonthesphere, D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials, D.E.PELINOVSKY
391 Fusionsystemsinalgebraandtopology, M.ASCHBACHER,R.KESSAR&B.OLIVER
392 Surveysincombinatorics2011, R.CHAPMAN(ed)
393 Non-abelianfundamentalgroupsandIwasawatheory, J.COATESetal(eds)
394 Variationalproblemsindifferentialgeometry, R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds)
395 Howgroupsgrow, A.MANN
396 Arithmeticdifferentialoperatorsoverthep-adicintegers, C.C.RALPH&S.R.SIMANCA
397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology, J.BOLTE&F.STEINER(eds)
398 Mathematicalmodelsincontactmechanics, M.SOFONEA&A.MATEI
399 Circuitdoublecoverofgraphs, C.-Q.ZHANG
400 Densespherepackings:ablueprintforformalproofs, T.HALES
401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory, B.DENG,J.DU&Q.FU
402 Mathematicalaspectsoffluidmechanics, J.C.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds)
403 Foundationsofcomputationalmathematics,Budapest2011, F.CUCKER,T.KRICK,A.PINKUS&
A.SZANTO(eds)
404 Operatormethodsforboundaryvalueproblems, S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds)
405 Torsors,étalehomotopyandapplicationstorationalpoints, A.N.SKOROBOGATOV(ed)
406 Appalachiansettheory, J.CUMMINGS&E.SCHIMMERLING(eds)
407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups, J.N.BRAY,D.F.HOLT&
C.M.RONEY-DOUGAL
408 Complexityscience:theWarwickmaster’scourse, R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds)
409 Surveysincombinatorics2013, S.R.BLACKBURN,S.GERKE&M.WILDON(eds)
410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups,
T.CECCHERINI-SILBERSTEIN,F.SCARABOTTI&F.TOLLI
411 Modulispaces, L.BRAMBILA-PAZ,O.GARCÍA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds)
412 Automorphismsandequivalencerelationsintopologicaldynamics, D.B.ELLIS&R.ELLIS
413 Optimaltransportation, Y.OLLIVIER,H.PAJOT&C.VILLANI(eds)
414 AutomorphicformsandGaloisrepresentationsI, F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
415 AutomorphicformsandGaloisrepresentationsII, F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
416 Reversibilityindynamicsandgrouptheory, A.G.O’FARRELL&I.SHORT
417 Recentadvancesinalgebraicgeometry, C.D.HACON,M.MUSTAT¸A˘&M.POPA(eds)
418 TheBloch–KatoconjecturefortheRiemannzetafunction, J.COATES,A.RAGHURAM,A.SAIKIA&
R.SUJATHA(eds)
419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations, J.C.MEYER&
D.J.NEEDHAM
420 Arithmeticandgeometry, L.DIEULEFAITetal(eds)
421 O-minimalityandDiophantinegeometry, G.O.JONES&A.J.WILKIE(eds)
422 GroupsStAndrews2013, C.M.CAMPBELLetal(eds)
423 Inequalitiesforgrapheigenvalues, Z.STANIC´
424 Surveysincombinatorics2015, A.CZUMAJetal(eds)
425 Geometry,topologyanddynamicsinnegativecurvature, C.S.ARAVINDA,F.T.FARRELL&
J.-F.LAFONT(eds)
426 Lecturesonthetheoryofwaterwaves, T.BRIDGES,M.GROVES&D.NICHOLLS(eds)
427 RecentadvancesinHodgetheory, M.KERR&G.PEARLSTEIN(eds)
428 GeometryinaFréchetcontext, C.T.J.DODSON,G.GALANIS&E.VASSILIOU
429 Sheavesandfunctionsmodulop, L.TAELMAN
430 RecentprogressinthetheoryoftheEulerandNavier-Stokesequations, J.C.ROBINSON,J.L.RODRIGO,
W.SADOWSKI&A.VIDAL-LÓPEZ(eds)
431 Harmonicandsubharmonicfunctiontheoryontherealhyperbolicball, M.STOLL
432 Topicsingraphautomorphismsandreconstruction(2ndEdition), J.LAURI&RSCAPELLATO
433 RegularandirregularholonomicD-modules, M.KASHIWARA&P.SCHAPIRA
434 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems(2ndEdition), K.TAIRA
435 GradedringsandgradedGrothendieckgroups, R.HAZRAT
LondonMathematicalSocietyLectureNoteSeries:435
Graded Rings and Graded
Grothendieck Groups
ROOZBEH HAZRAT
WesternSydneyUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316619582
©RoozbehHazrat2016
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Firstpublished2016
PrintedintheUnitedKingdombyClays,StIvesplc
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LibraryofCongressCataloguinginPublicationdata
Names:Hazrat,Roozbeh,1971–
Title:GradedringsandgradedGrothendieckgroups/RoozbehHazrat,
UniversityofWesternSydney.
Description:Cambridge:CambridgeUniversityPress,2016.|Series:London
MathematicalSocietylecturenoteseries;435|Includesbibliographical
referencesandindex.
Identifiers:LCCN2016010216|ISBN9781316619582(pbk.:alk.paper)
Subjects:LCSH:Grothendieckgroups.|Gradedrings.|Rings(Algebra)
Classification:LCCQA251.5.H392016|DDC512/.4–dc23LCrecordavailable
athttp://lccn.loc.gov/2016010216
ISBN978-1-316-61958-2Paperback
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accurateorappropriate.
Contents
Introduction page1
1 Gradedringsandgradedmodules 5
1.1 Gradedrings 6
1.1.1 Basicdefinitionsandexamples 6
1.1.2 Partitioninggradedrings 10
1.1.3 Stronglygradedrings 14
1.1.4 Crossedproducts 16
1.1.5 Gradedideals 20
1.1.6 Gradedprimeandmaximalideals 22
1.1.7 Gradedsimplerings 23
1.1.8 Gradedlocalrings 25
1.1.9 GradedvonNeumannregularrings 26
1.2 Gradedmodules 28
1.2.1 Basicdefinitions 28
1.2.2 Shiftofmodules 28
1.2.3 TheHomgroupsandcategoryofgradedmodules 30
1.2.4 Gradedfreemodules 33
1.2.5 Gradedbimodules 34
1.2.6 Tensorproductofgradedmodules 35
1.2.7 Forgettingthegrading 36
1.2.8 Partitioninggradedmodules 37
1.2.9 Gradedprojectivemodules 41
1.2.10 Gradeddivisiblemodules 47
1.3 Gradingonmatrices 49
1.3.1 Gradedcalculusonmatrices 50
1.3.2 Homogeneousidempotentscalculus 58
1.3.3 Gradedmatrixunits 59
v
vi Contents
1.3.4 Mixedshift 60
1.4 Gradeddivisionrings 64
1.4.1 Thezerocomponentringofgradedsimplering 71
1.5 StronglygradedringsandDade’stheorem 72
1.5.1 Invertiblecomponentsofstronglygradedrings 78
1.6 Gradingongraphalgebras 79
1.6.1 Gradingonfreerings 79
1.6.2 CornerskewLaurentpolynomialrings 81
1.6.3 Graphs 84
1.6.4 Leavittpathalgebras 85
1.7 ThegradedIBNandgradedtype 94
1.8 Thegradedstablerank 96
1.9 Gradedringswithinvolution 100
2 GradedMoritatheory 104
2.1 FirstinstanceofthegradedMoritaequivalence 105
2.2 Gradedgenerators 109
2.3 GeneralgradedMoritaequivalence 111
3 GradedGrothendieckgroups 122
3.1 ThegradedGrothendieckgroupKgr 124
0
3.1.1 Groupcompletions 124
3.1.2 Kgr-groups 126
0
3.1.3 Kgr ofstronglygradedrings 128
0
(cid:2)
3.1.4 ThereducedgradedGrothendieckgroupKgr 130
0
3.1.5 Kgr asaZ[Γ]-algebra 132
0
3.2 Kgr fromidempotents 132
0
3.2.1 Stabilityofidempotents 137
3.2.2 ActionofΓonidempotents 137
3.2.3 Kgr isacontinuousfunctor 138
0
3.2.4 TheHattori–Stallings(Chern)tracemap 139
3.3 Kgr ofgraded∗-rings 141
0
3.4 RelativeKgr-groups 142
0
3.5 Kgr ofnonunitalrings 145
0
3.5.1 Gradedinnerautomorphims 148
3.6 Kgr isapre-orderedmodule 149
0
3.6.1 Γ-pre-orderedmodules 149
3.7 Kgr ofgradeddivisionrings 152
0
3.8 Kgr ofgradedlocalrings 158
0
3.9 Kgr ofLeavittpathalgebras 160
0
3.9.1 K ofLeavittpathalgebras 161
0
Contents vii
3.9.2 ActionofZonKgr ofLeavittpathalgebras 163
0
3.9.3 Kgr of a Leavitt path algebra via its 0-
0
componentring 164
3.10 Ggr ofgradedrings 168
0
3.10.1 Ggr ofgradedArtinianrings 169
0
3.11 SymbolicdynamicsandKgr 172
0
3.12 Kgr-theory 177
1
4 GradedPicardgroups 180
4.1 Picgr ofagradedcommutativering 181
4.2 Picgr ofagradednoncommutativering 184
5 Gradedultramatricialalgebras,classificationviaKgr 192
0
5.1 Gradedmatricialalgebras 193
5.2 Gradedultramatricialalgebras,classificationviaKgr 198
0
6 Gradedversusnongraded(higher)K-theory 205
6.1 K∗gr ofpositivelygradedrings 206
6.2 ThefundamentaltheoremofK-theory 214
6.2.1 Quillen’sK-theoryofexactcategories 214
6.2.2 Basechangeandtransferfunctors 215
6.2.3 Alocalisationexactsequenceforgradedrings 216
6.2.4 Thefundamentaltheorem 217
6.3 RelatingK∗gr(A)toK∗(A0) 222
6.4 RelatingK∗gr(A)toK∗(A) 223
References 227
Index 232
Introduction
A bird’s eye view of the theory of graded modules over a graded ring might
give the impression that it is nothing but ordinary module theory with all its
statementsdecoratedwiththeadjective“graded”.Oncethegradingisconsid-
eredtobetrivial,thegradedtheoryreducestotheusualmoduletheory.From
thisperspective,thetheoryofgradedmodulescanbeconsideredasanexten-
sion of module theory. However, this simplistic overview might conceal the
pointthatgradedmodulescomeequippedwithashift,thankstothepossibility
ofpartitioningthestructuresandthenrearrangingthepartitions.Thisaddsan
extralayerofstructure(andcomplexity)tothetheory.Thismonographfocuses
onthetheoryofthegradedGrothendieckgroupKgr,thatprovidesasparkling
0
illustration of this idea. Whereas the usual K is an abelian group, the shift
0
provides Kgr with anatural structure ofa Z[Γ]-module, whereΓ is thegroup
0
used for the grading and Z[Γ] its group ring. As we will see throughout this
book,thisextrastructurecarriessubstantialinformationaboutthegradedring.
Let Γ and Δ be abelian groups and f : Γ → Δ a group homomorphism.
Then for any Γ-graded ring A, one can consider a natural Δ-grading on A
(see §1.1.2); in the same way, any Γ-graded A-module can be viewed as a Δ-
gradedA-module.Theseoperationsinducefunctors
U :GrΓ-A−→GrΔ-A,
f
(−)Ω :GrΓ-A−→GrΩ-AΩ,
(see§1.2.8),whereGrΓ-AisthecategoryofΓ-gradedrightA-modules,GrΔ-A
that of Δ-graded right A-modules, and GrΩ-A the category of Ω-graded right
AΩ-modulewithΩ=ker(f).
One aim of the theory of graded rings is to investigate the ways in which
these categories relate to one another, and which properties of one category
canbeliftedtoanother.Inparticular,inthetwoextremecaseswhenthegroup
1
