Graded Rings and Graded Grothendieck Groups

244 Pages·2016·2.806 MB·English

by  Roozbeh Hazrat

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
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LondonMathematicalSocietyLectureNoteSeries:435
Graded Rings and Graded
Grothendieck Groups
ROOZBEH HAZRAT
WesternSydneyUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316619582
©RoozbehHazrat2016
Thispublicationisincopyright.Subjecttostatutoryexception
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permissionofCambridgeUniversityPress.
Firstpublished2016
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
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
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyInternetWebsitesreferredtointhispublication,
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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