This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

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Hazrat, Roozbeh

RCOG Press

London Mathematical Society lecture note series, Cambridge, United Kingdom, 2016

Cambridge University Press, Cambridge, United Kingdom, 2016

United Kingdom and Ireland, United Kingdom

0

lg2164103

producers:
Acrobat Pro DC 17.12.20098

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Series page 2
Title page 4
Copyright 5
Table of contents 6
Introduction 10
1 Graded rings and graded modules 14
1.1 Graded rings 15
1.1.1 Basic definitions and examples 15
1.1.2 Partitioning graded rings 19
1.1.3 Strongly graded rings 23
1.1.4 Crossed products 25
1.1.5 Graded ideals 29
1.1.6 Graded prime and maximal ideals 31
1.1.7 Graded simple rings 32
1.1.8 Graded local rings 34
1.1.9 Graded von Neumann regular rings 35
1.2 Graded modules 37
1.2.1 Basic definitions 37
1.2.2 Shift of modules 37
1.2.3 The Hom groups and the category of graded modules 39
1.2.4 Graded free modules 42
1.2.5 Graded bimodules 43
1.2.6 Tensor product of graded modules 44
1.2.7 Forgetting the grading 45
1.2.8 Partitioning graded modules 46
1.2.9 Graded projective modules 50
1.2.10 Graded divisible modules 56
1.3 Grading on matrices 58
1.3.1 Graded calculus on matrices 59
1.3.2 Homogeneous idempotents calculus 67
1.3.3 Graded matrix units 68
1.3.4 Mixed shift 69
1.4 Graded division rings 73
1.4.1 The zero component ring of a graded central simple ring 80
1.5 Strongly graded rings and Dade’s theorem 81
1.5.1 Invertible components of strongly graded rings 87
1.6 Grading on graph algebras 88
1.6.1 Grading on free rings 88
1.6.2 Corner skew Laurent polynomial rings 90
1.6.3 Graphs 93
1.6.4 Leavitt path algebras 94
1.7 The graded IBN and graded type 103
1.8 The graded stable rank 105
1.9 Graded rings with involution 109
2 Graded Morita theory 113
2.1 First instance of the graded Morita equivalence 114
2.2 Graded generators 118
2.3 General graded Morita equivalence 120
3 Graded Grothendieck groups 131
3.1 The graded Grothendieck group Kgr0 133
3.1.1 Group completions 133
3.1.2 Kgr0-groups 135
3.1.3 Kgr0 of strongly graded rings 137
3.1.4 The reduced graded Grothendieck group Kgr0 139
3.1.5 Kgr0 as a Z[Γ]-algebra 141
3.2 Kgr0 from idempotents 141
3.2.1 Stability of idempotents 146
3.2.2 Action of Γ on idempotents 146
3.2.3 Kgr0 is a continuous functor 147
3.2.4 The Hattori–Stallings (Chern) trace map 148
3.3 Kgr0 of graded ∗-rings 150
3.4 Relative Kgr0-groups 151
3.5 Kgr0 of nonunital rings 154
3.5.1 Graded inner automorphims 157
3.6 Kgr0 is a pre-ordered module 158
3.6.1 Γ-pre-ordered modules 158
3.7 Kgr0 of graded division rings 161
3.8 Kgr0 of graded local rings 167
3.9 Kgr0 of Leavitt path algebras 169
3.9.1 K0 of Leavitt path algebras 170
3.9.2 Action of Z on Kgr0 of Leavitt path algebras 172
3.9.3 Kgr0 of a Leavitt path algebra via its 0-component ring 173
3.10 Ggr0 of graded rings 177
3.10.1 Ggr0 of graded Artinian rings 178
3.11 Symbolic dynamics and Kgr0 181
3.12 Kgr1-theory 186
4 Graded Picard groups 189
4.1 Picgr of a graded commutative ring 190
4.2 Picgr of a graded noncommutative ring 193
5 Graded ultramatricial algebras classification via Kgr0 201
5.1 Graded matricial algebras 202
5.2 Graded ultramatricial algebras, classification via Kgr0 207
6 Graded versus nongraded (higher) K-theory 214
6.1 Kgr∗ of positively graded rings 215
6.2 The fundamental theorem of K-theory 223
6.2.1 Quillen’s K-theory of exact categories 223
6.2.2 Base change and transfer functors 224
6.2.3 A localisation exact sequence for graded rings 225
6.2.4 The fundamental theorem 226
6.3 Relating Kgr∗ (A) to K∗(A0) 231
6.4 Relating Kgr∗ (A) to K∗(A) 232
References 236
Index 241

Graded Rings And Graded Modules -- Graded Morita Theory -- Graded Grothendieck Groups -- Graded Picard Groups -- Graded Ultramatricial Algebras, Classification Via K[superscript Gr]0 -- Graded Versus Nongraded (higher) K-theory. Roozbeh Hazrat, Western Sydney University. Includes Bibliographical References And Index.

This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra, and brings together results scattered across the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

2017-12-27