Brauer Groups and the Cohomology of Graded Rings 9781000103786, 0824779789, 1000103781
This book introduces various notions defined in graded terms extending the notions most frequently used as basic ingredi
202 78 35MB
English Pages 280 Year 2020
Table of contents :
Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 8
Copyright Page......Page 9
Preface......Page 10
Contents......Page 16
I.1 Graded Ring Theory......Page 20
I.2 Generalized Crossed Products......Page 22
II.1 Arithmetically Graded Rings......Page 38
Il.2 Separability and Graded Galois Extensions......Page 53
II.3 Graded Completion and Henselization......Page 65
II.4 The Join of gr-Henselian Rings......Page 80
III.1 Graded Faithfully Flat Descent......Page 98
III.2 Projective Graded Modules......Page 102
III.3 Grothendieck and Picard Groups of Graded Rings......Page 109
III.4 Brauer Groups of Graded Rings......Page 124
III.5 Graded Cohomology Groups and the Crossed Product Theorem......Page 138
IV.1 Brauer Groups of Graded Fields......Page 152
IV.2 Brauer Groups of gr-Local Rings......Page 160
IV.3 The Brauer Group of a Graded Ring Modulo a Graded Ideal......Page 168
IV.4 Brauer Groups of Regular Graded Rings......Page 170
V.1 Cohomology on the gr-Etale Site......Page 176
V.2 Hypercoverings and Verdier's Refinement Theorem......Page 185
V.3 Application to the Graded Brauer Group......Page 191
V.4 A Graded Version of Gabber's Theorem......Page 203
V.5 The Villamayor-Zelinsky Approach......Page 209
VI.1 The Brauer-Long Group......Page 220
VI.2 The Brauer-Wall Group......Page 247
VI.3 Graded Brauer Groups in a Geometrical Context......Page 254
References......Page 270
Index......Page 278
Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 8
Copyright Page......Page 9
Preface......Page 10
Contents......Page 16
I.1 Graded Ring Theory......Page 20
I.2 Generalized Crossed Products......Page 22
II.1 Arithmetically Graded Rings......Page 38
Il.2 Separability and Graded Galois Extensions......Page 53
II.3 Graded Completion and Henselization......Page 65
II.4 The Join of gr-Henselian Rings......Page 80
III.1 Graded Faithfully Flat Descent......Page 98
III.2 Projective Graded Modules......Page 102
III.3 Grothendieck and Picard Groups of Graded Rings......Page 109
III.4 Brauer Groups of Graded Rings......Page 124
III.5 Graded Cohomology Groups and the Crossed Product Theorem......Page 138
IV.1 Brauer Groups of Graded Fields......Page 152
IV.2 Brauer Groups of gr-Local Rings......Page 160
IV.3 The Brauer Group of a Graded Ring Modulo a Graded Ideal......Page 168
IV.4 Brauer Groups of Regular Graded Rings......Page 170
V.1 Cohomology on the gr-Etale Site......Page 176
V.2 Hypercoverings and Verdier's Refinement Theorem......Page 185
V.3 Application to the Graded Brauer Group......Page 191
V.4 A Graded Version of Gabber's Theorem......Page 203
V.5 The Villamayor-Zelinsky Approach......Page 209
VI.1 The Brauer-Long Group......Page 220
VI.2 The Brauer-Wall Group......Page 247
VI.3 Graded Brauer Groups in a Geometrical Context......Page 254
References......Page 270
Index......Page 278
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