Brauer Groups and the Cohomology of Graded Rings (Chapman & Hall/CRC Pure and Applied Mathematics Book 121)

This book introduces various notions defined in graded terms extending the notions most frequently used as basic ingredients in the theory of Azumaya algebras: separability and Galois extensions of commutative rings, crossed products and Galois cohomology, Picard groups, and the Brauer group. 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 Stefaan Caenepeel, Freddy Van Oystaeyen. Includes Index. Bibliography: P. 251-258.