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

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 Half Title Series Page Title Page Copyright Page Preface Contents I. Generalized Crossed Products I.1 Graded Ring Theory I.2 Generalized Crossed Products II. Some Results On Commutative Rings Graded II.1 Arithmetically Graded Rings Il.2 Separability and Graded Galois Extensions II.3 Graded Completion and Henselization II.4 The Join of gr-Henselian Rings III. Graded Brauer Groups And The Crossedproduct Theorems III.1 Graded Faithfully Flat Descent III.2 Projective Graded Modules III.3 Grothendieck and Picard Groups of Graded Rings III.4 Brauer Groups of Graded Rings III.5 Graded Cohomology Groups and the Crossed Product Theorem IV. Application To Some Special Cases IV.1 Brauer Groups of Graded Fields IV.2 Brauer Groups of gr-Local Rings IV.3 The Brauer Group of a Graded Ring Modulo a Graded Ideal IV.4 Brauer Groups of Regular Graded Rings V. Etale Cohomology For Graded Rings V.1 Cohomology on the gr-Etale Site V.2 Hypercoverings and Verdier's Refinement Theorem V.3 Application to the Graded Brauer Group V.4 A Graded Version of Gabber's Theorem V.5 The Villamayor-Zelinsky Approach VI. Application VI.1 The Brauer-Long Group VI.2 The Brauer-Wall Group VI.3 Graded Brauer Groups in a Geometrical Context References Index Stefaan Caenepeel, Freddy Van Oystaeyen. Includes Index. Bibliography: P. 251-258.