brauer groups, tamagawa measures, and rational points

CONTENTS
Table of contents . . ....................................................
iii
Notation and fundamental conventions . . .............................. vii
Notation and conventions . . ............................................ vii
References and Citation . . .............................................. ix
Introduction . . ..........................................................
xi
Part A. The Brauer group . . ............................................ 1
I. The Brauer-Severi variety associated with a central simple algebra . . 3
1. Introductory remarks . . .............................................. 3
2. Non-abelian group cohomology . . ................. . . ............... 5
3. Galois descent . . . ......................... .......................... 8
4. Central simple algebras and non-abelian H 1 . . . . . . . . . . . . . . . . . . . . . . . . 22
5. Brauer-Severi varieties and non-abelian H 1 . . ........................ 28
6. Central simple algebras and Brauer-Severi varieties . . . ............... 35
7. Functoriality . . ........................... ........................... 39
8. The functor of points . . .............................................. 53
II. On the Brauer group of a scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1. Azumaya algebras . . ........................ ........................ 61
2. The Brauer group . . .................................................. 65
3. The cohomological Brauer group . . .................................. 66
4. The relation to the Brauer group of the function field . . . . . . . . . . . . . . 71
5. The Brauer group and the cohomological Brauer group . . . . . . . . . . . . 74
6. Orders in central simple algebras . . ................. ................. 76
7. The Theorem of Auslander and Goldman . . . . . . . . . . . . . . . . . ......... 84