Real and Étale Cohomology

Claus Scheiderer (auth.)3540584366, 9783540584360, 0387584366

This book makes a systematic study of the relations between the étale cohomology of a scheme and the orderings of its residue fields. A major result is that in high degrees, étale cohomology is cohomology of the real spectrum. It also contains new contributions in group cohomology and in topos theory. It is of interest to graduate students and researchers who work in algebraic geometry (not only real) and have some familiarity with the basics of étale cohomology and Grothendieck sites. Independently, it is of interest to people working in the cohomology theory of groups or in topos theory.

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Table of contents :
Real spectrum and real étale site….Pages 1-8
Glueing étale and real étale site….Pages 9-17
Limit theorems, stalks, and other basic facts….Pages 18-29
Some reminders on Weil restrictions….Pages 30-41
Real spectrum of X and étale site of $$X[sqrt { – 1} ]$$ ….Pages 42-55
The fundamental long exact sequence….Pages 56-67
Cohomological dimension of X b , I: Reduction to the field case….Pages 68-86
Equivariant sheaves for actions of topological groups….Pages 87-95
Cohomological dimension of X b , II: The field case….Pages 96-106
G -toposes….Pages 107-127
Inverse limits of G -toposes: Two examples….Pages 128-160
Group actions on spaces: Topological versus topos-theoretic constructions….Pages 161-165
Quotient topos of a G -topos, for G of prime order….Pages 166-172
Comparison theorems….Pages 173-179
Base change theorems….Pages 180-190
Constructible sheaves and finiteness theorems….Pages 191-204
Cohomology of affine varieties….Pages 205-211
Relations to the Zariski topology….Pages 212-218
Examples and complements….Pages 219-243