Brauer Groups, Hopf Algebras and Galois Theory

ISBN-13: 9781402003462 / Angielski / Miękka / 2002 / 488 str.

S. Caenepeel; Stefaan Caenepeel

Opis

This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and etale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included. The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph. Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use.

Spis treści

I: The Brauer Group of a Commutative Ring. 1. Morita Theory for Algebras without a Unit. 2. Azumaya Algebras and Taylor-Azumaya Algebras. 3. The Brauer Group. 4. Central Separable Algebras. 5. Amitsur Cohomology and étale Cohomology. 6. Cohomological Interpretation of the Brauer Group. II: Hopf Algebras and Galois Theory. 7. Hopf Algebras. 8. Galois Objects. 9. Cohomology over Hopf Algebras. 10. The Group of Galois (co)Objects. 11. Some Examples. III: The Brauer-Long Group of a Commutative Ring. 12. H-Azumaya Algebras. 13. The Brauer-Long Group of a Commutative Ring. 14. The Brauer Group of Yetter- Drinfel'd Module Algebras. A: Abelian Categories and Homological Algebra. B: Faithfully Flat Descent. C: Elementary Algebraic K-Theory. Bibliography. Index.