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Full Description
This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of etale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups.
The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.
Contents
Background material on rings and modules
Modules over commutative rings
The Wedderburn-Artin theorem
Separable algebras, definition and first properties
Background material on homological algebra
The divisor class group
Azumaya algebras, I
Derivations, differentials and separability
Etale algebras
Henselization and splitting rings
Azumaya algebras, II
Galois extensions of commutative rings
Crossed products and Galois cohomology
Further topics
Acronyms
Glossary of notation
Bibliography
Index.
