The Separable Galois Theory of Commutative Rings, Second Edition provides a complete and self-contained account of the Galois theory of commutative rings from the viewpoint of categorical classification theorems and using solely the techniques of commutative algebra. Along with updating nearly every result and explanation, this edition contains a new chapter on the theory of separable algebras.
The book develops the notion of commutative separable algebra over a given commutative ring and explains how to construct an equivalent category of profinite spaces on which a profinite groupoid acts. It explores how the connection between the categories depends on the construction of a suitable separable closure of the given ring, which in turn depends on certain notions in profinite topology. The book also discusses how to handle rings with infinitely many idempotents using profinite topological spaces and other methods.
Table of Contents
Module projective algebras
Idempotents and Profinite Spaces
Boolean algebras and idempotents
Profinite group actions
Rings of functions
The Boolean Spectrum
Topology of the Boolean spectrum
The sheaf on the Boolean spectrum
Boolean spectra and rings of functions
Galois Theory over a Connected Base
Separable, strongly separable, locally strongly separable
Separably closed and separable closure
Infinite and locally weakly Galois extensions
Separable Closure and the Fundamental Groupoid
Componential strong separability
Correspondence for separably closed
Categorical Galois Theory and the Galois Correspondence
Subobjects, equivalence relations, and quotients
Splitting extensions and categorical correspondences
Bibliographic notes appear at the end of each chapter.
"This book provides a complete and self-contained account of the Galois theory of commutative rings …"
—Nikolay I. Kryuchkov, Zentralblatt MATH 1298