The Separable Galois Theory of Commutative Rings  book cover
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2nd Edition

The Separable Galois Theory of Commutative Rings




ISBN 9781482208054
Published July 14, 2014 by Chapman and Hall/CRC
184 Pages

 
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Book Description

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

Separability
Separable fields
Separable rings
Separable schemes
Separable polynomials
Module projective algebras

Idempotents and Profinite Spaces
Boolean algebras and idempotents
Profinite spaces
Covering spaces
Profinite group actions
Rings of functions

The Boolean Spectrum
Pierce’s representation
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
Separability idempotents
Infinite and locally weakly Galois extensions
Galois correspondence

Separable Closure and the Fundamental Groupoid
Componential strong separability
Separable closure
Correspondence for separably closed
Categorical correspondence

Categorical Galois Theory and the Galois Correspondence
Subobjects, equivalence relations, and quotients
Splitting extensions and categorical correspondences

Index

Bibliographic notes appear at the end of each chapter.

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Reviews

"This book provides a complete and self-contained account of the Galois theory of commutative rings …"
—Nikolay I. Kryuchkov, Zentralblatt MATH 1298