The Separable Galois Theory of Commutative Rings: 2nd Edition (Hardback) book cover

The Separable Galois Theory of Commutative Rings

2nd Edition

By Andy R. Magid

Chapman and Hall/CRC

184 pages

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pub: 2014-07-14
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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.

Reviews

"This book provides a complete and self-contained account of the Galois theory of commutative rings …"

—Nikolay I. Kryuchkov, Zentralblatt MATH 1298

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.

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT002000
MATHEMATICS / Algebra / General