2nd Edition

The Separable Galois Theory of Commutative Rings

By Andy R. Magid Copyright 2014
    184 Pages
    by Chapman & Hall

    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.

    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.

    Biography

    Andy R. Magid

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