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Galois Theories : Cambridge Studies in Advanced Mathematics (Hardcover) - Francis Borceux

Galois Theories

Cambridge Studies in Advanced Mathematics (Hardcover)

Hardcover Published: 19th March 2001
ISBN: 9780521803090
Number Of Pages: 356

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Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context, presenting work by Grothendieck in terms of separable algebras and then proceeding to the infinite-dimensional case, which requires considering topological Galois groups. In the core of the book, the authors first formalise the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience: the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit and adjoint functors. The first chapters are accessible to advanced undergraduates, with later ones at a graduate level. For all algebraists and category theorists this book will be a rewarding read.

From the hardback review: 'This book is a beautiful presentation of Janelidze's general categorical Galois theory ... a rewarding read.' Laszlo Marki, Acta Sci. Math. From the hardback review: '... highly recommended or anyone wishing to learn the mathematical side of category theory (rather than its computer-science aspect) ... I enjoyed reading it very much.' Proceedings of the Edinburgh Mathematical Society From the hardback review: 'A comprehensive account, which may well deepen one's understanding of the classical case.' Mathematika

Prefacep. vii
Classical Galois theoryp. 1
Algebraic extensionsp. 1
Separable extensionsp. 4
Normal extensionsp. 6
Galois extensionsp. 8
Galois theory of Grothendieckp. 15
Algebras on a fieldp. 15
Extension of scalarsp. 20
Split algebrasp. 23
The Galois equivalencep. 27
Infinitary Galois theoryp. 36
Finitary Galois subextensionsp. 36
Infinitary Galois groupsp. 39
Classical infinitary Galois theoryp. 44
Profinite topological spacesp. 47
Infinitary extension of the Galois theory of Grothendieckp. 56
Categorical Galois theory of commutative ringsp. 65
Stone dualityp. 65
Pierce representation of a commutative ringp. 72
The adjoint of the 'spectrum' functorp. 80
Descent morphismsp. 91
Morphisms of Galois descentp. 98
Internal presheavesp. 102
The Galois theorem for ringsp. 106
Categorical Galois theorem and factorization systemsp. 116
The abstract categorical Galois theoremp. 117
Central extensions of groupsp. 127
Factorization systemsp. 144
Reflective factorization systemsp. 149
Semi-exact reflectionsp. 156
Connected components of a spacep. 168
Connected components of a compact Hausdorff spacep. 170
The monotone-light factorizationp. 177
Covering mapsp. 186
Categories of abstract familiesp. 186
Some limits in Fam(A)p. 189
Involving extensivityp. 193
Local connectedness and etale mapsp. 197
Localization and covering morphismsp. 201
Classification of coveringsp. 207
The Chevalley fundamental groupp. 212
Path and simply connected spacesp. 216
Non-galoisian Galois theoryp. 225
Internal presheaves on an internal groupoidp. 225
Internal precategories and their presheavesp. 241
A factorization system for functorsp. 246
Generalized descent theoryp. 251
Generalized Galois theoryp. 258
Classical Galois theoriesp. 261
Grothendieck toposesp. 266
Geometric morphismsp. 274
Two dimensional category theoryp. 287
The Joyal-Tierney theoremp. 294
Final remarksp. 304
Separable algebrasp. 304
Back to the classical Galois theoryp. 310
Exhibiting some linksp. 316
A short summary of further results and developmentsp. 328
Bibliographyp. 331
Index of symbolsp. 336
General indexp. 338
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521803090
ISBN-10: 0521803098
Series: Cambridge Studies in Advanced Mathematics (Hardcover)
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 356
Published: 19th March 2001
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.39
Weight (kg): 0.69

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