Galois Theories

Galois Theories

by Francis Borceux, George Janelidze
     
 

ISBN-10: 0521070414

ISBN-13: 9780521070416

Pub. Date: 07/31/2008

Publisher: Cambridge University Press

Beginning with the classical finite dimensional Galois theory of fields, this book places the theory in a broader context. It presents work by Grothendieck in terms of separable algebras before moving on to the infinite dimensional case. The categorical context of a general Galois theorem is formulated, and applications are supplied. The first chapters are accessible…  See more details below

Overview

Beginning with the classical finite dimensional Galois theory of fields, this book places the theory in a broader context. It presents work by Grothendieck in terms of separable algebras before moving on to the infinite dimensional case. The categorical context of a general Galois theorem is formulated, and applications are supplied. The first chapters are accessible to upper-level undergraduates; later chapters are at the graduate level. Borceux teaches at the Universite Catholique de Luivain. Janelidze teaches at the Georgian Academy of Sciences, Tbilisi. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Product Details

ISBN-13:
9780521070416
Publisher:
Cambridge University Press
Publication date:
07/31/2008
Series:
Cambridge Studies in Advanced Mathematics Series, #72
Pages:
356
Product dimensions:
5.90(w) x 8.90(h) x 1.00(d)

Related Subjects

Table of Contents

Preface
1Classical Galois theory1
1.1Algebraic extensions1
1.2Separable extensions4
1.3Normal extensions6
1.4Galois extensions8
2Galois theory of Grothendieck15
2.1Algebras on a field15
2.2Extension of scalars20
2.3Split algebras23
2.4The Galois equivalence27
3Infinitary Galois theory36
3.1Finitary Galois subextensions36
3.2Infinitary Galois groups39
3.3Classical infinitary Galois theory44
3.4Profinite topological spaces47
3.5Infinitary extension of the Galois theory of Grothendieck56
4Categorical Galois theory of commutative rings65
4.1Stone duality65
4.2Pierce representation of a commutative ring72
4.3The adjoint of the 'spectrum' functor80
4.4Descent morphisms91
4.5Morphisms of Galois descent98
4.6Internal presheaves102
4.7The Galois theorem for rings106
5Categorical Galois theorem and factorization systems116
5.1The abstract categorical Galois theorem117
5.2Central extensions of groups127
5.3Factorization systems144
5.4Reflective factorization systems149
5.5Semi-exact reflections156
5.6Connected components of a space168
5.7Connected components of a compact Hausdorff space170
5.8The monotone-light factorization177
6Covering maps186
6.1Categories of abstract families186
6.2Some limits in Fam (A)189
6.3Involving extensivity193
6.4Local connectedness and etale maps197
6.5Localization and covering morphisms201
6.6Classification of coverings207
6.7The Chevalley fundamental group212
6.8Path and simply connected spaces216
7Non-galoisian Galois theory225
7.1Internal presheaves on an internal groupoid225
7.2Internal precategories and their presheaves241
7.3A factorization system for functors246
7.4Generalized descent theory251
7.5Generalized Galois theory258
7.6Classical Galois theories261
7.7Grothendieck toposes266
7.8Geometric morphisms274
7.9Two dimensional category theory287
7.10The Joyal-Tierney theorem294
App: Final remarks304
Bibliography331
Index of symbols336
General index338

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