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Galois Theories

Francis Borceux, George Janelidze
Format
Book
Published
Cambridge ; New York : Cambridge University Press, 2001.
Language
English
Series
Cambridge Studies in Advanced Mathematics
ISBN
0521803098 (hbk.)
Contents
  • 1 Classical Galois theory 1
  • 1.1 Algebraic extensions 1
  • 1.2 Separable extensions 4
  • 1.3 Normal extensions 6
  • 1.4 Galois extensions 8
  • 2 Galois theory of Grothendieck 15
  • 2.1 Algebras on a field 15
  • 2.2 Extension of scalars 20
  • 2.3 Split algebras 23
  • 2.4 Galois equivalence 27
  • 3 Infinitary Galois theory 36
  • 3.1 Finitary Galois subextensions 36
  • 3.2 Infinitary Galois groups 39
  • 3.3 Classical infinitary Galois theory 44
  • 3.4 Profinite topological spaces 47
  • 3.5 Infinitary extension of the Galois theory of Grothendieck 56
  • 4 Categorical Galois theory of commutative rings 65
  • 4.1 Stone duality 65
  • 4.2 Pierce representation of a commutative ring 72
  • 4.3 Adjoint of the 'spectrum' functor 80
  • 4.4 Descent morphisms 91
  • 4.5 Morphisms of Galois descent 98
  • 4.6 Internal presheaves 102
  • 4.7 Galois theorem for rings 106
  • 5 Categorical Galois theorem and factorization systems 116
  • 5.1 Abstract categorical Galois theorem 117
  • 5.2 Central extensions of groups 127
  • 5.3 Factorization systems 144
  • 5.4 Reflective factorization systems 149
  • 5.5 Semi-exact reflections 156
  • 5.6 Connected components of a space 168
  • 5.7 Connected components of a compact Hausdorff space 170
  • 5.8 Monotone-light factorization 177
  • 6 Covering maps 186
  • 6.1 Categories of abstract families 186
  • 6.2 Some limits in Fam(A) 189
  • 6.3 Involving extensivity 193
  • 6.4 Local connectedness and etale maps 197
  • 6.5 Localization and covering morphisms 201
  • 6.6 Classification of coverings 207
  • 6.7 Chevalley fundamental group 212
  • 6.8 Path and simply connected spaces 216
  • 7 Non-galoisian Galois theory 225
  • 7.1 Internal presheaves on an internal groupoid 225
  • 7.2 Internal precategories and their presheaves 241
  • 7.3 A factorization system for functors 246
  • 7.4 Generalized descent theory 251
  • 7.5 Generalized Galois theory 258
  • 7.6 Classical Galois theories 261
  • 7.7 Grothendieck toposes 266
  • 7.8 Geometric morphisms 274
  • 7.9 Two dimensional category theory 287
  • 7.10 Joyal-Tierney theorem 294
  • A.1 Separable algebras 304
  • A.2 Back to the classical Galois theory 310
  • A.3 Exhibiting some links 316.
Description
xiv, 341 p. : ill. ; 24 cm.
Notes
Includes bibliographical references (p. 331-335) and indexes.
Series Statement
Cambridge studies in advanced mathematics 72
Technical Details
  • Access in Virgo Classic
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    g| 1 t| Classical Galois theory g| 1 -- g| 1.1 t| Algebraic extensions g| 1 -- g| 1.2 t| Separable extensions g| 4 -- g| 1.3 t| Normal extensions g| 6 -- g| 1.4 t| Galois extensions g| 8 -- g| 2 t| Galois theory of Grothendieck g| 15 -- g| 2.1 t| Algebras on a field g| 15 -- g| 2.2 t| Extension of scalars g| 20 -- g| 2.3 t| Split algebras g| 23 -- g| 2.4 t| Galois equivalence g| 27 -- g| 3 t| Infinitary Galois theory g| 36 -- g| 3.1 t| Finitary Galois subextensions g| 36 -- g| 3.2 t| Infinitary Galois groups g| 39 -- g| 3.3 t| Classical infinitary Galois theory g| 44 -- g| 3.4 t| Profinite topological spaces g| 47 -- g| 3.5 t| Infinitary extension of the Galois theory of Grothendieck g| 56 -- g| 4 t| Categorical Galois theory of commutative rings g| 65 -- g| 4.1 t| Stone duality g| 65 -- g| 4.2 t| Pierce representation of a commutative ring g| 72 -- g| 4.3 t| Adjoint of the 'spectrum' functor g| 80 -- g| 4.4 t| Descent morphisms g| 91 -- g| 4.5 t| Morphisms of Galois descent g| 98 -- g| 4.6 t| Internal presheaves g| 102 -- g| 4.7 t| Galois theorem for rings g| 106 -- g| 5 t| Categorical Galois theorem and factorization systems g| 116 -- g| 5.1 t| Abstract categorical Galois theorem g| 117 -- g| 5.2 t| Central extensions of groups g| 127 -- g| 5.3 t| Factorization systems g| 144 -- g| 5.4 t| Reflective factorization systems g| 149 -- g| 5.5 t| Semi-exact reflections g| 156 -- g| 5.6 t| Connected components of a space g| 168 -- g| 5.7 t| Connected components of a compact Hausdorff space g| 170 -- g| 5.8 t| Monotone-light factorization g| 177 -- g| 6 t| Covering maps g| 186 -- g| 6.1 t| Categories of abstract families g| 186 -- g| 6.2 t| Some limits in Fam(A) g| 189 -- g| 6.3 t| Involving extensivity g| 193 -- g| 6.4 t| Local connectedness and etale maps g| 197 -- g| 6.5 t| Localization and covering morphisms g| 201 -- g| 6.6 t| Classification of coverings g| 207 -- g| 6.7 t| Chevalley fundamental group g| 212 -- g| 6.8 t| Path and simply connected spaces g| 216 -- g| 7 t| Non-galoisian Galois theory g| 225 -- g| 7.1 t| Internal presheaves on an internal groupoid g| 225 -- g| 7.2 t| Internal precategories and their presheaves g| 241 -- g| 7.3 t| A factorization system for functors g| 246 -- g| 7.4 t| Generalized descent theory g| 251 -- g| 7.5 t| Generalized Galois theory g| 258 -- g| 7.6 t| Classical Galois theories g| 261 -- g| 7.7 t| Grothendieck toposes g| 266 -- g| 7.8 t| Geometric morphisms g| 274 -- g| 7.9 t| Two dimensional category theory g| 287 -- g| 7.10 t| Joyal-Tierney theorem g| 294 -- g| A.1 t| Separable algebras g| 304 -- g| A.2 t| Back to the classical Galois theory g| 310 -- g| A.3 t| Exhibiting some links g| 316.
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