Galois TheoriesStarting 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 formalize 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. |
Contents
Classical Galois theory | 3 |
12 Separable extensions | 4 |
13 Normal extensions | 4 |
14 Galois extensions | 4 |
Galois theory of Grothendieck | 17 |
22 Extension of scalars | 22 |
23 Split algebras | 25 |
24 The Galois equivalence | 29 |
58 The monotonelight factorization | 179 |
Covering maps | 188 |
62 Some limits in Fam 4 We begin with | 191 |
63 Involving extensivity | 195 |
64 Local connectedness and etale maps | 199 |
65 Localization and covering morphisms | 203 |
66 Classification of coverings | 209 |
67 The Chevalley fundamental group | 214 |
Infinitary Galois theory | 38 |
32 Infinitary Galois groups | 41 |
33 Classical infinitary Galois theory | 46 |
34 Profinite topological spaces | 49 |
35 Infinitary extension of the Galois theory of Grothendieck | 58 |
Categorical Galois theory of commutative rings | 67 |
42 Pierce representation of a commutative ring | 74 |
43 The adjoint of the spectrum functor | 82 |
44 Descent morphisms | 93 |
45 Morphisms of Galois descent | 100 |
46 Internal presheaves | 104 |
47 The Galois theorem for rings | 108 |
Categorical Galois theorem and factorization systems | 118 |
51 The abstract categorical Galois theorem | 119 |
52 Central extensions of groups | 129 |
53 Factorization systems | 146 |
54 Reflective factorization systems | 151 |
55 Semiexact reflections | 158 |
56 Connected components of a space | 170 |
57 Connected components of a compact Hausdorff space | 172 |
68 Path and simply connected spaces | 218 |
7 Nongaloisian Galois theory | 227 |
72 Internal precategories and their presheaves | 243 |
73 A factorization system for functors | 248 |
74 Generalized descent theory | 253 |
75 Generalized Galois theory | 260 |
76 Classical Galois theories | 263 |
77 Grothendieck toposes | 268 |
78 Geometric rnorphisms | 276 |
79 Two dimensional category theory | 289 |
710 The JoyalTierney theorem | 296 |
AI Separable algebras | 306 |
A2 Back to the classical Galois theory | 312 |
A3 Exhibiting some links | 318 |
A4 A short summary of further results and developments | 330 |
Bibliography | 333 |
| 338 | |
| 340 | |
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Common terms and phrases
2-category adjunction arrows bijection boolean algebra C₁ central extension clopens coequalizer colim colimit commutative rings compact Hausdorff space composite conditions are equivalent connected components consider continuous map coproduct corollary corresponding covering map covering morphism defined definition diagram effective descent morphism element epimorphism equivalence of categories étale exists extension of fields factorization system Fam(A field extension filter finite dimensional Galois Fix G full subcategory functor G-set Galois descent Galois extension Galois group Galois theory geometric morphism given Grothendieck toposes Hausdorff space Homk homomorphism idempotent implies internal groupoid isomorphism K-algebra kernel pair left adjoint lemma Let K CL locally minimal polynomial monad monomorphism natural transformation notation open subsets p₁ phism polynomial p(X precategory presheaves Prof/Sp(S profinite space proposition pseudo-functor pullback quotient Sh(L split by ids subobjects surjective theorem topological spaces topos totally disconnected trivial unique yields