Galois theory for schemes

Introduction 1–5 Coverings of topological spaces. The fundamental group. Finité etale coverings of a scheme. An example. Contents of the sections. Prerequisites and conventions. 1. Statement of the main theorem 6–16 Free modules. Free separable algebras. Finité etale morphisms. Projective limits. Profinite groups. Group actions. Main theorem. The topological fundamental group. Thirty exercises. 2. Galois theory for fields 17–32 Infinite Galois theory. Separable closure. Absolute Galois group. Finite algebras over a field. Separable algebras. The main theorem in the case of fields. Twenty-nine exercises. 3. Galois categories 33–53 The axioms. The automorphism group of the fundamental functor. The main theorem about Galois categories. Finite coverings of a topological space. Proof of the main theorem about Galois categories. Functors between Galois categories. Twenty-seven exercises. 4. Projective modules and projective algebras 54–68 Projective modules. Flatness. Local characterization of projective modules. The rank. The trace. Projective algebras. Faithfully projective algebras. Projective separable algebras. Forty-seven exercises. 5. Finité etale morphisms 69–82 Affine morphisms. Locally free morphisms. The degree. Affine characterization of finité etale morphisms. Surjective, finite, and locally free morphisms. Totally split morphisms. Characterization of finité etale morphisms by means of totally split morphisms. Morphisms between totally split morphisms are locally trivial. Morphisms between finité etale morphisms are fi-nité etale. Epimorphisms and monomorphisms. Quotients under group actions. Verification of the axioms. Proof of the main theorem. The fundamental group. Twenty-three exercises. 6. Complements 83–100 Flat morphisms. Finitely presented morphisms. Unramified morphisms. ´ Etale morphisms. Finité etale is finite andétale. Separable algebras. Projective separable is projective and separable. Finité etale coverings of normal integral schemes. The fundamental group of such schemes. Dimension one. The projective line and the affine line. Finite rings. Forty exercises. Bibliography 101–102 Twenty-six references. Acknowledgements are due to Mrs. L. van Iterson for typing the manuscript; to Mr. W. Bosma for preparing the bibliography, the list of symbols, and the index; to Mrs. D. Craig for preparing the second edition and a first electronic version; and to Mr. T. Vorselen for preparing the third edition. Readers are requested to e-mail possible errors – mathematical, typographical, or otherwise – to hwl@math.leidenuniv.nl.