To Finite Group Schemes

1. Tate’s theorem 2 Exercises 5 2. Introduction to group schemes 5 Definition (as a functor) 6 Definition (as a group object) 6 Examples of group schemes 8 Rank and the augmentation ideal 9 Subgroup schemes, morphisms and kernels 11 Diagonalizable group schemes 13 Constant group schemes 14 Exercises 14 3. Duality and Deligne’s theorem 16 Cartier duality 16 Deligne’s theorem 19 Exercises 22 4. Étale schemes 22 Differentials 22 Étale group schemes (over a field) 23 Characteristic zero 24 Étale group schemes (over a ring) 26 Characteristic p 27 Connected and étale components 30 Exercises 32 5. Fontaine’s theorem 33 Ramification theory 34 Fontaine’s theorem: Statement and examples 36 A converse to Krasner’s lemma 36 Fontaine’s theorem: An overview 41 Example: Z[ζ7] 43 Reduction to the étale case 45 An equivalence of categories 46 Cokernels and sheaves 50 Nonexistence of abelian varieties 53 Exercises 57 6. Comments on the Exercises 58