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Finite Group
Schemes
Lecture course by Prof. Richard Pink
Wintersemester 2004/05
Lerneinheit Stunden Tag Zeit Ort
401-4205-00L 2 V Donnerstag
15-17 h HG G26.3
Begin: Thursday 21. October 2004
Participants: Students in higher semesters,
graduate students, etc.
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The main
aim of the course is the classification of finite commutative group schemes
over a perfect field of characteristic p, using the classical
approach by contravariant Dieudonné theory. The theory is developed from
scratch; emphasis is placed on complete proofs. No prerequisites other
than a good knowledge of algebra and the basic properties of categories
and schemes are required. The original plan included p-divisible
groups, but there was no time for this.
Here are the Complete Notes
in pdf format, the notes lecture by lecture are below. Material is added
as the course progresses.
| Title
and Contents Lecture 1 §1 Motivation §2 Group objects in a category Lecture 2 §3 Affine group schemes §4 Cartier duality §5 Constant group schemes Lecture 3 §6 Actions and quotients in a category §7 Quotients of schemes by finite group schemes, part I Lecture 4 §8 Quotients of schemes by finite group schemes, part II §9 Abelian categories §10 The category of finite commutative group schemes Lecture 5 §11 Galois descent §12 Étale group schemes §13 The tangent space Lecture 6 §14 Frobenius and Verschiebung §15 The canonical decomposition §16 Split local-local group schemes |
Lecture
7 §17 Group orders §18 Motivation for Witt vectors §19 The Artin-Hasse exponential Lecture 8 §20 The ring of Witt vectors over Z §21 Witt vectors in characteristic p Lecture 9 §22 Finite Witt group schemes Lecture 10 §23 The Dieudonné functor in the local-local case Lecture 11 §24 Pairings and Cartier duality §25 Cartier duality of finite Witt group schemes Lecture 12 §26 Duality and the Dieudonné functor Lecture 13 §27 The Dieudonné functor in the reduced-local case §28 The Dieudonné functor in the general case References |