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.



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