Finite Group Schemes (V5A4), Summer Term 2021

Lecturer: Dr. Gebhard Martin
Email: G. Martin

Lecture

Wednesday 12 c.t. - 14

Registration: Please register for the lecture on this course's eCampus page. The password is "Cartier".

Exams

• First Exam: Between July 27 and July 29
• Second Exam: Between September 21 and September 23

Both exams will be oral exams. More details will be available on eCampus at the end of June.

Content

Finite group schemes are to schemes what finite groups are to sets. As such, they appear in algebraic geometry whenever one studies automorphisms, for example,

• as subgroup schemes of the automorphism scheme of a proper scheme over a field,
• as kernels of isogenies of abelian varieties and linear algebraic groups,
• when studying algebraic analogues of fundamental groups (e.g. Nori's fundamental group scheme),
• in the study of quotient singularities in positive characteristic,
• etc.

In this lecture course, we will introduce and study the category of finite group schemes over a field. We will give precise descriptions of the following important subcategories of the category of finite group schemes:

• finite étale group schemes over arbitrary fields via finite groups with Galois action,
• finite abelian group schemes over perfect fields via Dieudonné theory, and
• finite connected group schemes of height 1 over fields of positive characteristic via restricted Lie algebras.

Even though the category of finite group schemes over a field is not abelian, kernels, images, and normal subgroup objects do exist, making it possible to talk about extensions and exact sequences of finite group schemes. We will use this to study properties of general finite group schemes by breaking them into simple(r) pieces contained in the three categories above.

Prerequisites

Basic knowledge in commutative algebra and algebraic geometry (e.g. from Algebra I and Algebraic Geometry I)

Recommended Literature

• R. Pink: Finite group schemes, Lecture Course at ETH Zürich (https://people.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf)
• J. Tate: Finite flat group schemes, in "Modular Forms and Fermat's Last Theorem"
• J. S. Milne: Algebraic Groups, Cambridge Studies 170
• W. Waterhouse, Introduction to Affine Group Schemes, GTM 66, Springer
• M. Demazure, P. Gabriel: Groupes Algébriques, North-Holland publishing Company
• M. Demazure, A. Grothendieck: SGA 3, LN 151-153, Springer