Algebraic Groups, summer term 2020

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News

Currently, all non-virtual teaching at the University of Regensburg is suspended: more information about dealing with 2019-nCoV at the University of Regensburg.
Details about the summer term 2020 and the exact organisation of this course will be announced on 15th April. Please stay safe and patient!

If you are interested in participating in this course (whenever/however it will be taught ...), please register on the GRIPS page.

Here you find some information on Weil restriction. If there are more questions on this subject, don't hesitate to ask during lecture.

Content

The theory of algebraic groups plays a fundamental role in many areas of mathematics, such as algebraic number theory (module forms, automorphic forms...). In algebraic geometry they provide useful tools to study group actions and to form quotient spaces; for example they are used for the construction of various moduli spaces.

The goal of this course is to give an introduction in the theory of algebraic groups. We will use the modern approach of the functor of points. The only prerequisite is therefore commutative algebra. Some basic knowledge about algebraic geometry will also be helpful but not necessary.

An important result that we will discuss is the theorem of Chevalley, which states that an algebraic group can be see as an extension of an abelian variety by an affine algebraic group. Another topic is the classification of reductive algebraic groups by root systems and Dynkin diagrams.

Format

In this course we will use a variation of the Moore method.
On the GRIPS page, the participants can access a latex template of our lecture notes. Here most of the proofs and examples are left blank with some hints.
Every week, the participants are asked to work through a part of the notes. During the lectures and exercise sessions we discuss problems and questions that come up and try to find solutions together.
The participants will acquire a good command of the topic by working through the notes and filling in the blanks.
The filled in lecture notes will be handed in every 2 or 3 weeks by email and checked. Every filled in place can receive 0, 1 or 2 points.

Ideally, by the end of the term every student has a complete set of lecture notes, written by him- or herself.

At a glance

Zeitplan

Week 1	21/04/2020      Discussion of the format	23/04/2020 Overview of the subject
Week 2	28/04/2020 Affine schemes	30/04/2020 General schemes
Week 3	04/05/2020 Projective space	07/05/2020 Base change, Weil restriction
Week 4	11/05/2020 Weil restriction, flat maps	14/05/2020 Faithfully flat descent
Week 5	18/05/2020 Grothendieck's generic freeness theorem	--
Week 6	25/05/2020 Chevalley's constructibility theorem	28/05/2020 Derivations
Week 7	--	04/06/2020 Total tangent space
Week 8	08/06/2020 Euler sequence	--
Week 9	15/06/2020 Properties of morphisms	18/06/2020 Properties of morphisms
Week 10	22/06/2020 Definition of group schemes	25/06/2020 Lie algebra of a group scheme
Week 11	29/06/2020 Lie algebra of a group scheme	02/07/2020 The identity component
Week 12	06/07/2020 Operations of group schemes	09/07/2020
Week 13	13/07/2020 Orbits, fixed scheme	16/07/2020 Affine group schemes, representations
Week 14	20/07/2020	23/07/2020
Week 15	-	-
Week 16	03.08.2020 Chevalley's structure theorem	06.08.2020 Chevalley's structure theorem