Time and place: on campus, Spring Semester 2022

Outline of the course:

The goal of this course is to understand the basic concepts of affine algebraic geometry and affine algebraic groups. This includes theorems like Hilbert's Nullstellensatz, Noether Normalisation and the Linearisation of affine algebraic groups. The outline of the course will be roughly as follows below. At the end of each chapter outline there is a list of fundamental mathematical terms and objects that will be used in that part and with whose definitions you should be familiar to understand the content of the lectures.

Chapter 1: Commutative algebra
In this section we discuss the basics in the theory of commutative algebras over an arbitrary base field. The main focus here are a number of very famous theorems from the late 19th and early 20th century, namely Noether's Normalisation theorem and Hilbert's Nullstellensatz and the Basis theorem.
Prerequisites: Rings and algebras over fields, ideals (including prime and maximal ideals), zero-divisors and nilpotent elements in a ring, modules for a ring, field extensions and algebraic closure of a field.

Chapter 2: Algebraic sets and Zariski topology
In this section we discuss the the concept of algebraic sets and the Zariski topology defined by them. We will also introduce the concept of the coordinate ring of an algebraic set and we will see how Hilbert's Nullstellensatz translates to these algebraic sets.
Prerequisites: Topology on a set, tensor product of vector spaces.

Chapter 3: Affine algebraic varieties
In this section we introduce affine algebraic varieties over an algebraically closed field and the maximal spectrum of an algebra as a topological space. We will look at the ring of regular functions of an affine algebraic variety. Finally we will recover an affine algebraic variety from its ring of regular functions making the concept of an affine algebraic variety more universal than the one of an algebraic set.

Chapter 4: Affine algebraic groups
In this section we discuss affine algebraic groups. We will see how affine algebraic groups are related to certain Hopf algebras. We will also see how a variety being a group will affect the various properties of morphisms of varieties as well as the topology. Finally we will prove the linearisation theorem, that every affine algebraic group is in fact isomorphic to a linear algebraic group.
Prerequisites: Groups, index of subgroups

Chapter 5+:
Depending on the progress of the lecture we will cover additional topics after the end of chapter 4.

Lecture notes:

There will be up-to date notes send out after the lecture on each Friday, covering the topic of the week’s lectures.

Exercises:

There will be two exercise sheets in total.

• Exercise sheet 1 (available in week 3, deadline end of week 4)
• Exercise sheet 2 (available in week 6, deadline end of week 7)

Examination:

The final mark for the lecture will be comprised of the following parts:

• Written report on sheet 1: Deadline: 2022-03-20
• Written report on sheet 2: Deadline: 2022-04-10
• Written exam after the end of the course

Each exercise report will make up 30% of the final score, while the exam will make up 40% of the score. The exercise reports can be handed in on paper or via email (in English).

References:

Commutative algebra

• Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra, Reading, Mass. Addison Wesley 1969
• Eisenbud, D.: Commutative Algebra with a view towards Algebraic Geometry, Springer Verlag, Graduate Texts in Mathematics, No. 150, New York 1995

Algebraic geometry and algebraic groups

• Borel, A.: Linear algebraic groups. Second edition. Graduate Texts in Mathematics, No. 126. Springer-Verlag, New York, 1991.
• Humphreys, J. E.: Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.