# Homological algebraSS 2022

Dietrich Burde

Lectures: Monday 12:30-13:15 and Wednesday 13:15-14:45 (or 13:30-15:00) Online with Moodle

Exercises: Wednesday 12:30-13:15 in Seminarraum 7

• This page contains informations and pdf-files for this lecture and its exercise class.
• Homological algebra is, to put it very briefly, the theory of abelian categories and the functors between them. It has played a very important role in algebraic topology. Its influence has gradually expanded and nowadays plays a vital role in commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and in the theory of partial differential equations.

• The chapters are as follows:

Chapter 1: Introduction to homological algebra.
Chapter 2: Rings and modules.
Chapter 3: Categories and functors.
Chapter 4: Resolutions and derived functors.
Chapter 5: Homology and cohomology of groups.

• Here is a syllabus and a bibliography available.

• Exercises: There will be exercises for this lecture. Participation is possible for those present in the first meeting at the beginning of March. In general, presence is mandatory for the exercise class. Students being absent more than once will be signed out. This will not apply in case of illness with a medical certificate. The rules may also change due to the present COVID-19 situation.
There are 33 exercises in total. For each week everyone is supposed to prepare three exercises, i.e., we start with 1,2,3 for the second week, then 4,5,6 for the third week, then 7,8,9, etc.

Exam 2022:

Wednesday, June 29-th, 13:15 - 14:30, Written exam, Seminarraum 12. No documents allowed.

## Files:

 No. Thema Datum file 1 Syllabus 01.03.2022 Contents.pdf 2 My Lecture Notes 01.06.2022 Homological Algebra 3 Exercises 01.03.2022 Exercises.pdf

Topics for the exam:

1. Free, projective, flat, torsion-free, injective and divisible modules
2. Categories and functors, abelian categories, additive categories
3. Injective and projective resolutions, derived functors, homology and homotopy, Ext and Tor, Double complexes, Yoneda Ext
4. Group cohomology and group homology, bar resolution, explicit coboundary map, inflation, restriction and cup product

Dietrich Burde