Modules over commutative rings occur almost everywhere in Mathematics. The first common examples are groups seen as modules over the ring of integers. Modules behave more richer than vector spaces, for example a module over a ring does not need to have a basis anymore. To understand ring extensions, Dedekind rings and group rings it needs the understanding of modules.

As an application of module theory we introduce a connection to geometry: We will study basics on homological algebra and group cohomology.

Course Task：To learn the basic notions and theorems for algebraic objects over commutative rings. The course is about

* modules (flat, free, projective, injective)

* tensor products of modules,

* localization

* completions and Hensel’s Lemma

* dimension theory (Noether normalization)

* homological algebra

* group cohomology

The final mark will be computed in the following way: 40% homework 20%quizes 40%final exam

**Literature:** 1) David Eisenbud, Commutative Algebra with a view towards Algebraic Geometry

2) Kenneth Brown, Cohomology of groups

The syllabus can be found here.

Problems to present to the tutors:

Week (20.9--): Problem 2 (Sheet 1)

Exercise sheets: Sheet 1 Sheet 2 Sheet 3 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7

Sheet 8 Sheet 9 Sheet 10 Sheet 11 Sheet 12