Lecture Commutative Algebra

 

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 TaskTo 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

 

 

Shanghaitech University

IMS S506

 

Every Tuesday and Thursday 13:00-13:45 and 13:55-14:40

Start 14th of September 2021

 

Office hours: Thursdays 11:30-12:30 IMS S412

 

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© Daniel Skodlerack