# MA3204 Homological algebra

## Lecturer

**Chrysostomos Psaroudakis**, Room 801, Sentralbygg II, chrysostomos [dot] psaroudakis [at] math [dot] ntnu [dot] no

## Timeplan

Lectures:

- Monday, 14:15-16:00, Room R81
- Tuesday, 14:15-16:00, Room 734

## Messages

**First meeting:**Monday, 22nd of August, 14:15 in R81 for the first lecture and to discuss the schedule.

**August 23, 2016:** Change of rooms, Wednesdays 8:15 - 10:00 in Room 734, Sentralbygg II.

**August 24, 2016:** Change of timeplan, Tuesdays 14:15 - 16:00 in Room 734, Sentralbygg II.

**September 2, 2016:** The first exercise sheet is now available.

**September 19, 2016:** The second exercise sheet is now available.

**September 30, 2016:** The third exercise sheet is available.

**October 25, 2016:** Students that are not attending the lectures but are willing to give exams are advised to contact me by email. The dates of the oral exams will be decided in the class.

**November 10, 2016:** The fourth exercise sheet is available.

**November 26, 2016:** The fifth exercise sheet is available.

## Content

**From the study hand book:**

The course deals with homological algebra for abelian categories in general, and modules over a ring in particular.

First category theory is introduced, both in the setup of categories in general and abelian categories in particular, and some basic properties are discussed (functors, natural transformations, limits and colimits, in particular kernels, cokernels, pullbacks, pushouts).

The main part of the course focuses on the study of derived functors, in particular the derived functors Ext and Tor. To this end, the concepts of complexes, homotopy, homology, projective and injective resolutions are introduced and studied. The discussion of the first Ext also involves comparison to short exact sequences (Yoneda-Ext).

Finally triangulated, and in particular derived categories are introduced, and Ext is interpreted as morphism set in the derived category.

A more detailed list of subjects discussed in this course 2 years ago can be found here: contents (the plan is to cover approximately the same content this year)

## Literature

The content of the course is what will be presented during the lectures. We will mainly follow the lecture notes of Steffen Oppermann: https://www.math.ntnu.no/~opperman/HomAlg.pdf.

If you would like to have a book, the following can be used:

**Joseph J. Rotman**,*An introduction to Homological Algebra*, first edition**Joseph J. Rotman**,*An introduction to Homological Algebra*, second edition**Charles A. Weibel**,*An introduction to Homological Algebra***Sergei I. Gelfand, Yuri I. Manin**,*Methods of Homological Algebra*, second edition**Peter J. Hilton, Urs Stammbach**,*A Course in Homological Algebra*, second edition