# Homological algebra

# Math 9144A

UWO Fall 2016 - MWF 2:30-3:30 @MC 108

Instructor: Cihan Okay

Office: MC 134 (by appointment)

Homological algebra started as a generalization of the idea of homology of a space. This elegant formalism now appears in many branches of mathematics. It can be used to study algebraic systems such as groups, rings, algebras, sheaves and so on. Click here for its history by C. Weibel.

Course Syllabus [ pdf]

In this course we will learn basic techniques of homological algebra with a focus on cohomology of groups. Basic understanding of groups, rings, and modules is required. Also we will use the language of category theory. It is recommended for students to familiarize themselves with basic definitions such as of categories, functors, natural transformations, limits and colimits, adjoint functors...

Tentatively, we will cover most of the following:

Abelian categories, Chain complexes, projective/injective resolutions, Derived functors, Ext/Tor functor, Change of rings theorems, Spectral sequences, Group (co)homology, Derived category

Recommended reading:

An introduction to homological algebra - C. A. Weibel

Cohomology of Groups - K. S. Brown

more reading:

An introduction to homological algebra - J. J. Rotman

Homological Algebra - by H. Cartan, S. Eilenberg

Categories for the working mathematician - S. Mac Lane

A user's guide to spectral sequences - J. McCleary

Cohomology of Finite Groups - A. Adem, R. J. Milgram

Assignments

Evaluation will be based on assignments (approx. every two weeks) which will be posted here. Assignments will be submitted in class.

Assignment 1 due on October 12

Assignment 2 due on October 31

Assignment 3 due on December 2

Assignment 4 due on December 12, 14:30 MC 134