BIREP — Representations of finite dimensional algebras at Bielefeld

• BIREP
• Fakultät für Mathematik
• Universität Bielefeld

 

Masters Course: Homological Algebra

Lectures in Summer Semester 2026
Lecturer: Prof. Dr. William Crawley-Boevey
Exercises: Raphael Bennett-Tennenhaus

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Contents and Literature

Homological algebra is the algebra that was invented in order to define and study the homology and cohomology of topological spaces, but it has applications all over mathematics. Planned contents (subject to change):

1. Abelian categories
2. Projective, injective and flat modules
3. Complexes
4. Resolutions, Ext and Tor
5. Group cohomology
6. Triangulated categories and derived categories

Students are expected to already have some familiarity with rings and modules - as background reading, I suggest my notes for Algebra II.

Some suggested books:

• C. A. Weibel, An introduction to homological algebra, CUP 1994, E-book.
• J. J. Rotman, An introduction to homological algebra, Springer 2009, E-book.
• M. S. Osborne, Basic homological algebra, Springer 2000.
• S. I. Gelfand and Yu. I. Manin, Methods of homological algebra, 2nd ed., Springer 2010.
• H. Krause, Homological theory of representations, CUP 2022, E-book.
• The Stacks Project, https://stacks.math.columbia.edu/.

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Exercises

By Raphael Bennett-Tennenhaus.

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Examination

Oral examination at the end of the semester, by arrangement with the lecturer.

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Notes

I taught the same course in Winter Semester 2024/25. The lecture notes are here. My plan this time is to cover less material and go more slowly, but in general I aim to follow these notes.

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• Last file change on 30 January 2026 by William Crawley-Boevey
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