Homological Algebra Seminar

The seminar will run every Wednesday from 10am to 11am during Summer C, via Zoom. We will cover the following topics:

It time permits I may cover some additional topics, such as group cohomology, Hochschild cohomology, or derived categories.

The first three of these can be found on the internet in PDF form:

Hilton and Stammbach, A Course in Homological Algebra. A good basic treatment but mostly focuses on module categories.
Weibel, An Introduction to Homological Algebra. In spite of the title, a more advanced treatment than Hilton-Stammbach. I will be using this a lot.
Grothendieck, Sur quelque points d’algebre homologique, Tohoku J. Math 9 no. 2, 1957. This is the classic reference, and the fact that it’s in French will be the least of your worries. The wikipedia page has a pointer to an English translation.
Gelfand and Manin, Methods of Homological Algebra. Lots of interesting stuff but very idiosyncratic.
Chapter 12 of the Stacks Project is devoted to homological algebra in the setting of abelian categories.

Lecture 1 – Additive Categories
Lecture 2 – Abelian Categories
Lecture 3 – Exactness in Abelian Categories
Lecture 4 – Projective and Injective objects
Lecture 5 – Exactness, Complexes, Homology
Lecture 6 – Additive Functors, Resolutions, Homotopies
Lecture 7 – Derived Functors
Lecture 8 – Universal properties of derived functors. Ext groups
Lecture 9 – Interpretation of the Ext groups. Tor functors
Lecture 10 (7/21/21) – Sheaves on a topological space
Lecture 11 (7/23/21) – Properties of abelian sheaves. Direct images, local and global Ext
Lecture 12 – Spectral Sequences
Lecture 13 – The Spectral Sequence of a Composite Functor. Applications.
Lecture 14 – Derived Categories (introduction)
Lecture 15 – Triangulated Categories

Lectures 14 and 15 will not actually be delivered but are posted here for your, ahem entertainment. Enjoy!

Richard Crew