# Homological Algebra Seminar

### Description

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

• Abelian Categories
• Derived Functors
• Ext and Tor functors
• Sheaf cohomology
• Spectral Sequences

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

### General references

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.

### Lectures

• 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!