# Homological Algebra (TCC 2019)

### Lectures

Friday mornings 10.00-12.00, from 11 Oct 2019 onwards (with 5 minute break around 11.00).

**No lecture 18 Oct.**

### Lecture Notes

- Lecture 1 (11/10/19): Categories, functors, natural transformations (definition & examples). Monomorphisms and epimorphisms. Equivalences of categories.
- Lecture 2 (25/10/19): Limits and colimits. Adjunctions. Additive categories, kernels and cokernels. Abelian categories.
- Lecture 3 (1/11/19): Left / right exact functors, injective and projective objects. Enough injective and projective modules. The category of complexes.
- Lecture 4 (8/11/19): The Snake Lemma. Injective and projective resolutions: existence, uniqueness up to homotopy. The Horseshoe Lemma. (See here for the proof of the Snake Lemma).
- Lecture 5 (15/11/19): Definition of derived functors. Examples: Ext, Tor, group cohomology and homology, and sheaf cohomology.
- Lecture 6 (22/11/19): Spectral sequences: definitions; spectral sequences of filtered and double complexes. Hypercohomology. Applications to real and complex de Rham cohomology.
- Lecture 7 (29/11/19): The Grothendieck spectral sequence. Applications: Hochschild–Serre for group cohomology; Leray for sheaf cohomology. Edge maps and degeneration of spectral sequences. The homotopy category of complexes.
- Lecture 8 (6/12/19): Localisation of categories. Derived categories and injective resolutions. Total left/right derived functors. Application: Grothendieck duality for non-smooth projective varieties.

## Coursework

Assessment of the course will be based on 3 problem sheets, to be distributed after lectures 3, 6 and 8.

- Sheet 1 (deadline 22/11/19)—Solutions
- Sheet 2 (deadline 16/12/19)—Solutions
- Sheet 3 (deadline 13/1/20) —Solutions

### Syllabus

- Review of categories and functors
- Additive and abelian categories
- Exact sequences
- Projective and injective objects, resolutions
- Derived functors
- Applications: Tor and Ext; group cohomology; sheaf cohomology
- Spectral sequences
- The derived category