# Homological Algebra (TCC 2016)

Syllabus —Lecture notes—Problem sheets

This is a home page for the Homological Algebra graduate course (Mathematics Taught Course Centre, October-December 2016). The lectures are taking place on **Fridays 10-12 am**, starting 14th October, in the Access Grid room (Zeeman Building B0.06).

### Syllabus

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

The course is intended to be accessible to a wide variety of backgrounds; in particular, expertise in algebraic topology or algebraic geometry will **not** be assumed.

### Lecture notes

- Lecture 1 (14/10/16): definitions and examples of categories, functors and natural transformations; adjoint functors; definition of an additive category.

- Lecture 2 (21/10/16): kernels and cokernels, abelian categories; exact sequences; exact, left exact, and right exact functors.
- Lecture 3 (28/10/16): (board notes missing last page; Alessandro's notes for last page)—projective & injective objects; the category of complexes; snake lemma; cohomology functors.
- Lecture 4 (4/11/16): Projective and injective resolutions, examples; null-homotopies, uniqueness of resolutions up to homotopy; definition and basic properties of derived functors; Tohuku viewpoint (universal delta-functors).
- Lecture 5 (11/11/16) (board notes (incomplete), Alessandro's notes): Examples of derived functors. Ext functor; balancing for Ext. Group cohomology; the bar resolution, cochains. Brief introduction to sheaves, global sections, and higher direct images.

- Lecture 6 (18/11/16): Introduction to spectral sequences; the two spectral sequences of a double complex.

- Lecture 7 (25/11/16): Hyper-derived functors; Grothendieck's spectral sequence; applications (Hodge spectral sequence, universal coefficients for cohomology, proof of Hochschild--Serre, derived functors of inverse limit).
- Lecture 8 (2/12/16): The homotopy category of complexes and the derived category.

### Problem sheets

Danger: (co)cycles!