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