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Homological Algebra (TCC 2019)


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.


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


  • 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