Homological Algebra

Basic information

In this course, we develop the necessary tools for the definition and the study of various (co)homology theories.

The plan is roughly:

  1. Introduction to category theory.
  2. Universal properties, representable functors and adjoint functors.
  3. Tensor product of bimodules.
  4. Additive categories and their categories of (co)chain complexes.
  5. Abelian categories and homology.
  6. Derived functors, Ext and Tor.
  7. Application to algebraic topology: singular homology.
  8. Morita theory.

There might be some changes, depending on our progess.

Handwritten notes (contain small mistakes and typos!)

12/09/2022
14/09/2022
19/09/2022
21/09/2022
26/09/2022
28/09/2022
03/10/2022
05/10/2022
10/10/2022
12/10/2022
17/10/2022
19/10/2022

Exercise sheets

Sheet 1
Sheet 2
Sheet 3
Sheet 4

Exam with correction

Exam

Hints and solutions for the exercises

  1. TD 2 Exercises 7 and 8 For exercise 7, you can skip question 3, which is quite technical. The solution proposed in the note is not correct. The 'natural' map \(S\) that you can build does not give a splitting. One has to twist it a little bit. Alternatively, one can check that \(SD+DS\), where \(D\) is the derivation of the cone is an endomorphism of the cone and is also an isomorphism (in matrix form, it is a triangular matrix with identities on the diagonal). Hence, an automorphism of the cone is homotopic to zero. Composing with the inverse isomorphism gives that the identity of the cone is homotopic to zero...
  2. TD 2 Exercises 2
  3. TD3 Exercise 3: you can look the beginning of chapter 9 in Weibel. Exercise 6 : Baer criterion is not so easy, you can find the proof in Assem for example or Weibel Page 39.
  4. TD 4 Exercises 1 and 2
  5. TD4 Exercise 6, Q3 : there is a Künneth formula for group cohomology. Exercice 7 : you can find the answer in any standard reference, e.g. in Weibel.

Progress

  1. 12/09/2022 Category theory: categories, functors, natural transformations, characterization of equivalences.
  2. 14/09/2022 Universal properties, initial/final objects, representable functors, Yoneda Lemma, adjoint functors, unit and counit of adjunction.
  3. 19/09/2022 Limits and colimits, examples product, coproduct, equalizer..., a category is complete iff it has products and equalizer (proof without all the details).
  4. 21/09/2022 Tensor product, isomorphisme cher à Cartan. Pre-additive category and k-category (=k-linear category for Maclane). Bibproduct.
  5. 26/09/2022 Additive categories, categories of (co)chain complexes. Simplicial methods : one can construct a chain complex starting with a (semi)simplicial abelian group.
  6. 28/09/2022 Application : singular chain complex, bar resolution. Homotopy relation for morphisms of chain complexes. Homotopy category.
  7. 03/10/2022 Abelian categories, exact functors, homology of chain complexes.
  8. 05/10/2022 Projective, injective objetcs. Projective and injective resolutions and their 'functoriality'.
  9. 10/10/2022 Derived functors: definitions and first properties, deriving bifunctors, Ext.
  10. 12/10/2022 Ext, Tor, universal coefficients theorems.
  11. 17/10/2022 Singular homology and the Eilenberg-Steenrod axioms. Homology of spheres.
  12. 19/10/2022 More properties of singular homology and singular cohomology. Appendix on Morita theory.

References

  1. For Chapters 1 and 2, Category theory in context, E. Riehl or Introduction au langage catégorique, I. Assem.
  2. For Chapters 3 and 4, Introduction au langage catégorique, I. Assem or An introduction to homological algebra, C. Weibel.
  3. For chapters 5 and 6, An introduction to homological algebra, C. Weibel.
  4. For chapter 7, An introduction to algebraic topology. J. Rotman.

Category: teaching