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:
- Introduction to category theory.
- Universal properties, representable functors and adjoint functors.
- Tensor product of bimodules.
- Additive categories and their categories of (co)chain complexes.
- Abelian categories and homology.
- Derived functors, Ext and Tor.
- Yoneda's Ext.
- Application to algebraic topology: singular homology.
There might be some changes, depending on our progess.
Handwritten notes
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Exercise sheets
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Solutions for the exercises
Sheet 1
Sheet 2
Sheet 3
Exercise 2
Sheet 4
Exercise 6
Progress
- Introduction to category theory.
- Universal properties, representable functors and adjoint functors.
- Tensor product of bimodules.
- Additive categories and their categories of (co)chain complexes.
- Abelian categories, chain complexes, homology, long exact sequence
- Projectives and injectives objects. Projective resolutions.
- Derived functors, long exact sequence, exemple of Ext and Tor.
References
- For Chapters 1 and 2, Category theory in context, E. Riehl or Introduction au langage catégorique, I. Assem.
- For Chapters 3 and 4, Introduction au langage catégorique, I. Assem or An introduction to homological algebra, C. Weibel.
- For chapters 5 and 6, An introduction to homological algebra, C. Weibel.
- For chapter 8, An introduction to algebraic topology. J. Rotman.