Homological Algebra

Rutgers MATH 560, Fall 2018.
Categories of modules, complexes and resolutions, derived functors, spectral sequences,...


Lecture 1: Introductory remarks
Lecture 2: Ad-Cats, Additive Categories, Abelian Categories
Lecture 3: Snakes, long exact sequences, chain homotopies (sections 1.3, 1.4)
Lecture 4: A look at some (derived) functors of interest, introduction to delta functors (section 2.1)
Lecture 5.1: Chasing diagrams with the Yoneda lemma
Lecture 5.2: Universal and effaceable delta functors
Lecture 6: Projective resolutions, projective modules (section 2.2)
Lecture 7: Left derived functors (section 2.4)
Lecture 9: Digression into quotients of Abelian categories
Lecture 10: There are enough injective modules
Lecture 11: Double complexes and the acyclic assembly lemma
Lecture 12: Mapping cones and balancing Tor
Lecture 13: Balancing (conclusion). Intro to group cohomology and sheaf cohomology
Lecture 14: Group cohomology
Lecture 15: Bar resolution and not quite spectral sequences
Lecture 16: What are spectral sequences?
Lecture 17: The spectral sequence of a filtered complex
Lecture 18: The spectral sequence of a double complex
Lecture 19: Hyperderived functors
Lecture 20: The hyperderived functor spectral sequences
Lecture 21: From the hyperderived functor spectral sequence to the Grothendieck spectral sequence
Lecture 22: Sheaf cohomology, de Rham cohomology
Lecture 23: Overview of derived couples. Introduction to homological dimension of rings.
Lecture 24: Some things about rings.