Homological Algebra

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

daniel.krashen@rutgers.edu

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.