Introduction to Homological Algebra
This course develops the fundamental concepts and techniques of homological algebra. We begin with chain complexes, homology, exact sequences, connecting homomorphisms, and chain homotopies. We then study projective, injective, and flat modules, resolutions, and the construction of derived functors. Particular attention will be given to the functors Tor and Ext, their computational properties, and their roles in measuring failures of exactness and classifying extensions. The course concludes with homological dimension, spectral sequences, and an introduction to derived categories.
Lecturer
Date
15th September ~ 8th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 13:30 - 16:55 | Shuangqing | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Abstract algebra: groups, rings, modules.
Syllabus
1) Categorical and algebraic preliminaries (Categories, functors, natural transformations, additive and abelian categories, kernels, cokernels, and exact sequences).
2) Chain complexes and homology (Chain and cochain complexes, cycles, boundaries, homology, cohomology, exact complexes, and basic examples).
3) Exact sequences of complexes (Short and long exact sequences, connecting homomorphisms, the Snake Lemma, and naturality).
4) Chain maps and chain homotopies (Homotopy equivalences, quasi-isomorphisms, mapping cones, and the homotopy category of complexes).
5) Projective, injective, and flat modules (Characterizations and examples, projective and injective resolutions, and uniqueness of resolutions up to homotopy).
6) Derived functors (Left and right derived functors, the comparison theorem, long exact sequences, dimension shifting, and acyclic resolutions).
7) The functor Tor (Construction using projective and flat resolutions, computational methods, long exact sequences, and the relation between Tor and flatness).
8) The functor Ext (Construction using projective and injective resolutions, long exact sequences, extensions of modules, and the Yoneda interpretation of Ext).
9) Homological dimension (Projective, injective, and flat dimensions, global dimension, dimension-shifting arguments, examples, and computations).
10) Introduction to spectral sequences (Filtered complexes, exact couples, pages and differentials, convergence, and edge homomorphisms).
11) Applications of spectral sequences (Spectral sequences of double complexes, comparison results, and computations involving derived functors).
12) Introduction to derived categories (The homotopy category, localization at quasi-isomorphisms, distinguished triangles, derived functors, and a review of the principal ideas of the course).
2) Chain complexes and homology (Chain and cochain complexes, cycles, boundaries, homology, cohomology, exact complexes, and basic examples).
3) Exact sequences of complexes (Short and long exact sequences, connecting homomorphisms, the Snake Lemma, and naturality).
4) Chain maps and chain homotopies (Homotopy equivalences, quasi-isomorphisms, mapping cones, and the homotopy category of complexes).
5) Projective, injective, and flat modules (Characterizations and examples, projective and injective resolutions, and uniqueness of resolutions up to homotopy).
6) Derived functors (Left and right derived functors, the comparison theorem, long exact sequences, dimension shifting, and acyclic resolutions).
7) The functor Tor (Construction using projective and flat resolutions, computational methods, long exact sequences, and the relation between Tor and flatness).
8) The functor Ext (Construction using projective and injective resolutions, long exact sequences, extensions of modules, and the Yoneda interpretation of Ext).
9) Homological dimension (Projective, injective, and flat dimensions, global dimension, dimension-shifting arguments, examples, and computations).
10) Introduction to spectral sequences (Filtered complexes, exact couples, pages and differentials, convergence, and edge homomorphisms).
11) Applications of spectral sequences (Spectral sequences of double complexes, comparison results, and computations involving derived functors).
12) Introduction to derived categories (The homotopy category, localization at quasi-isomorphisms, distinguished triangles, derived functors, and a review of the principal ideas of the course).
Reference
Charles A. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
No
Language
English
Lecturer Intro
Prof. Sergei Ivanov is a mathematician from St. Petersburg, Russia. His research interests include homological algebra, algebraic topology, group theory, simplicial homotopy theory, simplicial groups.