Fall 2025
Lev Borisov
Course Description:
This will be an introduction to Homological Algebra. Homological Algebra is used in various branches of mathematics, especially in Algebra, Topology and Algebraic Geometry.
The first part of the course will cover Chain Complexes, Projective and Injective Modules, Derived Functors, Ext and Tor. Some basic notions of Category Theory will be presented: adjoint functors, abelian categories, natural transformations, limits and colimits.
The second part of the course will study Spectral Sequences, and their applications depending of the interests of the students in the class.
Text:
An introduction to homological algebra, by C. Weibel, Cambridge University Press, paperback edition
Prerequisites:
First semester of graduate algebra.
Schedule of Sections:
16:640:560 Schedule of Sections
Previous Semesters:
Fall 2025 - Borisov, Lev
Fall 2022 - Retakh, Vladimir
Spring 2021 - Coley, Ian
Fall 2018
Daniel Krashen
Text:
An introduction to homological algebra, by C. Weibel, Cambridge U. Press
Prerequisites:
First-year knowledge of groups and modules.
Course Description:
From some perspectives, homological algebra is the study of the failure of modules over rings (and related objects) to behave like vector spaces. Somewhat more precisely, homological algebra collects a number of ideas and tools with origins in topology, such as chain complexes and derived functors, to study categories of modules and other Abelian categories. Homological algebra is fairly ubiquitous, and finds applications in
module theory over rings
representations of groups and Lie algebras
sheaves on algebraic varieties
algebraic topology
various other subjects