Course Descriptions

16:640:560 - Homological Algebra

Fall 2022

Vladimir Retakh

Course Description:

This will be an introduction to Homological Algebra. Homological Algebra is a tool used in many 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. In addition, 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 apply this to several topics such as Homology of Groups and Lie Algebras.


An introduction to homological algebra, by C. Weibel, Cambridge University Press


Math 551, 552


Spring 2021

Ian Coley

Course Description:

Homological algebra is an essential tool in algebraic geometry and algebraic topology and find use in representation theory, Lie theory, and much more. Roughly, it is a first step in applying some category theory to questions of concrete interest. Main topics: chain complexes, abelian and triangulated categories, derived functors, (co)homology. Actual computations will be emphasized!


An introduction to Homological Algebra, Weibel [other resources TBD]


451-2 (or permission)


Schedule of Sections:


Previous Semesters:

Fall 2018

Daniel Krashen


An introduction to homological algebra, by C. Weibel, Cambridge U. Press


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