Math 750 -- Homological algebra

Spring 2018

Homework assignments

Course content (probably overly optimistic)

This course is an introduction to homological algebra. I hope to cover the following topics.

Derived functors

Many constructions in mathematics lead to functors that fail to be exact (do not respect exact sequences). This issue can often be corrected by introducing the so called derived functors; we will look at the construction and numerous examples from commutative algebra/algebraic geometry, representation theory, and topology. Important ideas: injective/projective resolutions, acyclic objects, (co)homological dimension, spectral sequences.

Derived categories

Derived functors can be computed as cohomology objects of certain complexes. However, in this process we lose some information: it is often important to remember the complex itself. This path naturally leads to derived categories; roughly speaking, the idea is to identify an object with all of its resolutions. Important ideas: quasi-isomorphism, cone of morphism, triangulated categories, resolutions of complexes, homotopy category of complexes.

Beyond triangulated categories

As we will see, viewing the derived categories as triangulated categories is sometimes less than ideal. To resolve this issue, one has to replace traingulated categories by better frameworks, such as dg-categories or infinity-categories. Important ideas: dg-algebras and dg-categories, homotopy category of a dg-category.

There are many other important and interesting topics, which I would be happy to discuss if the time permits. However, this seems incredibly unlikely, so perhaps they may be more appropriate for, say, a talk at a seminar for graduate students. Some such topics are compact objects, compactly generated categories, the Brown Representability Theorem, model categories, dg-Lie algberas and deformation theory, and others.

Plan (probably overly optimistic)

1. Naive homological algebra: derived functors.
   1. Ext for abelian groups
   2. Derived functors in category of modules. Examples: Ext, Tor
   3. Abelian categories. Examples.
   4. Spectral sequences.
2. Modern homological algebra: derived categories.
   1. Resolution of complexes.
   2. Derived categories and derived functors. Examples
   3. Triangulated categories.
3. Sophisticated homological algebra: beyond triangulated categories.
   1. DG categories.