Math 750 -- Homological algebra
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)
- Naive homological algebra: derived functors.
- Ext for abelian groups
- Derived functors in category of modules. Examples: Ext, Tor
- Abelian categories. Examples.
- Spectral sequences.
- Modern homological algebra: derived categories.
- Resolution of complexes.
- Derived categories and derived functors. Examples
- Triangulated categories.
- Sophisticated homological algebra: beyond triangulated categories.
- DG categories.