MATH 6290

Homological Algebra

This document was last modified on: UTC

This is a rough guide to the topics we will cover.

Homework will be assigned in class.

Back to main page

Date Topics covered
Monday January 13
Wednesday January 15
Review category theory and algebra
Friday January 17
Review cont.
Monday January 20
Wednesday January 22
Chain complexes
Complexes of R-modules, operations on chain complexes, long exact sequences.
Friday January 24
Chain homotopies.
Monday January 27
Mapping cones and mapping cylinders.
Wednesday January 29
Mapping cones and mapping cylinders cont.
Friday January 31
More on abelian categories.
Monday February 3
Derived Functors
Delta functors, projective resolutions, injective resolutions.
Wednesday February 5
Left derived functors, right derived functors.
Friday February 7
Adjoint functors and left/right exactness.
Monday February 10
Balancing Tor and Ext.
Wednesday February 12
Derived categories (preview).
Friday February 14
Tor and Ext
Tor for abelian groups, Tor and flatness.
Monday February 17
Ext for nice rings.
Wednesday February 19
Ext and extensions.
Friday February 21
Universal coefficients theorem
Monday February 24
Cup product
Wednesday February 26
Homological dimension
Dimensions, change of rings theorems.
Friday February 28
Local rings.
Monday March 3
Koszul complexes.
Wednesday March 5
Local cohomology.
Friday March 7
Cohen--Macaulay rings.
Monday March 10
Cohen--Macaulay rings.
Wednesday March 12
Spectral sequences
Introduction, terminology.
Friday March 14
The Leray--Serre spectral sequence.
Monday March 17
The spectral sequence of a filtration.
Wednesday March 19
Friday March 21
Spectral sequences of a double complex.
Week of March 24-March 28
Monday March 31
Hypercohomology (derived categories preview).
Wednesday April 2
Grothendieck Spectral Sequences, applications.
Friday April 4
Exact couples.
Monday April 7
The derived category
Introduction, categories of complexes.
Wednesday April 9
Triangulated categories.
Friday April 11
Triangulated categories continued.
Monday April 14
Localization and the calculus of fractions.
Wednesday April 16
The derived category.
Friday April 18 Derived functors.
Monday April 21
The total tensor product.
Wednesday April 23
Ext and RHom.
Friday April 25
Replacing spectral sequences.
Monday April 28
Wednesday April 30
Friday May 2
Wednesday May 7

Back to main page

I strongly encourage everyone to use LaTex for typing their homework.  If you have a mac, one possible easy way to get started is with texshop.  If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc.  If you are using windows, you're on your own, but I'm sure there's something online.  Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file).  It is probably easiest to just look at the .tex file, and start to experiment.  There is also the (Not so) short introduction to latex, which will answer most of your questions (although typing your question into your favorite search engine will probably work well, too).  You may also want to try for a cloud version.