Homological Algebra | UCR Spring 2021

Homological Algebra

Spring 2021

University of California, Riverside

Instructor: Eloísa Grifo (please address me as Eloísa)

Course notes (no longer maintaned)

Welcome to Homological Algebra!

This course is both an introduction to homological algebra and a part 2 of sorts to the commutative algebra course I taught in Winter 2021.

Homological algebra is the study of homology in a general algebraic setting. While you may have encountered some homological ideas in algebraic topology, homological tools play a central role in other fields, and in particular in commutative algebra. We will learn about some of these tools, building on examples and motivation primarily from commutative algebra. Topics will include complexes and short exact sequences, Hom and tensor, free and projective modules, free and projective resolutions, exact functors, and Ext and Tor. Once we have built enough of these things over R-modules, we will talk a little bit about abelian categories in general, and how to generalize these constructions. We will also discuss some commutative algebra topics that have a strong homological flavor: the Koszul complex and regular sequences, depth, and Cohen-Macaulay rings.

Computing homology of complexes in general can be quite difficult. We will learn a little bit about how we can use Macaulay2 to do homological calculations. As surprising as it may sound, we can use a computer to do very abstract calculations!


Prerequisites: The official prerequisite is the Algebra sequence at UCR. If you have taken an introduction to ring theory somewhere else, I will allow you to enroll as long as you convince me you have enough background to follow. If you were not in my Winter 2021 Commutative Algebra course, you should still be able to follow most of the course, but some of the later topics rely on concepts we discussed in Commutative Algebra. If you'd like to enroll anyway, please contact me.

Course expectations: Attendance is expected. There will be 5 problem sets, which you are very welcome to (read: encouraged to!) work on together in groups; you should however write your own solutions to the problem sets. Your final grade will be based on the problem sets, although attendance will also be taken into account.

Office Hours: Mondays 3 — 4 pm, Thursdays 1 — 2 pm, and by appointment.


Course notes

Here are the Course notes. I will be updating these throughout the quarter. If you find any typos at all, however small, please let me know.

Apart from these notes, we will not follow any particular textbook, but recommended sources include:

  • Rotman's An Introduction to Homological Algebra, second edition
  • Weibel's Homological Algebra
  • Mac Lane's Categories for the working mathematician
  • Emily Riehl's Category Theory in context


    • Problem Sets

    Instructions: For each problem set, pick 5 problems to submit, including at least one problem that involves some Macaulay2 work. You are welcome to work together with your classmates on the problems, and I will be happy to give you hints or discuss the problems with you, but you should write up your solutions by yourself. Your submission should have two parts:
  • The written problems should be in a pdf file, but you are not required to type your solutions. If you prefer to handwrite your work on paper, there are apps like Scannable or Genius Scan that can scan your work into one pdf.
  • Any Macaulay2 work you want to submit should preferably be in a .m2 file. If you are a windows user and have to use Macaulay2 online, please copy your work (inputs only) to a txt file, and then rename it to be a .m2 file. You can include written comments on your Macaulay2 work in your pdf as well.

    • Schedule



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    University of California, Riverside