Homological Algebra | UNL Fall 2023

Basic information

Welcome to Homological Algebra!

This course is an introduction to homological algebra; we will assume that nobody has seen homological algebra before.

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. We will learn about some of these tools, building on examples and motivation primarily from the world of modules. Before we start discussing homological algebra, we will need an introduction to category theory; again, no familiarity with categories and categorical notation will be assumed. The remainder of the class will be an introduction to homological algebra. Topics will include complexes and short exact sequences, homology, 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. Towards the end of the course, we will introduce spectral sequences.

Computing homology of complexes in general can be quite difficult. As surprising as it may sound, we can use a computer to do very abstract calculations! Sadly, we will not have the chance to learn how we can use Macaulay2 to do homological calculations; those students who are familiar with Macaulay2 are welcome to ask me about this outside of class.

Prerequisites: The official prerequisite is the Algebra sequence at UNL (Math 817-818). If you have taken an introduction to ring and module theory somewhere else, I will allow you to enroll as long as you convince me you have enough background to follow.

Course expectations: Attendance is expected, as much as possible, but please do NOT attend class if you are feeling ill or may have been exposed to COVID-19. You are encouraged to work on the problem sets together in groups, or discuss them with me; you should however write up your own solutions. The only other resources you are allowed to use to solve the problem sets are our class notes. The midterm and final exam will be take-home and look a lot like the problem sets, except you cannot collaborate on the problems, and you are encouraged to try to solve the problems without checking the class notes.

Office Hours: Mondays 5 — 6 pm, Fridays 2 — 3 pm, and by appointment.

Course notes

Here are the Course notes. I will be updating these throughout the semester. 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

(important! The first and second edition are two completely different books)

Weibel's Homological Algebra

Mac Lane's Categories for the working mathematician

Emily Riehl's Category Theory in context

Eisenbud's Commutative Algebra with a view towards algebraic geometry (see the appendices)

Problem Sets

Instructions: 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. You will submit your work on canvas; your submission should be 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.

Problem Set 1: due September 12 (tex)

Problem Set 2: due September 29 (tex)

Problem Set 3: due October 13 (tex)

Midterm: due October 30 (tex)

Problem Set 4: due December 1 (tex)

Final Exam: due December 11 (tex)

Schedule

Lecture 1 (Monday, August 21): Complexes and homology. Exact sequences.

Lecture 2 (Wednesday, August 23): Categories for the working homological algebraist: an introduction.

Lecture 3 (Friday, August 25): Functors.

Lecture 4 (Monday, August 28): Natural transformations.

Lecture 5 (Wednesday, August 30): Optional class on the Yoneda Lemma.

Lecture 6 (Friday, September 1): Products and coproducts.

Lecture 7 (Wednesday, September 6): Limits.

Lecture 8 (Friday, September 8): Colimits. Adjointness.

Lecture 9 (Monday, September 11): The category Ch(R).

Lecture 10 (Wednesday, September 13): More about Ch(R). Split short exact sequences.

Lecture 11 (Friday, September 15): The Snake Lemma.

Lecture 12 (Monday, September 18): Finishing the proof of the Snake Lemma. The long exact sequence in homology.

Lecture 13 (Wednesday, September 20): The category R-Mod. Additive functors.

Lecture 14 (Friday, September 22): Right and left exact functors. Hom is not right exact.

Lecture 15 (Monday, September 25): Hom is left exact.

Lecture 16 (Wednesday, September 27): Tensor products.

Lecture 17 (Friday, September 29): Properties of the tensor product.

Lecture 18 (Monday, October 2): More properties of the tensor product. Extension of scalars.

Lecture 19 (Wednesday, October 4): Tensor is right exact but not left exact. Localization as a tensor product.

Lecture 20 (Friday, October 6): Localization as a tensor product. Module structures on Hom over noncommutative rings.

Lecture 21 (Monday, October 9): Hom-tensor adjunction.

Lecture 22 (Wednesday, October 11): Projectives.

Lecture 23 (Friday, October 13): For finitely generated modules over a local commutative ring, projective=free.

Lecture 25 (Wednesday, October 18): Injectives.

Lecture 26 (Friday, October 20): Baer's criterion. More about injectives.

Lecture 27 (Monday, October 23): Divisible vs injective modules.

Lecture 28 (Wednesday, October 25): Every module embeds into an injective module.

Lecture 29 (Friday, October 27): Flat modules.

Lecture 30 (Monday, October 30): More about flat modules.

Lecture 31 (Wednesday, November 1): Every module has a free resolution. A sidenote about graded stuff.

Lecture 32 (Friday, November 3): Minimal free resolutions.

Lecture 33 (Monday, November 6): Some useful but technical facts about free resolutions.

Lecture 34 (Wednesday, November 8): Minimal free resolutions are unique up to isomorphism.

Lecture 35 (Friday, November 10): The Horseshoe Lemma.

Lecture 36 (Monday, November 13): Derived Functors.

Lecture 36 (Wednesday, November 15): How to recognize a sequence of functors as the derived functors of a given F.

Lecture 37 (Friday, November 17): Ext and Tor: computations and examples.

Lecture 38 (Monday, November 20): Ext and Tor: a rough outline of the proofs that Ext and Tor are balanced.

Lecture 39 (Monday, November 27): Other examples of derived functors.

Lecture 40 (Wednesday, November 29): Additive categories.

Lecture 41 (Friday, December 1): Abelian categories.

Lecture 42 (Monday, December 4): Spectral Sequences.

Lecture 43 (Wednesday, December 6): Spectral Sequences.

Lecture 44 (Friday, December 8): Spectral Sequences.

Here are some resources regarding spectral sequences:

Clover May's videos on spectral sequences

Spectral Sequences: friend or foe? by Ravi Vakil

Hatcher's extra chapter on spectral sequences as a late addition to his algebraic topology book

McCleary's A user's guide to spectral sequences

Weibel's An introduction to homological algebra

Mel Hochster's webpage, which includes various old course notes about many topics, including spectral sequences

Here are some resources regarding tensor products:

How to conquer tensorphobia (warning: this article only talks about tensor products of vector spaces)

Why are tensor products scary?

What is the point of tensor products?

Here are some other random resources:

Why every ring should have a 1

What exactly is an arrow?

nLab, a math wiki written from the perspective of (higher) category theory

Macaulay2, a commutative algebra software

Macaulay2 online, a way to run Macaulay2 without installing it

University of Nebraska-Lincoln