# Prerequisites and references for homological algebra

I'm very interested in learning Homological Algebra, but I'm not sure about the prerequisites for learning it.

My current knowledge in algebra consists of Abstract Algebra (groups, rings, and fields), Linear Algebra, Galois Theory, Basic Module Theory and some introduction to Category Theory. Also I'm currently enrolled in a Commutative Algebra Course (using Atiyah's classical text).

Most of you are experts of mathematics, so, according to you, what are some great reasons to study Homological Algebra? I'd prefer that you recommend books (for self-study) as well as abstract subjects that I should learn before learning Homological Algebra. Also, what are the good texts/notes/video lectures on Homological Algebra according to my background?

• It seems like you have enough background. I don't know that you should get convinced to study something like this by random people on the internet. Homological algebra is used in many parts of mathematics but I'm not sure that many people sit down and read a book about it. You learn something that needs parts of it and you pick some stuff up and use it, and then one day you wake up and you're an "expert". If you are using Atiyah (and let's not forget Macdonald) then you'll find a few exercises that assume knowledge of $\operatorname{Tor}$ and stuff like that.
– Hoot
Aug 22, 2015 at 4:04
• If those exercises pique your interest then look into those applications. As a dumb example: prove that if you have an exact sequence $\mathscr{C}: 0 \to M' \to M \to M'' \to 0$ and $M''$ is flat then $\mathscr{C} \otimes N$ is always exact. Do it with and without $\operatorname{Tor}$. Somehow the homological algebra encapsulates a diagram chase that you should really only ever have to do once. Part III of Eisenbud's book has way more sophisticated examples. It's filled with typos but it's very interesting.
– Hoot
Aug 22, 2015 at 4:08
• @Hoot Thank you for your detailed comment.Regards, Aug 22, 2015 at 4:25
• If you want a quick introduction to homological algebra, then J.J. Rotman, An Introduction to Homological Algebra, 1979 is a marvelous textbook. If you want to spend more time on homological algebra, then the second edition of the same book (published in 2009) is also a good choice. Aug 22, 2015 at 18:16
• @user26857 Thank you,Regards, Aug 23, 2015 at 5:52

At Cornell, we had recently a class on Homological algebra taught by Yuri Berest. I think you have enough background for reading the notes to that class. You can find them here. I think he did quite a good job of carefully going through the main basic things like abelian and triangulated categories, derived functors et.c. But at the same time, he was trying to give plenty of non-trivial examples and applications.

I helped with editing the notes, so I am sorry if I self-advertise a bit.

• These notes are too advanced for a beginner in homological algebra. I suppose that one wants to know first what are Ext and Tor, not to learn about Hochschild (co)homology. (Or at least this is the way I did it some time ago.) Aug 22, 2015 at 20:51
• @user26857 Well, first of all, Hochschild homology and cohomology are just Tor and Ext! You might be right, though... Maybe they are a bit advanced. But I think if one have read Atiyah-Macdonald, one should probably know what Tor and Ext are (if I remember correctly, they talk about it a bit). Aug 22, 2015 at 23:04

Weibel's An Introduction to Homological Algebra is a standard text.

• Not that good book, there are plenty of typos and misprints. I had to use Robert Ash's notes from his website to clarify me some stuff from Weibel's book which wasn't explain good enough. Aug 21, 2015 at 12:09
• @MathematicalPhysicist So can you suggest a "good" text on Homological Algebra? Aug 21, 2015 at 12:14
• I said that Robert Ash's notes are good introduction: math.uiuc.edu/~r-ash/Algebra/Chapterr10.pdf you can also try Saunders Maclane classic: Homology - amazon.com/Homology-Classics-Mathematics-Saunders-MacLane/dp/… Aug 21, 2015 at 12:19
• @MathematicalPhysicist ,@TorsionSquid I'll Surely have a look in the suggested texts.Regards, Aug 21, 2015 at 12:27
• How about Gel'fand Manin? There are some typos, but I remember it was not too bad (more approachable than MacLane, I think) Aug 22, 2015 at 4:15