Mathematics Asked by GFR on August 11, 2020

I’d like to learn some homological algebra. Being a physicist my abstract algebra background is not particularly strong so I find most of the usual books a bit forbidding. I recently tried Rotman’s book "An introduction to homological algebra" and that’s pitched at exactly the right level for me.

However Rotman’s book walks a pretty long walk and getting to Tor or Ext takes forever. While that is surely interesting and conductive to a deeper understanding, for now I would prefer to have a much more superficial and instrumental understanding – being able to recognise/use homological algebra arguments where they emerge even if I don’t grasp the subject at a deeper level or have not seen a proof of the big theorems.

To give an explicit example, I would like to be able to follow the argument in the proof of Lemma 6.22 of Kirillov "An introduction to Lie groups and Lie algebras", where Ext and some homological algebra are used to prove that $H^1(mathfrak{g},V)=0$ for any representation $V$. Note that I am not especially interested in the result/proof itself, which surely can be reformulated to avoid the use of homological algebra, this is just an example of the kind of understanding of the subject I would like to get to.

Not being familiar with the subject I find it hard to devise by myself a shorter route through Rotman’s book.

Can anyone recommend a homological algebra book or a set of notes which starts at about the same level as Rotman’s but takes a quicker if less in depth route?

I think that chapter 2 of the following notes may be enough. Depending on your background, perhaps at some point you will have to consult a few pages of chapter 0, but most of the contents of chapter 1 are not necessary for chapter 2. So I think you will need to read less than 30 pages in total. It is much faster for your purposes than Rotman's while the level is similar:

S. RAGHAVAN, R. BALWANT SINGH and R. SRIDHARAN, Homological mathods in Commutative Algebra

Correct answer by SlavaM on August 11, 2020

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