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?

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    $\begingroup$ You could check out the first 100 pages or so of Weibel. For the argument you cite, all you need is the long exact sequence on homology which is very accessible once you have some basic algebra. $\endgroup$ – Justin Young Aug 7 '20 at 11:47
  • $\begingroup$ @JustinYoung thanks I'll have a look $\endgroup$ – GFR Aug 7 '20 at 16:33

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

  • $\begingroup$ Thanks, this seems to be along the lines I had in mind $\endgroup$ – GFR Aug 7 '20 at 11:17

Try Robert B. Ash's notes entitled Abstract Algebra: The Basic Graduate Year, especially Chapter 10 on homology and the accompanying Supplement. Also available in book form as Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates. In the preface he writes:

I have attempted to communicate the intrinsic beauty of the subject. Ideally, the reasoning underlying each step of a proof should be completely clear, but the overall argument should be as brief as possible, allowing a sharp overview of the result. These two requirements are in opposition, and it is my job as expositor to try to resolve the conflict.

See also the end of the preface for his remarks on homological algebra:

In Chapter 10, we introduce some of the tools of homological algebra. Waiting until the last chapter for this is a deliberate decision. Students need as much exposure as possible to specific algebraic systems before they can appreciate the broad viewpoint of category theory. Even experienced students may have difficulty absorbing the abstract definitions of kernel, cokernel, product, coproduct, direct and inverse limit. To aid the reader, functors are introduced via the familiar examples of hom and tensor. No attempt is made to work with general abelian categories. Instead, we stay within the category of modules and study projective, injective and flat modules.

In a supplement, we go much farther into homological algebra than is usual in the basic algebra sequence. We do this to help students cope with the massive formal machinery that makes it so difficult to gain a working knowledge of this area. We concentrate on the results that are most useful in applications: the long exact homology sequence and the properties of the derived functors Tor and Ext. There is a complete proof of the snake lemma, a rarity in the literature. In this case, going through a long formal proof is entirely appropriate, because doing so will help improve algebraic skills. The point is not to avoid difficulties, but to make most efficient use of the finite amount of time available.


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