Getting comfortable with homological algebra

[Tiger99]

Hello, I've been trying to learn about cohomology theory (particularly in relation to Lie algebras, the Hochschild cohomology and deformations of algebras) and I never seem to get very far.

It seems there's a bit of a jump between knowing what Ext and Tor mean and actually feeling comfortable with them and feeling confident that I can actually calculate them, and knowing when you can calculate them. Maybe I'm just being lazy. I have to keep looking up the definitions of each every time I see them being used in something (I'm thinking of making myself a little card to put in my wallet). Does anyone have some tips? Are there some exercises that you worked through to get the hang of it?

 

[morphism]

Are you using Weibel's book? It has a lot of good exercises, and the text itself is really nice. It even has a chapter on Lie algebra cohomology and another on Hochschild stuff.

In any case, it's pretty common to feel overwhelmed when you start dabbling in homological algebra. I think one of the main reasons for this is that the question "What the hell is this crap good for?" doesn't have a convincing answer for the beginner. The only way to get a convincing answer is, in my experience, by simply keeping at it and learning a lot of concrete math which uses homological algebra constantly--two obvious candidates here are topology (both differential and algebraic) and algebraic geometry.

For example, Ext and Tor show up in topology as obstructions to what would be the naive change of base-group operation in cohomology and homology. Look up the "universal coefficient theorem" in any algebraic topology book (IIRC Hatcher had a good exposition of this) to see what I mean. I found this helped me really appreciate the "obstruction"ness of Ext and Tor, although the usual spiel about extension of modules and free resolutions is actually pretty illuminating.

I also recommend the article by Burt Totaro on Algebraic Topology in the Princeton Companion to Mathematics. It's really well-written and I think it gives the reader a good general idea of why homology and cohomology are useful.

In the same vein I recommend the book by Dieudonné called something like "A History of Differential and Algebraic Topology". I wouldn't read this with the goal of understanding what each sentence means, but rather I'd use the book to help me draw a mental image of how homological algebra came to be.

Also, for Lie algebra cohomology, have you checked out the Chevalley-Eilenberg paper? It might be instructive to go over it and compare with Weibel.

 

[Tiger99]

Thanks Morphism for all your advice.

Yes, I've been looking at Weibel's book. I went away for the summer and couldn't take it with me, but now I'm back home so I will try to read it again, and look at the book by Cartan and Eilenberg too, and the Chevalley and Eilenberg paper.

I suppose I was just being lazy. There are many little things that trip me up and probably shouldn't. For example, I had trouble getting comfortable with projective resolutions, because I don't think I would recognise a non-projective module when I see one. At first, actually I couldn't think of modules that weren't projective (by looking at the diagram definition).

Now with the help of the internet, I know that Q is not projective as a Z-module (and I understand that one), and abelian groups are projective as Z-modules only if they're free.

It seems like I need to know a bit more of everything.

Thanks again.