I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra.
Now, this sort of question has been asked many times on forums such as mathoverflow (viz. here
and here) so, let me try to make more specific what I'd like to understand, which isn't covered in either of the responses to these questions. I have a (with no small amount of help from the first link I posted) a fairly good intuition for the uses of homology in topology, what I am more interested in is the intuition for the use of homological algebra in "pure algebra". I want to understand better why studying what we study in homological algebra gives us valuable information about the ring over which we are concerned–what information does it exactly give us? Of course, for commutative rings the question is answered definitively by Morita's theorem which tells us that $R\text{-}\mathbf{Mod}$ is categorially equivalent to $S\text{-}\mathbf{Mod}$ implies that $R$ and $S$ are isomorphic as rings. Fine, I can see why this obviously gives us motivation to study some of the things we study in homological algebra, but the only problem is I have no intuition for why Morita's theorem should be true. Can anyone elucidate this, in the most simple terms possible?
While this very well may be equivalent to what I have asked in the above paragraph (if so, feel free to concatenate answers) I was wondering if someone could more fully explain Eisenbud's analogy that homological algebra is to ring theory as representation theory is to group theory (in your own opinion, I know you don't know what he was thinking).
Lastly, for me, to get a basic motivation/intuition it is necessary for me to see how powerful a subject $X$ can be, in the sense that it can answer questions which (ostensibly!) have nothing to do with subject $X$. The classic example being that Burnside's theorem has nothing directly to do with representation theory (there is no use of character theoretic language in its statement) yet the only "simple" proof uses character theory. Unfortunately, in the realm of pure algebra I have been able to find very few examples of such uses of homological algebra–the only exception being the Schur-Zassenhaus theorem. So, any (as elementary possible) applications of homological algebra to problems in more elementary algebra (group theory, module theory, ring theory, and to some extent [but preferably less so] commutative algebra) where the statements would seem to suggest that the proof could be self-contained, yet realistically requires homological algebra would be great.
Thank you very much friends, help with any of these questions would go a LONG way to helping a very excited, and eager learner of homological algebra.
Best Answer
I certainly won't try to give a general philosophical answer to your question, but I'll mention a success story that persuaded specialists that homological algebra was an amazingly powerful tool in commutative algebra.
Auslander, Buchsbaum and Serre proved that a local noetherian ring is regular if and only if it has finite global dimension.
From this it is easy to deduce that the localization at a prime ideal of a regular local ring is still regular.
The statement of that result has nothing to do with homological algebra but since nobody had managed to prove it before, without homological algebra, this duly impressed algebraists.
Optional technicalities
Let me give some relevant definitions here.
A noetherian local ring $(R,\mathfrak m)$ is called regular if its maximal ideal can be generated by $dim (A)$ (=Krull dimension of $A$) elements.
This definition, due to Zariski, is a purely algebraic way of ensuring that an algebraic variety has no singularities.
The global dimension of the ring $A$ is the supremum of the projective dimensions $pd_AM$ of its modules $M$.
And $pd_AM$ is the infimum of the lengths of resolutions $0\to P_n\to...\to P_0\to M\to0$ of $M$ by projective $A$-modules $P_i$.