# Founding of homological without quite involving derived categories

I am looking at the foundations of homological algebra, e.g. the introduction of Ext and Tor, and am unsatisfied. The references I look at start with "this is called a projective module, this is called a projective resolution, now pick one and use it to define the right derived functors of your left exact functor". I would like to see a presentation more along the following lines:

1. The functor Hom(A,*), applied to a short exact sequence of modules, doesn't produce another such. An oracle tells us that it does produce a long exact sequence; what could it be?

2. We already know (from antiquity) that a short exact sequence of complexes induces a long exact sequence on cohomology.

3. But in #1 we put in modules, not complexes. So let's fix that by hoping that Hom(A,*) can be extended in a natural way to the category of complexes (and really, to descend to the derived category).

4. Such an extension might be required to have the following properties: ???

5. Now I'd like it to be easy to see that the extension is unique if it exists. When is it easy to compute? At this point I'd like the definition of "projective module" to suggest itself.

6. Finally, the usual boring checks that using projective resolutions to define it, the extension does indeed exist.

One way to answer this is to say "In part 4, define the derived category, and its t-structure, then ask that the extension be exact in the appropriate sense". I'm hoping to avoid going quite that far, or at least, doing it in a way that doesn't involve introducing too many more definitions.

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Grothendieck Tohoku lecture and B.Keller's lecture on Stable categories might be the proper reference – Shizhuo Zhang Jan 7 '10 at 2:38
I know you already saw Anton's great answer to (essentially) this question, but I just have to recommend Gunter Harder's excellent book "Lectures on Algebraic Geometry 1" (which is actually primarily about homological algebra). His approach is exactly Anton's, and if you ever need to teach this stuff it might be a nice one to use. Check it out! – Steven Gubkin Jun 10 '10 at 1:31