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So as the question suggests, I am looking for a reference on homological algebra. I am aware of the a lot of the standard texts available, and they have been very helpful for me getting familiar with the concepts. However, I was looking for someone with a particular perspective. I am in the process of learning the homological algebra as it pertains to algebraic geometry. I am starting to realize that there are some technical gaps in my background that I wanted to fill.

What I am really looking for is a book/article/lectures that develops homological algebra from the simplest possible setting and gradually generalizes. What I had in mind was something that starts with the category of abelian groups and develops the homological algebra there, then moves to the category of, say, modules over a PID, then moves, say, to modules over a general ring, and finally (this may be too much to ask) homological algebra in a Grothendieck category. Is there any such text available?

I've also tagged it as a soft question since I am looking for suggestions as to whether people think such an approach would even be fruitful or helpful at all?

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The number of theorems in homological algebra that don't involve results for specific rings is small. Those theorems wouldn't be any clearer if you assumed yourself to be in the category of abelian groups versus a general ring. You should just read Weibel's book in detail. There he does what you say: he develops some fundamental machinery and in Chapter 3 and 4 he proceeds through various rings, indeed starting with abelian groups. Those chapters basically applies the machinery of homological algebra to see what can be said about specific rings and modules.

In addition, you should check out Beilinson's notes. Lubkin's book "Cohomology of Completions" does homological alegebra in a general abelian category and for most people this approach is unreadable, but curious and of academic interest.

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While Weibel's book is the first references everyone gives, I do prefer Rotman's "An introduction to homological algebra". There are two versions of it, one from 79' and the other from 2010. The second one is far more complete, also he does ALL the details that Weibel omits or gives as (very non-trivial) exercises. He has a hole chapter to specific rings, and another one introducing sheaves, also working all the details.

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