Cartan-Eilenberg was written long before it was published the preface is dated and copies of the manuscript were available to Serre and Grothendieck in the early fifties as you can read in their Correspondance - they call it Cartan-Sammy. The main thrust of the book is that the various cohomology theories could be cast in terms of derived functors. It was only Grothendieck who proved the existence of enough injectives in his Tohoku paper, for example. If they could not come up with the proof of the existence of enough injective objects, I think it could be an answer to my question. He worked on a special kind of abelian categories called Grothendieck categories nowadays.
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Main An introduction to homological algebra An introduction to homological algebra Joseph J. Rotman With a wealth of examples as well as abundant applications to algebra, this is a must-read work: an easy-to-follow, step-by-step guide to homological algebra. The author provides a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology.
In this brand new edition the text has been fully updated and revised throughout and new material on sheaves and abelian categories has been added.
Learning homological algebra is a two-stage affair. Firstly, one must learn the language of Ext and Tor, and what this describes. Secondly, one must be able to compute these things using a separate language: that of spectral sequences. The basic properties of spectral sequences are developed using exact couples.
All is done in the context of bicomplexes, for almost all applications of spectral sequences involve indices. Applications include Grothendieck spectral sequences, change of rings, Lyndon-Hochschild-Serre sequence, and theorems of Leray and Cartan computing sheaf cohomology.