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Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at the same time as) the publication of the book "Homological algebra". Cartan was an expert on sheaf theory. So I think he was well aware of the possibility. I think it's strange.

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Just some quick thoughts: 1. Cartan-Eilenberg was written long before it was published (the preface is dated 1953 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). 2. The main thrust of the book is that the various cohomology theories could be cast in terms of derived functors. At that point in time sheaf theory simply wasn't that far developed. It was only Grothendieck who proved the existence of enough injectives in his Tohoku paper, for example. –  t.b. Jul 4 '12 at 11:13
    
See also Cartan's contribution to this collection of articles on Eilenberg in the Notices for some interesting background (also on the genesis of the book). –  t.b. Jul 4 '12 at 11:16
    
@t.b. 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. But it's difficult to imagine. The Grothendieck's proof mimicked their proof on modules over rings. He worked on a special kind of abelian categories(called Grothendieck categories nowadays). However, one can prove the theorem on sheaves of modules over a ringed space directly, i.e. wihout the theory of abelian categories. –  Makoto Kato Jul 4 '12 at 21:08
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In retrospect it seems quite easy, yes, and I'm sure if they had decided to do so they could have done it. But: Were ringed spaces even conceived of at that point? Was it clear that it would be worthwhile to pursue such a study? I'm not sure. They had their students and collaborators working on it. There was Cartan's seminar and Buchsbaum's thesis, for example. Later Heller and Grothendieck. Godement's book grew out of Bourbaki's efforts to come to grips with sheaf theory in which Cartan was directly involved. Besides, there was a lot of other things the two of them pursued at the same time... –  t.b. Jul 4 '12 at 22:20
    
Of course, Grothendieck's Tohoku paper was influenced greatly by their idea. However, it seems that he hadn't seen the manuscript of the book before he submitted the paper. In his letter to Serre, Grothendieck seemed to be worried if a part or most of the content of his paper was included in their book. –  Makoto Kato Jul 5 '12 at 5:03
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