6
$\begingroup$

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.

  • 6
    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.2012-07-04
  • 0
    See also Cartan's contribution to [this collection of articles](http://www.ams.org/notices/199810/mem-eilenberg.pdf) on Eilenberg in the Notices for some interesting background (also on the genesis of the book).2012-07-04
  • 0
    @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.2012-07-04
  • 2
    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...2012-07-04
  • 0
    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.2012-07-05
  • 0
    You're right. I can't vouch for complete accuracy in the details in what I wrote (hence a comment, not an answer). I don't have the time to do the leg-work and collect all the sources. It seems to me that it is rather safe to say that the time just wasn't quite ripe for including sheaf theory in the book at the beginning of the fifties. Maybe the historical references I give in my answer [here](http://math.stackexchange.com/a/152341) are of some use for you if you want to investigate the matter further.2012-07-05
  • 0
    Thanks. I find history of sheaf theory is interesting.2012-07-05
  • 0
    @t.b. They(Cartan et al.) had a notion of sheaves of abelian groups and rings. They had a notion of coherent sheaves on analytic spaces. They had cohomology theory of coherent sheaves on analytic spaces. I think these facts indicate that they might have enough motivation to develop homological algebra on sheaves at that time.2012-07-06
  • 0
    Colin McLarty’s essay is a great read on the history of all this: http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty1.pdf2012-07-08
  • 0
    @AdeelAhmadKhan Thanks. "Cartan was aware of it and told Buchsbaum to work on it, but he seems not to have done it." So most likely they did not come up with the notion of injective resolutions of sheaves even on Hausdorff paracompact spaces. This is surprising to me.2012-07-10
  • 0
    I think the following is related, but I am not sure how to link the two (I just begun using this program). Here it is here: http://math.stackexchange.com/questions/650275/cartan-and-eilenberg-homological-algebra Hopefully I did not cause an international incident by inserting the url...2014-01-25

0 Answers 0