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The question Consider a topological space $X$ and a family of sheaves (of abelian groups, say) $\; \mathcal F_i \;(i\in I)$ on $X$. Is it true that $$H^*(X,\prod \limits_{i \in I} \mathcal F_i)=\prod \limits_{i \in I} H^*(X,\mathcal F_i) \;?$$ According to Godement's and to Bredon's monographs this is correct if the family of sheaves is locally finite (In particular if $I$ is finite). [Bredon also mentions in an exercise that equality holds for spaces in which every point has a smallest open neighbourhood.]

What about the general case?

A variant Same question for $\check{C}$ech cohomology: is it true that $$\check{H}^*(X,\prod \limits_{i \in I} \mathcal F_i)=\prod \limits_{i \in I} \check{H}^*(X,\mathcal F_i) \;?$$ (Of course, $\check{C}$ech cohomology often coincides with derived functor cohomology but still the question should be considered independently)

A prayer Godement's book Topologie algébrique et théorie des faisceaux was published in 1960 and is still, with Bredon's, the most complete book on the subject. I certainly appreciate the privilege of working in a field where a book released half a century ago is still relevant: programmers and molecular biologists are not so lucky. Still I feel that a new treatise is due, in which naïve/foundational questions like the above would be addressed, and which would take the research and shifts in emphasis of half a century into account: one book on sheaf theory every 50 years does not seem an unreasonable frequency. So might I humbly suggest to one or several of the awesome specialists on MathOverflow to write one? I am sure I'm not the only participant here whose eternal gratitude they would earn.

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Why don't you like Kashiwara and Shapira's "Categories and sheaves"? By the way, Godement's promised second volume that will allow the reader to compute cohomologies of the sphere never materialized, has it? –  Victor Protsak Jun 16 '10 at 23:27
    
Dear Victor, I certainly don't dislike Kashiwara/Schapira: I'm just not familiar with that book. I'll try to check it in the future: thanks for the reminder. And no,the second tome of Godement's treatise never appeared, unfortunately. –  Georges Elencwajg Jun 17 '10 at 8:03

1 Answer 1

up vote 7 down vote accepted

The answer to the first question is almost always no, see Roos, Jan-Erik(S-STOC) Derived functors of inverse limits revisited. (English summary) J. London Math. Soc. (2) 73 (2006), no. 1, 65--83. .

Addendum: The crucial point is that infinite products are not exact. The most precise counterexample statement is Cor 1.11 combined with Prop 1.6 which identifies the stalks of the higher derived functors of the product with what you are interested in. Formally, it doesn't give a counter example for a single $X$ but Cor 1.11 shows that for any paracompact space with positive cohomological dimension there is some open subset for which your question has a negative answer. It seems clear that one could examples for specific $X$.

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The answer to the last question ("Would you please write a book on the subject?") is almost always no, too! –  Allen Knutson Jun 16 '10 at 16:49
    
The point is that infinite product of sheaves does not preserve exact sequences, right? –  Tom Goodwillie Jun 16 '10 at 19:45
    
Dear Torsten, thank you very much for the reference. The article "proves,corrects and extends" results of a note by the author published in 1961 (!), which tends to confirm the feelings I expressed at the end of my question. However because of the numerous cross-references and the general structure of the paper I could not locate a counter-example to the first equality, nor the assertion that the answer is almost always no. Needless to say, that didn't prevent me from upvoting you ! –  Georges Elencwajg Jun 16 '10 at 19:51
    
Thanks for the addendum –  Georges Elencwajg Jun 17 '10 at 22:28

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