**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.



The cohomology of a product of sheaves and a plea.
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 $X$.


