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Given a sheaf $\mathcal{F}$ with respect to some Grothendieck topology, is the cohomological dimension for this sheaf less than or equal to the cohomological dimension of a finer topology?

Example: $cd_{Zar} \leq cd_{Nis} \leq cd_{├ęt}$.

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There's a tag for just this sort of question, so I added it –  David White Aug 2 '11 at 16:07
    
I added ArXiv subject tags. I hope you don't mind. –  S. Carnahan Aug 2 '11 at 16:14
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Tom: I think our interpretations of the question were different. If $X$ is a finer topology than $Y$, then we have a continuous map $f:X\rightarrow Y$. For any sheaf $F$ on $Y$, we can then consider the sheaf pull-back $f^*F$ obtained by sheafifying the presheaf pullback. I thought the question concerned the comparison between the cohomology of $F$ and that of $f^*F$. At least, this is the 'standard' comparison that comes up in practice. It seems to me that the constant sheaf always pulls back to the constant sheaf. –  Minhyong Kim Aug 3 '11 at 5:17
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Of course, if $G$ is a sheaf on $X$, then the comparison between $G$ and $f_*(G)$, which is what you were doing, is also important. –  Minhyong Kim Aug 3 '11 at 5:21
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"Suppose we have one category with two Grothendieck topologies, one finer than the other. Suppose some presheaf is a sheaf with respect to both of them. If Hn=0 for all n>d in the finer case, must the same be true in the other case?" Yes, that is what i meant. –  user12832 Aug 3 '11 at 13:28

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