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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: $cd_{Zar} \leq cd_{Nis} \leq cd_{ét}$.

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  • $\begingroup$ There's a tag for just this sort of question, so I added it $\endgroup$ Aug 2, 2011 at 16:07
  • $\begingroup$ I added ArXiv subject tags. I hope you don't mind. $\endgroup$
    – S. Carnahan
    Aug 2, 2011 at 16:14
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    $\begingroup$ 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. $\endgroup$ Aug 3, 2011 at 5:17
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    $\begingroup$ 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. $\endgroup$ Aug 3, 2011 at 5:21
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    $\begingroup$ "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. $\endgroup$
    – user12832
    Aug 3, 2011 at 13:28

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