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

