Consider the Nisnevich site of a noetherian scheme $S$ of finite Krull dimension (the objects are schemes $U$ smooth and of finite type over $S$), let $A$ be a sheaf of abelian groups on this site. I want to know:



> 
> Is the Nisnevich sheafification of the presheaf $$U\mapsto\mathrm{H}\_{\mathrm{Nis}}^{n}(U, A)$$
>  trivial for $n>0$ (or does this presheaf have trivial stalks)?
> 
> 
> 


One can also assume $S=\mathrm{Spec}(k)$ for $k$ a field if needed.


The case of $n=1$ is trivial since $\mathrm{H}\_{\mathrm{Nis}}^{1}$ classifies torsors, which are Nisnevich locally trivial.



The sheafification of taking cohomology is trivial?
This is true (in any site), you are deriving the identity functor, which is exact, so you get zero.


More generally, for a map of sites $$ \varepsilon\colon C \to D $$
(which by definition is a functor $\varepsilon^{-1}$ in the opposite direction satisfying some properties), the higher direct image functors $R^i \varepsilon\_\* \mathscr{F}$ coincide with the sheafification of $U\mapsto H^i(\varepsilon^{-1}(U), \mathscr{F})$. If $C=D$ and $\varepsilon = {\rm id}$, we have $R^i \varepsilon\_\* = 0$ for $i>0$.


