Cohomology of sheaves of abelian groups on affine space Is it true that $$\mathrm{H}^p_\mathrm{Zar}(\mathbb{A}^n_\mathbb{K}, \mathcal{F})=0$$ for any $p>1$ and $\mathcal{F}$ (non constant) sheaf of abelian groups?  
If not, is it true for some fields/sheaves but not others? Which ones?
Do you know some counterexamples?
I'd really appreciate some reference, thank you in advance.
 A: No, this is not true unless you restrict to quasicoherent sheaves.  For instance, just as a topological space, $\mathbb{A}^1$ is homeomorphic to $\mathbb{P}^1$, and there are coherent sheaves on $\mathbb{P}^1$ with nontrivial $H^1$.  In fact, there are even sheaves of $\mathcal{O}_{\mathbb{A}^n}$-modules on $\mathbb{A}^n$ which have nontrivial cohomology.  For instance, let $p\in\mathbb{A}^n$ be a closed point and let $\mathcal{G}$ be the skyscraper sheaf at $p$ on $\mathbb{A}^n$ with stalk $\mathcal{O}_{\mathbb{A}^n, p}$.  This sheaf has a global section corresponding to $1\in\mathcal{O}_{\mathbb{A}^n,p}$ and there is a corresponding map of sheaves $\mathcal{O}_{\mathbb{A}^n}\to\mathcal{G}$.  In fact, this map is surjective, as is easily seen by looking at stalks.  If you let $\mathcal{F}$ be its kernel, then there is an exact sequence $$H^0(\mathcal{O}_{\mathbb{A}^n})\to H^0(\mathcal{G})\to H^1(\mathcal{F}).$$
But the map $\mathcal{O}_{\mathbb{A}^n}\to\mathcal{G}$ is not surjective on global sections, so it follows that $H^1(\mathcal{F})\neq 0$.
