Is there a topological space $X$ with a nonzero sheaf $\mathcal{F}$ of abelian groups such that $H^i(X,\mathcal{F})=0$ for all $i=0,1,2...$?



Is there a nonzero sheaf with all cohomologies vanish?
What about the "Moebius" local system over $S^1$ with fibers $\mathbb{Q}$ and monodromy $-1$? (It has $H^0=H^1$ by Poincare duality with coefficients).


