Cohomology of sheaf of Schwartz distributions with support in a submanifold

Question

Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely for any open subset $U\subset M$, $F(U)$ is the space of generalized functions on $U$.

Is it true that $H^i_Z(M,F)=0$ for any $i>0$?

Remark. I think that if one considers the sheaf of smooth functions instead of generalized, then the anologous statement is not true in general even when $M =\mathbb{R}$ and $Z$ is a point.

Answer 1

The long exact sequence cotains $H^0(M,F)\to H^0(M\smallsetminus Z,F|_{M\smallsetminus Z})\to H_Z^1(M,F)\to H^1(X,F)$. The last term is zero, as $F$ is a fine sheaf. But the first arrow is not surjective in general, so the first cohomology group is not trivial. The same argument shows that for $i>1$ the cohomologies are indeed trivial.