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 $U$. 



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



Cohomology of sheaf of Schwartz distributions with support in a submanifold
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.


