Hi.

Can any one me say if there is a simple proof of this claim which i can prove it by localization and no easy technique of nuclear spaces...

Let $f:X\rightarrow S$ be an open, surjective morphism of complex reduced spaces with constant fiber dimension $n$ (or universally open morphism with $n$-fibers between locally noetherian excellent without embedded componnet schemes on field of charact.0). Let $G$ be a coherent sheaf on $X$.

Claim: ${\rm R}^{n}f_{*}G=0$ if and only if the support of $G$ satisfies:

${\rm Supp}(G)\cap X_{s}$ is nowhere dense in the fiber $X_{s}$.

Remark: 1) We can assume $X$ $n$-complete space(i.e there exist a smooth strongly $n$-convex exhaustion function on $X$) and then all cohomology group $H^{k}(X, F)$ vanish for all coherent sheaf $F$ and all $k>n$.

2) For a projection $f:S\times U\rightarrow S$ where $U$ is an open polydisque of $C^{n}$, the result is true for ${\rm R}^{n}f_{!}$. By right exactness of the $n$-higher direct image functor, we can localize and deduce from the projection case the general case.... But, it is no simple proof...

Thank you very much.