  In this article, we show that if $\Pi: X\rightarrow Y$ is an unbranched
Riemann domain over an $r$-complete complex space $Y$, then for any coherent
analytic sheaf ${\mathcal{F}}$ on $X$ the cohomology group
$H^{p}(X,{\mathcal{F}})$ vanishes for all $p\geq q+r-1.$ In particular, one
gets a positive answer to a generalization of the local Stein problem and new
vanishing theorems for the cohomology of locally $q$-complete open sets in
$r$-complete spaces.
