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Let $Z$ be a subset of the spectrum of a local ring $R$ stable under specialization and let $N$ be a $d$-dimensional finitely
generated $R$-module. It is shown that $H_Z^d(N)$, the $d$-th local cohomology module of the sheaf associated to $N$ with support in $Z$, vanishes if and only if for every $d$-dimensional $\fp \in \T{Ass}_{\hat{R}} \hat{N}$, there is a $\fq\in Z$ such that $\dim \hat{R}/(\fq\hat{R}+\fp)>0$. Applying this criterion for vanishing of $H_Z^d (N)$, several connectedness results for certain
algebraic varieties are proved.
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