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Let $\varphi: (R, \frak{m}) \rightarrow S$ be a flat ring
homomorphism such that $\frak{m}S \neq S$. Assume that $M$ is a
finitely generated $S$-module with dim$_{R}(M) = d$. If the set of
support of $M$ has a special property, then it is shown that
$H^{d}_\frak{a}(M)=0$ if and only if for each prime ideal
$\frak{p} \in$ Supp$_{\widehat{R}}(M\otimes_{R}\widehat{R})$
satisfying dim $\widehat{R}/\frak{p}=d$, we have
dim$(\widehat{R}/(\frak{a}\widehat{R}+\frak{p}))>0$. This gives a
generalization of the LichtenbauM-HaRTshorne vanishing theorem for
modules which are finite over a ling homomorphism. Furthermore, we
provide two extensions of Grothendieck's non-vanishing theorem.
Applications to connectedness properties of the suppon are given.
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