“School of Mathematics”
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| Paper IPM / M / 8719 | ||||||||||||||||||||||
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| Abstract: | ||||||||||||||||||||||
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Let φ: (R, \frakm) → S be a flat ring
homomorphism such that \frakmS ≠ S. Assume that M is a
finitely generated S-module with dimR(M) = d. If the set of
support of M has a special property, then it is shown that
Hd\fraka(M)=0 if and only if for each prime ideal
\frakp ∈ Supp∧R(M⊗R∧R)
satisfying dim ∧R/\frakp=d, we have
dim(∧R/(\fraka∧R+\frakp)) > 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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