The Lichtenbaum–Hartshorne theorem for modules which are finite over a ring homomorphism

Let image be a flat ring homomorphism such that image. 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 image if and only if for each prime ideal image satisfying image, we have image. This gives a generalization of the Lichtenbaum–Hartshorne vanishing theorem for modules which are finite over a ring homomorphism. Furthermore, we provide two extensions of Grothendieck’s non-vanishing theorem. Applications to connectedness properties of the support are given.