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

Let φ:(R,m)→Sφ:(R,m)→S be a flat ring homomorphism such that mS≠SmS≠S. Assume that MM is a finitely generated SS-module with dimR(M)=ddimR(M)=d. If the set of support of MM has a special property, then it is shown that Had(M)=0 if and only if for each prime ideal p∈SuppR̂(M⊗RR̂) satisfying dimR̂/p=d, we have dim(R̂/(aR̂+p))>0. 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.