Local cohomology and Gorenstein injective dimension over local   homomorphisms
  A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull dimension and is equal to the supremum of the depths of the localizations of the ring over primes in the support of the module. 
Leila Khatami, Massoud Tousi and Siamak Yassemi