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Paper   IPM / M / 8719
School of Mathematics
  Title:   The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism
1.  M. Tousi
2.  S. Yassemi
  Status:   Published
  Journal: J. Pure Appl. Algebra
  Vol.:  212
  Year:  2008
  Pages:   1222-1228
  Supported by:  IPM
Let φ: (R, \frakm) → S be a flat ring homomorphism such that \frakmSS. 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 ∈ SuppR(MRR) satisfying dim R/\frakp=d, we have dim(R/(\frakaR+\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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