COFINITE MODULES AND GENERALIZED LOCAL COHOMOLOGY
2009; University of Houston; Volume: 35; Issue: 4 Linguagem: Inglês
ISSN
0362-1588
Autores Tópico(s)Algebraic Geometry and Number Theory
ResumoAbstract. Let R be a commutative Noetherian ring, a an ideal of R, andM, N two nitely generated R-modules. We prove that the generalized localcohomology modules H ta (M;N) are a-co nite; that is, Ext iR (R=a;H at (M;N))is nitely generated for all i;t 0, in the following cases:(i) cd(a) = 1, where cd is the cohomological dimension of a in R.(ii) dimR 2.Additionally, we show that if cd(a) = 1 then Ext iR (M;H ta (N)) is a-co nitefor all i;t 0. 1. IntroductionThroughout this paper, we assume that Ris a commutative Noetherian ringwith non-zero identity, a an ideal of R, and M, N two nitely generated R-modules. Let tbe a non-negative integer. Grothendieck [5] introduced the localcohomology modules H ta (N) of Nwith respect to a. He proved their basic proper-ties. For example, he proved that Hom R (R=m;H tm (N)) is nitely generated when-ever Ris local with maximal ideal m. Later Grothendieck asked in [6] whether asimilar statement is valid if m is replaced by an arbitrary ideal. Hartshorne gavea counterexample in [7], also he de ned that an R-module S (not necessarily nitely generated) is a-co nite, if Supp(S) V(a) and Ext
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