On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one

抄録

Let R be a commutative Noetherian ring and I be an ideal of R with cd(I, R) ≤ 1. For an R-module M, we introduce a class of prime ideals, say M, as the set of all prime ideals of R such that . We show that if R is a Noetherian complete local ring and M is an I-cofinite R-module, then M is finite. Also, we prove that for each I-cofinite R-module M, , where Λ_R(I, M) is the set of all maximal elements of M\V(I) with respect to inclusion. Subsequently, for each a ∈ I, the R-module (0 :_M a) is finitely generated if and only if .