Hartshorne's questions and weakly cofiniteness

Original · EN

Let R be a commutative Noetherian ring, be an ideal of R and M be an R-module. The main purpose of this paper is to answer the Hartshorn's questions in the class of weakly Laskerian modules. It is shown that if s≥ 1 is a positive integer such that ʲᵣ(R/, M) is weakly Laskerian for all j≤ s and the R-module Hⁱ(M) is FD≤ ₁ for all i < s, then the R-module Hⁱ(M) is -weakly cofinite for all i <s. In addition, we show that the category of all -weakly cofinite FD≤ ₁ R-modules is an Abelian subcategory of the category of all R-modules. Also, we prove that if ⁱᵣ(R/,M) is weakly Laskerian for all i≤ M, then the R-module ⁱᵣ(N,M) is weakly Laskerian for all i≥ 0 and for any finitely generated R-module N with ᵣ(N) V () and N ≤ 1.