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Local cohomology, cofiniteness and homological functors of modules

Kamal Bahmanpour — 2022

Czechoslovak Mathematical Journal

Let I be an ideal of a commutative Noetherian ring R . It is shown that the R -modules H I j ( M ) are I -cofinite for all finitely generated R -modules M and all j 0 if and only if the R -modules Ext R i ( N , H I j ( M ) ) and Tor i R ( N , H I j ( M ) ) are I -cofinite for all finitely generated R -modules M , N and all integers i , j 0 .

Artinian cofinite modules over complete Noetherian local rings

Behrouz SadeghiKamal BahmanpourJafar A'zami — 2013

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R -module. In this paper it is shown that if 𝔭 is a prime ideal of R such that dim R / 𝔭 = 1 and ( 0 : M 𝔭 ) is not finitely generated and for each i 2 the R -module Ext R i ( M , R / 𝔭 ) is of finite length, then the R -module Ext R 1 ( M , R / 𝔭 ) is not of finite length. Using this result, it is shown that for all finitely generated R -modules N with Supp ( N ) V ( I ) and for all integers i 0 , the R -modules Ext R i ( N , M ) are of finite length, if and only if, for all finitely generated R -modules N with Supp ( N ) V ( I ) and...

Cofiniteness of torsion functors of cofinite modules

Reza NaghipourKamal BahmanpourImaneh Khalili Gorji — 2014

Colloquium Mathematicae

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules T o r i R ( N , M ) are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules T o r i R ( N , H I j ( M ) ) are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules T o r i R ( N , M ) are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it is shown that...

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