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A generalization of the finiteness problem of the local cohomology modules

Ahmad Abbasi, Hajar Roshan-Shekalgourabi (2014)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring and 𝔞 an ideal of R . We introduce the concept of 𝔞 -weakly Laskerian R -modules, and we show that if M is an 𝔞 -weakly Laskerian R -module and s is a non-negative integer such that Ext R j ( R / 𝔞 , H 𝔞 i ( M ) ) is 𝔞 -weakly Laskerian for all i < s and all j , then for any 𝔞 -weakly Laskerian submodule X of H 𝔞 s ( M ) , the R -module Hom R ( R / 𝔞 , H 𝔞 s ( M ) / X ) is 𝔞 -weakly Laskerian. In particular, the set of associated primes of H 𝔞 s ( M ) / X is finite. As a consequence, it follows that if M is a finitely generated R -module and N is an 𝔞 -weakly...

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar 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...

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