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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...

A new version of Local-Global Principle for annihilations of local cohomology modules

K. Khashyarmanesh, M. Yassi, A. Abbasi (2004)

Colloquium Mathematicae

Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of f ( N ) relative to in the context of generalized local cohomology modules as f ( M , N ) : = i n f i 0 | ( 0 : R H i ( M , N ) ) , where M is an R-module. We also show that f ( N ) f ( M , N ) for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.

Adic-completion and some dual homological results.

Anne-Marie Simon (1992)

Publicacions Matemàtiques

Let a be an ideal of a commutative ring A. There is a kind of duality between the left derived functors Uia of the a-adic completion functor, called local homology functors, and the local cohomology functors Hai.Some dual results are obtained for these Uia, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the Ua and Ha-global dimensions of A are finite.

Annihilators of local homology modules

Shahram Rezaei (2019)

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a local ring, 𝔞 an ideal of R and M a nonzero Artinian R -module of Noetherian dimension n with hd ( 𝔞 , M ) = n . We determine the annihilator of the top local homology module H n 𝔞 ( M ) . In fact, we prove that Ann R ( H n 𝔞 ( M ) ) = Ann R ( N ( 𝔞 , M ) ) , where N ( 𝔞 , M ) denotes the smallest submodule of M such that hd ( 𝔞 , M / N ( 𝔞 , M ) ) < n . As a consequence, it follows that for a complete local ring ( R , 𝔪 ) all associated primes of H n 𝔞 ( M ) are minimal.

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...

Artinianness of formal local cohomology modules

Shahram Rezaei (2019)

Commentationes Mathematicae Universitatis Carolinae

Let 𝔞 be an ideal of Noetherian local ring ( R , 𝔪 ) and M a finitely generated R -module of dimension d . In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to 𝔪 . Also we prove that for an arbitrary local ring ( R , 𝔪 ) (not necessarily complete), we have Att R ( 𝔉 𝔞 d ( M ) ) = Min V ( Ann R 𝔉 𝔞 d ( M ) ) .

Associated primes of local cohomology modules of generalized Laskerian modules

Dawood Hassanzadeh-Lelekaami, Hajar Roshan-Shekalgourabi (2019)

Czechoslovak Mathematical Journal

Let be a set of ideals of a commutative Noetherian ring R . We use the notion of -closure operation which is a semiprime closure operation on submodules of modules to introduce the class of -Laskerian modules. This enables us to investigate the set of associated prime ideals of certain -closed submodules of local cohomology modules.

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