Displaying similar documents to “Local cohomology, cofiniteness and homological functors of modules”

Cominimaxness of local cohomology modules

Moharram Aghapournahr (2019)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, I an ideal of R . Let t 0 be an integer and M an R -module such that Ext R i ( R / I , M ) is minimax for all i t + 1 . We prove that if H I i ( M ) is FD 1 (or weakly Laskerian) for all i < t , then the R -modules H I i ( M ) are I -cominimax for all i < t and Ext R i ( R / I , H I t ( M ) ) is minimax for i = 0 , 1 . Let N be a finitely generated R -module. We prove that Ext R j ( N , H I i ( M ) ) and Tor j R ( N , H I i ( M ) ) are I -cominimax for all i and j whenever M is minimax and H I i ( M ) is FD 1 (or weakly Laskerian) for all i .

S -depth on Z D -modules and local cohomology

Morteza Lotfi Parsa (2021)

Czechoslovak Mathematical Journal

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Let R be a Noetherian ring, and I and J be two ideals of R . Let S be a Serre subcategory of the category of R -modules satisfying the condition C I and M be a Z D -module. As a generalization of the S - depth ( I , M ) and depth ( I , J , M ) , the S - depth of ( I , J ) on M is defined as S - depth ( I , J , M ) = inf { S - depth ( 𝔞 , M ) : 𝔞 W ˜ ( I , J ) } , and some properties of this concept are investigated. The relations between S - depth ( I , J , M ) and H I , J i ( M ) are studied, and it is proved that S - depth ( I , J , M ) = inf { i : H I , J i ( M ) S } , where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

On the minimaxness and coatomicness of local cohomology modules

Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi (2022)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, I an ideal of R and M an R -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and 𝒞 -minimaxness of local cohomology modules. We show that if M is a minimax R -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if n is a nonnegative integer such that ( H I i ( M ) ) 𝔪 is a minimax R 𝔪 -module for all 𝔪 Max ( R ) and for all i < n , then the set Ass R ( H I n ( M ) ) is finite. Also, if H I i ( M ) is...

Recollements induced by good (co)silting dg-modules

Rongmin Zhu, Jiaqun Wei (2023)

Czechoslovak Mathematical Journal

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Let U be a dg- A -module, B the endomorphism dg-algebra of U . We know that if U is a good silting object, then there exist a dg-algebra C and a recollement among the derived categories 𝐃 ( C , d ) of C , 𝐃 ( B , d ) of B and 𝐃 ( A , d ) of A . We investigate the condition under which the induced dg-algebra C is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

Matlis dual of local cohomology modules

Batoul Naal, Kazem Khashyarmanesh (2020)

Czechoslovak Mathematical Journal

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Let ( R , 𝔪 ) be a commutative Noetherian local ring, 𝔞 be an ideal of R and M a finitely generated R -module such that 𝔞 M M and cd ( 𝔞 , M ) - grade ( 𝔞 , M ) 1 , where cd ( 𝔞 , M ) is the cohomological dimension of M with respect to 𝔞 and grade ( 𝔞 , M ) is the M -grade of 𝔞 . Let D ( - ) : = Hom R ( - , E ) be the Matlis dual functor, where E : = E ( R / 𝔪 ) is the injective hull of the residue field R / 𝔪 . We show that there exists the following long exact sequence 0 H 𝔞 n - 2 ( D ( H 𝔞 n - 1 ( M ) ) ) H 𝔞 n ( D ( H 𝔞 n ( M ) ) ) D ( M ) H 𝔞 n - 1 ( D ( H 𝔞 n - 1 ( M ) ) ) H 𝔞 n + 1 ( D ( H 𝔞 n ( M ) ) ) H 𝔞 n ( D ( H ( x 1 , ... , x n - 1 ) n - 1 ( M ) ) ) H 𝔞 n ( D ( H ( n - 1 M ) ) ) ... , where n : = cd ( 𝔞 , M ) is a non-negative integer, x 1 , ... , x n - 1 is a regular sequence in 𝔞 on M and, for an R -module L , H 𝔞 i ( L ) is the i th local cohomology module...

Some results on ( n , d ) -injective modules, ( n , d ) -flat modules and n -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let n , d be two non-negative integers. A left R -module M is called ( n , d ) -injective, if Ext d + 1 ( N , M ) = 0 for every n -presented left R -module N . A right R -module V is called ( n , d ) -flat, if Tor d + 1 ( V , N ) = 0 for every n -presented left R -module N . A left R -module M is called weakly n - F P -injective, if Ext n ( N , M ) = 0 for every ( n + 1 ) -presented left R -module N . A right R -module V is called weakly n -flat, if Tor n ( V , N ) = 0 for every ( n + 1 ) -presented left R -module N . In this paper, we give some characterizations and properties of ( n , d ) -injective modules and ( n , d ) -flat modules in...

On τ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of τ -extending modules by τ -rational submodules and study some properties of such modules. It is shown that the set of all τ -rational left ideals of R R is a Gabriel filter. An R -module M is called τ -extending if every submodule of M is τ -rational in a direct summand of M . It is proved that M is τ -extending if and only if M = R e j M E ( R / τ ( R ) ) N , such that N is a τ -extending submodule of M . An example is given to show that the direct sum of τ -extending modules need not...

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

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If X is a smooth scheme over a perfect field of characteristic p , and if 𝒟 X ( ) is the sheaf of differential operators on X [7], it is well known that giving an action of 𝒟 X ( ) on an 𝒪 X -module is equivalent to giving an infinite sequence of 𝒪 X -modules descending via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X S of a smooth morphism in characteristic p , endowed with Frobenius liftings. We also show that it...

Some homological properties of amalgamated modules along an ideal

Hanieh Shoar, Maryam Salimi, Abolfazl Tehranian, Hamid Rasouli, Elham Tavasoli (2023)

Czechoslovak Mathematical Journal

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Let R and S be commutative rings with identity, J be an ideal of S , f : R S be a ring homomorphism, M be an R -module, N be an S -module, and let ϕ : M N be an R -homomorphism. The amalgamation of R with S along J with respect to f denoted by R f J was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of ( R f J ) -module called the amalgamation of M and N along J with respect to ϕ , and denoted by M ϕ J N . We study some homological properties of the ( R f J ) -module M ϕ J N . Among...

α -modules and generalized submodules

Rafiquddin Rafiquddin, Ayazul Hasan, Mohammad Fareed Ahmad (2019)

Communications in Mathematics

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A QTAG-module M is an α -module, where α is a limit ordinal, if M / H β ( M ) is totally projective for every ordinal β < α . In the present paper α -modules are studied with the help of α -pure submodules, α -basic submodules, and α -large submodules. It is found that an α -closed α -module is an α -injective. For any ordinal ω α ω 1 we prove that an α -large submodule L of an ω 1 -module M is summable if and only if M is summable.

Coherence relative to a weak torsion class

Zhanmin Zhu (2018)

Czechoslovak Mathematical Journal

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Let R be a ring. A subclass 𝒯 of left R -modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let 𝒯 be a weak torsion class of left R -modules and n a positive integer. Then a left R -module M is called 𝒯 -finitely generated if there exists a finitely generated submodule N such that M / N 𝒯 ; a left R -module A is called ( 𝒯 , n ) -presented if there exists an exact sequence of left R -modules 0 K n - 1 F n - 1 F 1 F 0 M 0 such that F 0 , , F n - 1 are finitely generated free and K n - 1 is 𝒯 -finitely generated;...

Strongly ( 𝒯 , n ) -coherent rings, ( 𝒯 , n ) -semihereditary rings and ( 𝒯 , n ) -regular rings

Zhanmin Zhu (2020)

Czechoslovak Mathematical Journal

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Let 𝒯 be a weak torsion class of left R -modules and n a positive integer. A left R -module M is called ( 𝒯 , n ) -injective if Ext R n ( C , M ) = 0 for each ( 𝒯 , n + 1 ) -presented left R -module C ; a right R -module M is called ( 𝒯 , n ) -flat if Tor n R ( M , C ) = 0 for each ( 𝒯 , n + 1 ) -presented left R -module C ; a left R -module M is called ( 𝒯 , n ) -projective if Ext R n ( M , N ) = 0 for each ( 𝒯 , n ) -injective left R -module N ; the ring R is called strongly ( 𝒯 , n ) -coherent if whenever 0 K P C 0 is exact, where C is ( 𝒯 , n + 1 ) -presented and P is finitely generated projective, then K is ( 𝒯 , n ) -projective; the ring R is called...

Enveloping algebras of Slodowy slices and the Joseph ideal

Alexander Premet (2007)

Journal of the European Mathematical Society

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Let G be a simple algebraic group over an algebraically closed field 𝕜 of characteristic 0, and 𝔤 = Lie G . Let ( e , h , f ) be an 𝔰 𝔩 2 -triple in 𝔤 with e being a long root vector in 𝔤 . Let ( · , · ) be the G -invariant bilinear form on 𝔤 with ( e , f ) = 1 and let χ 𝔤 * be such that χ ( x ) = ( e , x ) for all x 𝔤 . Let 𝒮 be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of 𝒮 ; see [31]. In this article we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains...

Annihilators of local homology modules

Shahram Rezaei (2019)

Czechoslovak Mathematical Journal

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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.

Rings whose nonsingular right modules are R -projective

Yusuf Alagöz, Sinem Benli, Engin Büyükaşık (2021)

Commentationes Mathematicae Universitatis Carolinae

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A right R -module M is called R -projective provided that it is projective relative to the right R -module R R . This paper deals with the rings whose all nonsingular right modules are R -projective. For a right nonsingular ring R , we prove that R R is of finite Goldie rank and all nonsingular right R -modules are R -projective if and only if R is right finitely Σ - C S and flat right R -modules are R -projective. Then, R -projectivity of the class of nonsingular injective right modules is also considered....

On the symmetric algebra of certain first syzygy modules

Gaetana Restuccia, Zhongming Tang, Rosanna Utano (2022)

Czechoslovak Mathematical Journal

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Let ( R , 𝔪 ) be a standard graded K -algebra over a field K . Then R can be written as S / I , where I ( x 1 , ... , x n ) 2 is a graded ideal of a polynomial ring S = K [ x 1 , ... , x n ] . Assume that n 3 and I is a strongly stable monomial ideal. We study the symmetric algebra Sym R ( Syz 1 ( 𝔪 ) ) of the first syzygy module Syz 1 ( 𝔪 ) of 𝔪 . When the minimal generators of I are all of degree 2, the dimension of Sym R ( Syz 1 ( 𝔪 ) ) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.