Melkersson condition on Serre subcategories

Reza Sazeedeh; Rasul Rasuli

Displaying similar documents to “Melkersson condition on Serre subcategories”

Artinianness of formal local cohomology modules

Shahram Rezaei (2019)

Commentationes Mathematicae Universitatis Carolinae

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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

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

Some bounds for the annihilators of local cohomology and Ext modules

Ali Fathi (2022)

Czechoslovak Mathematical Journal

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Let $𝔞$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a nonnegative integer. Let $M$ and $N$ be two finitely generated $R$ -modules. In certain cases, we give some bounds under inclusion for the annihilators of ${Ext}_{R}^{t} (M, N)$ and $H_{𝔞}^{t} (M)$ in terms of minimal primary decomposition of the zero submodule of $M$ , which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.

Local cohomology, cofiniteness and homological functors of modules

Kamal Bahmanpour (2022)

Czechoslovak Mathematical Journal

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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 $j \in ℕ_{0}$ 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 $i, j \in ℕ_{0}$ .

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 $𝔪 \in Max (R)$ 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....

On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals

Thiago H. Freitas, Victor H. Jorge Pérez (2019)

Czechoslovak Mathematical Journal

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Let $𝔞$ , $I$ , $J$ be ideals of a Noetherian local ring $(R, 𝔪, k)$ . Let $M$ and $N$ be finitely generated $R$ -modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H_{I, J}^{t} (M)$ and $D (H_{I, J}^{t} (M))$ , where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D (-) : = {Hom}_{R} (-, E_{R} (k))$ is the Matlis dual functor. We show that if $R$ is a $d$ -dimensional complete Cohen-Macaulay...

The correspondence between Barsotti-Tate groups and Kisin modules when $p = 2$

Tong Liu (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a finite extension over $ℚ_{2}$ and $𝒪_{K}$ the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over $𝒪_{K}$ .

Cohomological dimension filtration and annihilators of top local cohomology modules

Ali Atazadeh, Monireh Sedghi, Reza Naghipour (2015)

Colloquium Mathematicae

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Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration $= {M_{i}}_{i = 0}^{c}$ , where c = cd(,M) and $M_{i}$ denotes the largest submodule of M such that $c d (, M_{i}) \leq i$ . Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module $H_{}^{c} (M)$ , namely $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / M_{c - 1})$ . As a consequence, there exists an ideal of R such that $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / H ⁰_{} (M))$ . This generalizes the...

Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$ -module. The notion of $C$ -tilting $R$ -modules is introduced as the relative setting of the notion of tilting $R$ -modules with respect to $C$ . Some properties of tilting and $C$ -tilting modules and the relations between them are mentioned. It is shown that every finitely generated $C$ -tilting $R$ -module is $C$ -projective. Finally, we investigate some kernel subcategories related to $C$ -tilting 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 $t \in ℕ_{0}$ be an integer and $M$ an $R$ -module such that ${Ext}_{R}^{i} (R / I, M)$ is minimax for all $i \leq t + 1$ . We prove that if $H_{I}^{i} (M)$ is ${FD}_{\leq 1}$ (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 ${FD}_{\leq 1}$ (or weakly Laskerian) for all $i$ .

Some Results About Holonomic $ε$ -Modules

Jan-Erik Björk (1983)

Recherche Coopérative sur Programme n°25

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$S$ -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 $S$ - $depth (I, J, M) = inf {S$ - $depth (𝔞, M) : 𝔞 \in \tilde{W} (I, J)}$ , 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 $S$ - $depth (I, J, M) = inf {i : H_{I, J}^{i} (M) \notin S}$ , where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

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 $M = R e j_{M} E (R / τ (R)) \oplus N$ , 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...

Annihilator ideals of finite dimensional simple modules of two-parameter quantized enveloping algebra $U_{r, s} ({𝔰𝔩}_{2})$

Yu Wang, Xiaoming Li (2023)

Czechoslovak Mathematical Journal

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Let $U$ be the two-parameter quantized enveloping algebra $U_{r, s} ({𝔰𝔩}_{2})$ and $F (U)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F (U)$ in the case when $r s^{- 1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.

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 $𝒟_{X}^{(\infty)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $𝒟_{X}^{(\infty)}$ on an $𝒪_{X}$ -module $ℰ$ is equivalent to giving an infinite sequence of $𝒪_{X}$ -modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$ , endowed with Frobenius liftings. We also show that it...

Cofiniteness and finiteness of local cohomology modules over regular local rings

Jafar A&#039;zami, Naser Pourreza (2017)

Czechoslovak Mathematical Journal

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Let $(R, 𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${mAss}_{R} (R / I) = {Assh}_{R} (I)$ . It is shown that the $R$ -module $H_{I}^{ht (I)} (R)$ is $I$ -cofinite if and only if $cd (I, R) = ht (I)$ . Also we present a sufficient condition under which this condition the $R$ -module $H_{I}^{i} (R)$ is finitely generated if and only if it vanishes.