Finiteness of local homology modules

Shahram Rezaei

Displaying similar documents to “Finiteness of local homology modules”

Characterizations of incidence modules

Naseer Ullah, Hailou Yao, Qianqian Yuan, Muhammad Azam (2024)

Czechoslovak Mathematical Journal

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Let $R$ be an associative ring and $M$ be a left $R$ -module. We introduce the concept of the incidence module $I (X, M)$ of a locally finite partially ordered set $X$ over $M$ . We study the properties of $I (X, M)$ and give the necessary and sufficient conditions for the incidence module to be an IN-module, -module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of -modules is closed under direct product and upper triangular matrix modules.

On generalized CS-modules

Qingyi Zeng (2015)

Czechoslovak Mathematical Journal

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An $𝒮$ -closed submodule of a module $M$ is a submodule $N$ for which $M / N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$ -closed submodule $N$ of $M$ is a direct summand of $M$ . Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$ -modules are projective if and only if all right $R$ -modules are GCS-modules.

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi, Kamran Divaani-Aazar, Massoud Tousi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a local ring and $C$ a semidualizing module of $R$ . We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$ -modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$ . In particular, we show that generalized Cohen-Macaulay $R$ -modules are invariant under this equivalence and if $M$ is a finitely generated $R$ -module in the Auslander class with respect to $C$ such that $C \otimes_{R} M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum. ...

Relative Gorenstein injective covers with respect to a semidualizing module

Elham Tavasoli, Maryam Salimi (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$ -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$ -injective module $G$ , the character module $G^{+}$ is $G_{C}$ -flat, then the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

Explicit cogenerators for the homotopy category of projective modules over a ring

Amnon Neeman (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $𝐊 (R - Proj)$ of projective $R$ -modules. We produced a set of generators for this category, proved that the category is $ℵ_{1}$ -compactly generated for any ring $R$ , and showed that it need not always be compactly generated, but is for sufficiently nice $R$ . We furthermore analyzed the inclusion $j_{!}^{} : 𝐊 (R - Proj) \to 𝐊 (R - Flat)$ and the orthogonal subcategory $𝒮 = {𝐊 (R - Proj)}^{⊥}$ . And we even showed that the inclusion $𝒮 \to 𝐊 (R - Flat)$ has a right adjoint; this forces some natural map to be...

Dual modules and reflexive modules with respect to a semidualizing module

Lixin Mao (2024)

Czechoslovak Mathematical Journal

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Let $C$ be a semidualizing module over a commutative ring. We first investigate the properties of $C$ -dual, $C$ -torsionless and $C$ -reflexive modules. Then we characterize some rings such as coherent rings, $Π$ -coherent rings and FP-injectivity of $C$ using $C$ -dual, $C$ -torsionless and $C$ -reflexive properties of some special modules.

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.

Some results on $G_{C}$ -flat dimension of modules

Ramalingam Udhayakumar, Intan Muchtadi-Alamsyah, Chelliah Selvaraj (2019)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we study some properties of $G_{C}$ -flat $R$ -modules, where $C$ is a semidualizing module over a commutative ring $R$ and we investigate the relation between the $G_{C}$ -yoke with the $C$ -yoke of a module as well as the relation between the $G_{C}$ -flat resolution and the flat resolution of a module over $G F$ -closed rings. We also obtain a criterion for computing the $G_{C}$ -flat dimension of modules.

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.

Generalized tilting modules over ring extension

Zhen Zhang (2019)

Czechoslovak Mathematical Journal

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Let $Γ$ be a ring extension of $R$ . We show the left $Γ$ -module $U = Γ \otimes_{R} C$ with the endmorphism ring End $_{Γ} U = Δ$ is a generalized tilting module when $_{R} C$ is a generalized tilting module under some conditions.

Some results on top local cohomology modules with respect to a pair of ideals

Saeed Jahandoust, Reza Naghipour (2020)

Mathematica Bohemica

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Let $I$ and $J$ be ideals of a Noetherian local ring $(R, 𝔪)$ and let $M$ be a nonzero finitely generated $R$ -module. We study the relation between the vanishing of $H_{I, J}^{dim M} (M)$ and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian $R$ -module $M / J M$ is equal to its integral closure relative to the Artinian $R$ -module $H_{I, J}^{dim M} (M)$ .

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We investigate the following quasilinear and singular problem, $t o 2.7 c m \{\begin{matrix} - Δ_{p} u = \frac{λ}{u^{δ}} + u^{q} & in Ω; \\ {u |}_{\partial Ω} = 0, u > 0 & in Ω, \end{matrix} t o 2.7 c m (P)$ where $Ω$ is an open bounded domain with smooth boundary, $1 < p < \infty$ , $p - 1 < q \leq p^{*} - 1$ , $λ > 0$ , and $0 < δ < 1$ . As usual, $p^{*} = \frac{N p}{N - p}$ if $1 < p < N$ , $p^{*} \in (p, \infty)$ is arbitrarily large if $p = N$ , and $p^{*} = \infty$ if $p > N$ . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_{0}^{1, p} (Ω)$ . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...