Finiteness of local homology modules

Shahram Rezaei

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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...