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On the equivalence of fsf and weakly Laskerian classes

K. Bahmanpour, A. Khojali

TL;DR

The paper investigates whether the FSF and weakly Laskerian module classes coincide over Noetherian rings and analyzes their behavior under key operations. It develops technical preliminaries on $I$-torsion injectives and a prime-filtering construction to bound associated primes and supports, establishing a path to finiteness results. The main result proves the equivalence of FSF and weakly Laskerian modules, and it further shows that this property ascends under completion via $M\otimes_R R^{*}$ and descends under finite local extensions $R\to S$. These findings clarify the structure of these module classes and their stability under standard ring-theoretic constructions, with implications for local cohomology and related finiteness phenomena.

Abstract

It is proved that, over a Noetherian ring R, the class of weakly Laskerian and FSF modules are the same classes. By using this characterization we proved that the property of being weakly Laskerian descends by finite integral extensions of local ring homomorphisms and ascends by tensoring under the completion.

On the equivalence of fsf and weakly Laskerian classes

TL;DR

The paper investigates whether the FSF and weakly Laskerian module classes coincide over Noetherian rings and analyzes their behavior under key operations. It develops technical preliminaries on -torsion injectives and a prime-filtering construction to bound associated primes and supports, establishing a path to finiteness results. The main result proves the equivalence of FSF and weakly Laskerian modules, and it further shows that this property ascends under completion via and descends under finite local extensions . These findings clarify the structure of these module classes and their stability under standard ring-theoretic constructions, with implications for local cohomology and related finiteness phenomena.

Abstract

It is proved that, over a Noetherian ring R, the class of weakly Laskerian and FSF modules are the same classes. By using this characterization we proved that the property of being weakly Laskerian descends by finite integral extensions of local ring homomorphisms and ascends by tensoring under the completion.

Paper Structure

This paper contains 2 sections, 11 theorems, 30 equations.

Table of Contents

  1. Preliminaries
  2. Main Result

Key Result

Theorem 1

Assume that $\frak a$ is an ideal of the Noetherian ring R, and let $M$ be a finitely generated R-module. Let $t\in \Bbb{N}_0$ be such that either $H^i_{\frak a}(M)$ is finite or ${\rm Supp}(H^i_{\frak a} (M))$ is finite for all $i<t$. Then ${\rm Ass}(H^t_{\frak a}(M))$ is finite.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Theorem 1.4
  • proof
  • ...and 14 more