Stability of the depth function of good filtrations

Document Type : Research Paper

Authors

1 Laboratoire de Mathématiques et Informatique, Université Nangui Abrogoua‎, ‎UFR‎ - ‎SFA‎, ‎‎Côte d'Ivoire

2 Departement de Sciences et Technologie, Ecole Normale Supérieure d'Abidjan‎, ‎‎‎Côte d'Ivoire

Abstract

Let $A$ be a Noetherian local ring‎, ‎and let $I \subset J$ be two ideals of $A$‎. ‎Let $M$ be a finitely generated $A$-module‎. ‎Brodmann proved that the function $n \mapsto \mathrm{depth}_{J}(\frac{M}{I^{n}M})$ is constant for large $n$‎.
‎In this paper‎, ‎we consider a filtration $\phi = (M_n)_{n \in \mathbb{N}}$ of $M$ and a filtration $f = (I_n)_{n \in \mathbb{N}}$ of $A$‎. ‎Generalizing Brodmann's result‎, ‎we first show that the function $n \mapsto \mathrm{depth}_{J}(\frac{M}{M_{n}})$ is constant for large $n$ of value $\mathrm{depth}_{J}(f‎, ‎M)$‎, ‎provided that $\phi$ is $f$-good and $f$ is strongly Noetherian‎. ‎Secondly‎, ‎we establish the inequality‎ ‎$\gamma_{J}(f‎, ‎M) \leq \dim_A(M)‎ - ‎\mathrm{depth}_{J}(f‎, ‎M)$‎, ‎where $\gamma_{J}(f‎, ‎M)$ denotes the analytic spread of $f$ at $J$ with respect to $M$‎. 

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References

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Journal of Algebra and Related Topics
Volume 13, Issue 2
December 2025
Pages 199-208

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