Quintasymptotic sequences over an ideal and quintasymptotic cograde

TL;DR

This paper introduces the concepts of quintasymptotic sequences and cograde over ideals in Noetherian rings, establishing their properties and applications in characterizing local rings.

Contribution

It defines and studies quintasymptotic sequences and cograde, extending the theory of quintessential sequences and providing tools for ring characterization.

Findings

Quintasymptotic cograde is well-defined in local rings.

Properties of quintasymptotic cograde are established under ring extensions.

Used to characterize specific classes of local rings.

Abstract

Let $I$ denote an ideal of a Noetherian ring $R$. The purpose of this article is to introduce the concepts of quintasymptotic sequences over $I$ and quintasymptotic cograde of $I$, and it is shown that they play a role analogous to quintessential sequences over $I$ and quintessential cograde of $I$, given in \cite{Ra1}. Also, we show that, if $R$ is local, then the quintasymptotic cograde of $I$ is unambiguously defined and behaves well when passing to certain related local rings. Finally, we use this cograde to characterize of two classes of local rings.