Quintasymptotic sequences over an ideal and quintasymptotic cograde

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 to show that they play a role analogous to quintessential sequences over $I$ and quintessential cograde of $I$. 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. Also, we use this cograde to characterize two classes of local rings.