Quasi-Hilbert rings and Ratliff-Rush filtrations

Abstract

Let $A$ be a non Gorenstein Cohen Macaulay ring of dimension $d\ge 1$, $I$ an ideal of $A$, and suppose ${\omega }_A$ is a canonical $A$-module. Set $$r(I,{\omega }_A)=\bigcup _{n\ge 0}(I^{n+1}{\omega }_A:I^n{\omega }_A)\subseteq A.$$ We show that the ideal $r(I,-)$ is ${\omega }_A$ invariant. Motivated by this property, we introduce a new class of rings, which we call quasi Hilbert rings. We provide several examples of quasi Hilbert rings and discuss a number of their applications. Let $A$ be a local ring with maximal ideal $\mathfrak{m}$. We prove that $A$ is quasi Hilbert iff $\hat{A}$ is quasi Hilbert, where $\hat{A}$ is the completion of $A$ w.r.t. $\mathfrak{m}.$ If $d\ge 2$ and $x\in \mathfrak{m}\setminus {\mathfrak{m}}^2$ is an $A⨁{\omega }_A$ superficial element, we prove that if $A$ is quasi Hilbert, then so is $A/(x)$. Writing $\tilde{I}$ for the Ratliff Rush closure of an ideal $I$, we also provide sufficient conditions ensuring the vanishing of $r(I^n,{\omega }_A)/\widetilde{I^n}$ for all $n\ge 1.$

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