On generalized Narita ideals

TL;DR

This paper studies generalized Narita ideals in Cohen-Macaulay local rings, showing their properties extend to modules and establishing bounds on the regularity of associated graded modules.

Contribution

It introduces the concept of generalized Narita ideals and proves their properties for maximal Cohen-Macaulay modules, including vanishing of certain Hilbert coefficients and bounds on regularity.

Findings

Vanishing of Hilbert coefficients for modules over generalized Narita ideals

Associated graded modules are generalized Cohen-Macaulay

Existence of a uniform bound on the regularity of associated graded modules

Abstract

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$. An $\mathfrak{m}$-primary ideal $I$ is said to be a generalized Narita ideal if $e_i^I(A) = 0$ for $2 \leq i \leq d$. If $I$ is a generalized Narita ideal and $M$ is a maximal Cohen-Macaulay $A$-module then we show $e_i^I(M) = 0$ for $2 \leq i \leq d$. We also have $G_I(M)$ is generalized Cohen-Macaulay. Furthermore we show that there exists $c_I$ (depending only on $A$ and $I$) such that $\text{reg} \ G_I(M) \leq c_I$.