On the depth of blow-up rings of ideals of minimal mixed multiplicity

TL;DR

This paper investigates the depth properties of blow-up rings of ideals with minimal mixed multiplicity in Cohen-Macaulay rings, establishing conditions under which these rings are Cohen-Macaulay or have related depth properties.

Contribution

It proves a depth inequality linking the fiber and associated graded rings for ideals of minimal mixed multiplicity and characterizes the Cohen-Macaulayness of these blow-up rings in specific cases.

Findings

Depth of the fiber ring is at least d-1 when the associated graded ring has depth at least d-1.

In two-dimensional regular local rings, the depths of the Rees algebra, associated graded ring, and fiber ring are equal.

An infinite class of ideals where the Rees algebra is Cohen-Macaulay but the fiber ring is not.

Abstract

We show that if $(R, \m)$ is a Cohen-Macaulay local ring and $I$ is an ideal of minimal mixed multiplicity, then $\depth G(I) \geq d- 1$ implies that $\depth F(I) \geq d-1$. We use this to show that if $I$ is a contracted ideal in a two dimensional regular local ring then $\depth R[It]-1= \depth G(I) = \depth F(I)$. We also give an infinite class of ideals where $R[It]$ is Cohen-Macaulay but $F(I)$ is not.