On the lengths of quotients of ideals and depths of fiber cones

TL;DR

This paper investigates the depth properties of fiber cones of ideals in Cohen-Macaulay local rings, linking these properties to the lengths of specific quotient modules and depth conditions of associated graded rings.

Contribution

It provides new depth criteria for fiber cones based on length conditions of ideal quotients and depth assumptions on associated graded rings.

Findings

Depth of fiber cones determined by length conditions

Connections between depth of fiber cones and associated graded rings

New criteria for Cohen-Macaulayness of fiber cones

Abstract

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition on quotients of powers of $I$ and $J$, namely $\sum_{n\geq0}\lambda(\mathfrak{m}I^{n+1}/\mathfrak{m}JI^n)$ and $\sum_{n\geq0}\lambda(\mathfrak{m}I^{n+1} \cap J/\mathfrak{m}JI^n)$.