A filtration of the Sally module and the First normal Hilbert Coefficient

TL;DR

This paper investigates the structure of the Sally module in relation to Hilbert coefficients, providing new insights and characterizations for the first normal Hilbert coefficient in Cohen-Macaulay local rings.

Contribution

It introduces a new filtration of the Sally module and characterizes the minimal value of the first Hilbert coefficient for normal filtrations.

Findings

New structural insights into the Sally module.

Characterization of the almost minimal first Hilbert coefficient.

Applications to normal filtrations in Cohen-Macaulay rings.

Abstract

The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results characterizing the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen-Macaulay local ring.