Graded filtrations and ideals of reduction number two

TL;DR

This paper introduces a method to construct graded filtrations of modules and applies it to analyze Hilbert functions, leading to inequalities for ideals with reduction number 2 or 3.

Contribution

It provides a new construction technique for graded filtrations and derives inequalities for Hilbert coefficients in specific ideal cases.

Findings

Constructed graded filtrations for modules.

Applied filtrations to Sally modules and Hilbert functions.

Established inequalities for Hilbert coefficients of certain ideals.

Abstract

In this paper, we give a way to construct graded filtrations of graded modules. We then apply it to the Sally module, which describes a correction term of the Hilbert function. As a result, we obtain the inequality of the Hilbert coefficients for ideals of reduction number $2$ or $3$.