Defect Functions Between Filtrations of Ideals

TL;DR

This paper introduces the defect function for pairs of ideal filtrations, showing it is asymptotically a quasi-polynomial and analyzing its structure for specific filtrations, extending the understanding of symbolic and ordinary powers.

Contribution

It generalizes the symbolic defect to a broader class of filtrations and establishes asymptotic polynomial behavior under certain algebraic conditions.

Findings

Defect function is asymptotically a quasi-polynomial.

The defect function becomes polynomial when Rees algebra is standard graded.

Top coefficients of the quasi-polynomial are constant under natural hypotheses.

Abstract

We introduce and study the defect function associated to a pair of filtrations of ideals, which generalizes the symbolic defect of ideals. Under the assumption that the Rees algebra of one filtration is Noetherian and that a natural graded module measuring the interaction between the filtrations is finitely generated over it, we show that the corresponding defect function is asymptotically a quasi-polynomial. Moreover, the defect function becomes eventually polynomial when the Rees algebra of the first filtration is standard graded. For filtrations arising from saturations and ordinary powers of monomial ideals, we further analyze the structure of the quasi-polynomial. We prove that the top two coefficients of the eventual quasi-polynomial are constant under natural hypotheses.