The Asymptotic Samuel Function of a Filtration

TL;DR

This paper extends the asymptotic Samuel function from ideals to filtrations, exploring their properties, differences, and special cases like discrete valued filtrations, revealing new insights into their structure.

Contribution

It introduces the asymptotic Samuel function for filtrations, analyzes their properties, and compares them to ideals, including special cases like discrete valued filtrations.

Findings

Many properties of the asymptotic Samuel function for ideals hold for filtrations.

Differences between ideals and filtrations are identified and studied.

Discrete valued filtrations exhibit particularly nice properties.

Abstract

We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties.