On the Chern number of $I$-admissible filtrations of ideals

TL;DR

This paper investigates the Chern number of $I$-admissible filtrations in Noetherian local rings, providing formulas involving Koszul complexes and applying these to derive new proofs of known results in low-dimensional cases.

Contribution

It introduces a formula for the Chern number using Euler characteristics of Koszul subcomplexes and applies it to obtain unified proofs in rings of dimension up to two.

Findings

Derived a formula for the Chern number involving Koszul complexes.

Provided explicit formulas for rings of dimension at most two.

Used these formulas to give new proofs of existing results.

Abstract

Let $I$ be an $\m$-primary ideal of a Noetherian local ring $(R, \m)$ of positive dimension. The coefficient $e_1(\mathcal I)$ of the Hilbert polynomial of an $I$-admissible filtration $\mathcal I$ is called the Chern number of $\mathcal I$. A formula for the Chern number has been derived involving Euler characteristic of subcomplexes of a Koszul complex. Specific formulas for the Chern number have been given in local rings of dimension at most two. These have been used to provide new and unified proofs of several results about $e_1(\mathcal I)$.