Variation of Hilbert Coefficients

TL;DR

This paper investigates how the first two Hilbert coefficients of an ideal's filtration relate to properties of Noetherian local rings and their blowups, providing estimations when the ideal is enlarged.

Contribution

It offers new estimations for Hilbert coefficients in Noetherian local rings when ideals are enlarged, extending understanding of their algebraic and geometric properties.

Findings

Provided bounds for $e_0$ and $e_1$ under ideal enlargement.

Connected Hilbert coefficients to properties of blowups and normalizations.

Extended estimations to general Noetherian local rings.

Abstract

For a Noetherian local ring $(\RR, \m)$, the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an $\m$-primary ideal $I$ are known to code for properties of $\RR$, of the blowup of $\spec(\RR)$ along $V(I)$, and even of their normalizations. We give estimations for these coefficients when $I$ is enlarged (in the case of $e_1$ in the same integral closure class) for general Noetherian local rings.