The Chern coefficients of local rings

TL;DR

This paper investigates the Chern coefficients, specifically the first Hilbert coefficient, of local rings, providing new techniques to understand their positivity and bounds in general geometric local domains.

Contribution

It introduces novel methods based on Cohen-Macaulay modules and extended multiplicity functions to analyze the positivity and bounds of the first Hilbert coefficient in arbitrary geometric local domains.

Findings

Established positivity criteria for $e_1(\

Derived bounds for $e_1(\

Abstract

The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring $(R, \mathfrak{m})$. For a Noetherian (good) filtration $\mathcal{A}$ of $\mathfrak{m}$-primary ideals, the positivity and bounds for $e_1(\mathcal{A})$ are well-studied if $R$ is Cohen-Macaulay, or more broadly, if $R$ is a Buchsbaum ring or mild generalizations thereof. For arbitrary geometric local domains, we introduce techniques based on the theory of maximal Cohen-Macaulay modules and of extended multiplicity functions to establish the meaning of the positivity of $e_1(\mathcal{A})$, and to derive lower and upper bounds for $e_1(\mathcal{A})$.