The Signature of the Chern Coefficients of Local Rings

TL;DR

This paper investigates a conjecture linking the sign of the Chern coefficient to Cohen-Macaulayness in local rings, proving it in specific cases and providing criteria for detecting Cohen-Macaulayness.

Contribution

It proves the conjecture for rings that are homomorphic images of Gorenstein rings and for certain integral domains, and develops criteria for Cohen-Macaulayness detection.

Findings

Proved the conjecture for Gorenstein homomorphic images.

Established the equivalence between negative Chern coefficient and non Cohen-Macaulayness.

Derived criteria for Cohen-Macaulayness in graded modules.

Abstract

This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient $e_1(J)< 0$ is equivalent to $R$ being non Cohen-Macaulay. The conjecture is established if $R$ is a homomorphic image of a Gorenstein ring, and for all universally catenary integral domains containing fields. Criteria for the detection of Cohen-Macaulayness in equi-generated graded modules are derived.