On the set of Chern numbers in local rings

TL;DR

This paper characterizes Noetherian local rings based on the boundedness and finiteness of Chern numbers of certain ideals, linking these properties to key ring properties like Gorensteinness and Cohen-Macaulayness.

Contribution

It provides a new characterization of local rings through the behavior of their Chern numbers, connecting algebraic properties to numerical invariants.

Findings

Chern numbers are bounded above in certain local rings.

Finiteness of Chern numbers characterizes Gorenstein and Cohen-Macaulay properties.

Behavior of Chern numbers determines generalized Cohen-Macaulayness.

Abstract

This paper purposes to characterize Noetherian local rings $(R, \mathfrak{m})$ such that the Chern numbers of certain $\mathfrak{m}$-primary ideals in $R$ bounded above or range among only finitely many values. Consequently, we characterize the Gorensteinness, Cohen-Macaulayness, generalized Cohen-Macaulayness of local rings in terms of the behavior of its Chern numbers.