On the Set of Chern Numbers in Local Rings

Abstract

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