On the set of Chern numbers in local rings (Q6563085)

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d.$ Assume that $M$ is a finitely generated $R$-module of dimension $d.$ Let $I$ be an $\mathfrak{m}$-primary ideal of $M.$ The behavior of the values of Hilbert coefficients $\{e_i(I,M)_{i=0}^d\}$ can be used to characterize Cohen-Macaulayness, Buchsbaumness, and generalized Cohen-Macaulayness of modules $M.$ In this paper, the authors investigated the characterizations of Cohen-Macaulay rings in terms of their Chern numbers and $C$-parameter ideals provided that $R$ is unmixed of dimension $d \ge 2.$ Moreover, the paper showed some results on the finiteness of sets $$\Xi_i(R)=\{e_i(\mathfrak{q}:\mathfrak{m})\mid \mathfrak{q} \text{ is a $C$-parameter ideal of }R\},$$ and $$\Omega_1^t(R)=\{e_1(\mathfrak{q}:\mathfrak{m})\mid \mathfrak{q} \text{ is a $C$-parameter ideal of $R$ contained in }\mathfrak{m}^t\}.$$