On some cconjectures about the Chern numbers of filtrations

Abstract

Let I be an m-primary ideal of a Noetherian local ring (R,m) of positive dimension. The coefficient $e_1(A)$ of the Hilbert polynomial of an I-admissible filtration A is called the Chern number of A. The Positivity Conjecture of Vasconcelos for the Chern number of the integral closure filtration $\overline{I^n}$ is proved for a 2-dimensional complete local domain and more generally for any analytically unramified local ring R whose integral closure in its total ring of fractions is Cohen-Macaulay as an R-module. It is proved that if I is a parameter ideal then the Chern number of the I-adic filtration is non-negative. Several other results on the Chern number of the integral closure filtration are established, especially in the case when R is not necessarily Cohen-Macaulay.

arXiv comment20 pages

Cited by in corpus (5)

• Cohen--Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals
• A Note on the Buchsbaum-Rim function of a parameter module
• Complexity Degrees of Algebraic Structures
• Negativity Conjecture for the First Hilbert Coefficient
• Negativity of the Chern number of parameter ideals