Higher iterated Hilbert coefficients of the graded components of bigraded modules

TL;DR

This paper investigates the polynomial behavior of higher iterated Hilbert coefficients of certain bigraded modules derived from graded ideals in polynomial rings, providing bounds on their degrees.

Contribution

It introduces a method to analyze the polynomial nature of Hilbert coefficients of bigraded modules related to powers of ideals, with explicit degree bounds.

Findings

Higher iterated Hilbert coefficients are polynomial in k.

Degree bounds are established for these polynomial functions.

Results apply to modules like Tor and Ext involving ideal powers.

Abstract

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over the field $K$, and let $I\subset S$ be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded $S$-modules $\Tor_i^S(M,I^k)$ and $\Ext^i_S(M,I^k)$ are polynomial functions in $k$, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules.