Graded Betti numbers of powers of ideals

TL;DR

This paper studies the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in polynomial rings with positive grading, showing they are governed by finitely many polynomials in a multigraded setting.

Contribution

It extends the understanding of Betti numbers of ideal powers by proving they are described by finitely many polynomials in multigraded contexts, refining previous results.

Findings

Betti numbers are encoded by finitely many polynomials in the graded case.

The polynomial ring's grading induces a finite partition of the parameter space.

The results generalize Kodiyalam's theorem to multigraded algebras.

Abstract

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive $\ZZ$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of $\ZZ$-grading, $\ZZ^2$ can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu, t)$, $\dim_k \left(\tor_i^S(I^t, k)_{\mu} \right)$ is equal to one of these polynomials in $(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a $\ZZ^d$-graded algebra, for a positive grading.