Graded Betti numbers of powers of ideals

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 ${\mathbb{Z}}^d$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials.

Specially, in the case of $\mathbb{Z}$-grading, for each homological degree $i$ we can split ${\mathbb{Z}}^2=\left\{\left(\mu ,t\right)\mid t,\mu \in \mathbb{Z}\right\}$ in a finite number of regions such that for each region there is a polynomial in $\mu$ and $t$ that computes ${dim}_k\left({Tor}_i^S{\left(I^t,k\right)}_{\mu }\right)$. 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 ${\mathbb{Z}}^d$-graded algebra, for a positive grading.