Amir Bagheri, Kamran Lamei. Source: Journal of Commutative Algebra, Volume 12, Number 2, 153--169.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 [math] -grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials.
Specially, in the case of [math] -grading, for each homological degree [math] we can split [math] in a finite number of regions such that for each region there is a polynomial in [math] and [math] that computes [math] . 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 [math] -graded algebra, for a positive grading.