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 -grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials.
Specially, in the case of -grading, for each homological degree we can split in a finite number of regions such that for each region there is a polynomial in and that computes . 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 -graded algebra, for a positive grading.
"Graded Betti numbers of powers of ideals." J. Commut. Algebra 12 (2) 153 - 169, Summer 2020. https://doi.org/10.1216/jca.2020.12.153