GRADED BETTI NUMBERS OF POWERS OF IDEALS

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 Z(d)-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. Specially, in the case of Z-grading, for each homological degree i we can split Z(2) = {(mu, t) vertical bar t, mu is an element of Z} in a finite number of regions such that for each region there is a polynomial in mu and t that computes dim(k)(Tor(i)(S) (I-t, k)(mu). 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 Z(d)-graded algebra, for a positive grading.