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 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 = { ( μ , t ) ∣ t , μ ∈ ℤ } in a finite number of regions such that for each region there is a polynomial in μ and t that computes dim k ( Tor i S ( I t , k ) μ ) . 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.