GRADED BETTI NUMBERS OF POWERS OF IDEALS AMIR BAGHERI AND KAMRAN LAMEI 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 posi- tive Z-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of Z-grading, Z 2 can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree (μ, t), dim k ( Tor S i (I t ,k) μ ) is equal to one of these polynomials in (μ, t). This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals in [18]. Our main statement treats the case of a power products of homogeneous ideals in a Z d -graded algebra, for a positive grading, in the sense of [20]. 1. Introduction The study of homological invariants of powers of ideals goes back, at least, to the work of Brodmann in the 70’s and attracted a lot of attention in the last two decades. One of the most important results in this area is the result on the asymptotic linearity of Castelnuovo-Mumford regularity obtained by Kodiyalam [19] and Cutkosky, Herzog and Trung [12], independently. The proof of Cutkosky, Herzog and Trung further describes the eventual linearity in t of end ( Tor S i (I t ,k) ) := max{µ| Tor S i (I t ,k) µ =0}. It is natural to concern the asymptotic behavior of Betti numbers β i (I t ) := dim k Tor S i (I t ,k) as t varies. In [21], Northcott and Rees already investigated the asymptotic behavior of β k 1 (I t ). Later, using the Hilbert-Serre theorem, Kodiyalam [18, Theorem 1] proved that for any non-negative integer i and sufficiently large t, the i-th Betti number, β k i (I t ), is a polynomial Q i in t of degree at most the analytic spread of I minus one. Recently, refining the result of [12] on end ( Tor S i (I t ,k) ) ,[4] gives a precise picture of the set of degrees γ such that Tor S i (I t ,A) γ = 0 when t runs over N. In [4], the authors consider a polynomial ring S = A[x 1 ,...,x n ] over a Noetherian ring A graded by a finitely generated abelian group G (see [4, Theorem 4.6]). 2000 Mathematics Subject Classification. 13A30, 13D02, 13D40. Key words and phrases. Betti numbers, Nonstandard Hilbert function, Vector partition function. The research of Amir Bagheri was in part supported by a grant from IPM (No. 93130024). 1 arXiv:1308.0943v2 [math.AC] 5 Aug 2015