On the Chern Number of an Ideal (original) (raw)

Hilbert polynomials and powers of ideals

Mathematical Proceedings of the Cambridge Philosophical Society, 2008

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k≥0 and (Jk)k≥0 with Jk ⊂ Ik for all k with the property that JkJℓ ⊂ Jk+ℓ and IkIℓ ⊂ Ik+ℓ for all k and ℓ, and such that the algebras A=Dirsumkgeq0JkA=\Dirsum_{k\geq 0}J_kA=Dirsumkgeq0Jk and B=Dirsumkgeq0IkB=\Dirsum_{k\geq 0}I_kB=Dirsumkgeq0Ik are finitely generated, we show the function k ↦ e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg−1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that limktoinftylength(Gammamm(S/Ik))/kninmathbbQ\lim_{k\to \infty}\length(\Gamma_\mm(S/I^k))/k^n \in \mathbb{Q}limktoinftylength(Gammamm(S/Ik))/kninmathbbQ, if I is a monomial ideal. We also study analogous statements in the local case.