Hilbert polynomials and powers of ideals

Abstract

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert coefficients $e_i(M/I^{k}M)$ are polynomial functions. Given two families of graded ideals $(I_k{)}_{k\ge 0}$ and $(J_k{)}_{k\ge 0}$ with $J_k\subset I_k$ for all $k$ with the property that $J_k{J}_{\ell }\subset J_{k+\ell }$ and $I_k{I}_{\ell }\subset I_{k+\ell }$ for all $k$ and $\ell$, and such that the algebras $A=\Dirsum_{k\geq 0}J_k$ and $B=\Dirsum_{k\geq 0}I_k$ are finitely generated, we show the function $k{\mapsto }_0(I_k/J_k)$ is of quasi-polynomial type, say given by the polynomials $P_0,...,P_{g-1}$. If $J_k=J^k$ for all $k$ then we show that all the $P_i$ have the same degree and the same leading coefficient. As one of the applications it is shown that $\lim_{k\to \infty}\length(Γ_\mm(S/I^k))/k^n \in \mathbb{Q}.$ We also study analogous statements in the local case.

24 pages