Hilbert polynomials and powers of ideals

By Jürgen Herzog, Tony J. Puthenpurakal, J. Verma. Mathematical Proceedings of the Cambridge Philosophical Society, 2007

65 citations recorded by the source corpus.

Abstract 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 $A=\Dirsum_{k\geq 0}J_k$ and $B=\Dirsum_{k\geq 0}I_k$ 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 $\lim_{k\to \infty}\length(\Gamma_\mm(S/I^k))/k^n \in \mathbb{Q}$, if I is a monomial ideal. We also study analogous statements in the local case.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7298dce43319b03c30818d0c9d84be1bf7fd38c7

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In Algebra and Number Theory

  1. Steven Dale Cutkosky
  2. Shreeram S. Abhyankar
  3. Chi Li
  4. Kiumars Kaveh
  5. Askold Khovanskii
  6. Franz-Viktor Kuhlmann

In Geometry and Topology

  1. Steven Dale Cutkosky
  2. Shreeram S. Abhyankar
  3. Chi Li
  4. Kiumars Kaveh
  5. Askold Khovanskii
  6. Franz-Viktor Kuhlmann

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