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Asymptotic Behavior of the Length of Local Cohomology

  Published:2005-12-01
 Printed: Dec 2005
  • Steven Dale Cutkosky
  • Huy Tài Hà
  • Hema Srinivasan
  • Emanoil Theodorescu
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Abstract

Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring, and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in $R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show that $$ \lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d} =\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d} $$ always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals $I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth.
Keywords: powers of ideals, local cohomology, Hilbert function, linear growth powers of ideals, local cohomology, Hilbert function, linear growth
MSC Classifications: 13D40, 14B15, 13D45 show english descriptions Hilbert-Samuel and Hilbert-Kunz functions; Poincare series
Local cohomology [See also 13D45, 32C36]
Local cohomology [See also 14B15]
13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincare series
14B15 - Local cohomology [See also 13D45, 32C36]
13D45 - Local cohomology [See also 14B15]
 

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