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Asymptotic Behavior of the Length of Local Cohomology

Published online by Cambridge University Press:  20 November 2018

Steven Dale Cutkosky
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri, 65201 USA, email: cutkoskys@missouri.edu, tai@math.missouri.edu, srinivasanh@missouri.edu, theodore@math.missouri.edu
Huy Tài Hà
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri, 65201 USA, email: cutkoskys@missouri.edu, tai@math.missouri.edu, srinivasanh@missouri.edu, theodore@math.missouri.edu
Hema Srinivasan
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri, 65201 USA, email: cutkoskys@missouri.edu, tai@math.missouri.edu, srinivasanh@missouri.edu, theodore@math.missouri.edu
Emanoil Theodorescu
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri, 65201 USA, email: cutkoskys@missouri.edu, tai@math.missouri.edu, srinivasanh@missouri.edu, theodore@math.missouri.edu
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Abstract

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Let $k$ be a field of characteristic $0,\,R\,=\,k\left[ {{x}_{1}},\,\ldots ,\,{{x}_{d}} \right]$ be a polynomial ring, and $m$ 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

$$\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( H_{m}^{0}\left( R/{{I}^{n}} \right) \right)}{{{n}^{d}}}\,=\,\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( \text{Ext}_{R}^{d}\left( R/{{I}^{n}},\,R\left( -d \right) \right) \right)}{{{n}^{d}}}$$

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 extension modules may not have polynomial growth.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2005

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