Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals → CJM
Abstract view

# Asymptotic Behavior of the Length of Local Cohomology

 Read article[PDF: 184KB]
Published:2005-12-01
Printed: Dec 2005
• Steven Dale Cutkosky
• Huy Tài Hà
• Hema Srinivasan
• Emanoil Theodorescu
 Format: HTML LaTeX MathJax PDF PostScript

## 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
 MSC Classifications: 13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincare series 14B15 - Local cohomology [See also 13D45, 32C36] 13D45 - Local cohomology [See also 14B15]

 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2016 : https://cms.math.ca/