Asymptotic Behavior of the Length of Local Cohomology

By Steven Dale Cutkosky, Huy Tài Hà, H. Srinivasan, Emanoil Theodorescu. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2004

48 citations recorded by the source corpus.

Abstract 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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a2b92ded767f20c5aa6bc3a2989fb16286938f97

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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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