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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules
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by M. Brodmann and F. Rohrer PDF
Proc. Amer. Math. Soc. 133 (2005), 987-993 Request permission

Abstract:

Let $R = \bigoplus _{n \geq 0} R_n$ be a Noetherian homogeneous ring with one-dimensional local base ring $(R_0, {\mathfrak m}_0)$. Let ${\mathfrak q}_0 \subseteq R_0$ be an ${\mathfrak m}_0$-primary ideal, let $M$ be a finitely generated graded $R$-module and let $i \in {\mathbb N}_0$. Let $H^i_{R_+}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+:= \bigoplus _{n > 0} R_n$ of $R$. We show that the first Hilbert-Samuel coefficient $e_1 \big ( {\mathfrak q}_0, H^i_{R_+}(M)_n \big )$ of the $n$-th graded component of $H^i_{R_+}(M)$ with respect to ${\mathfrak q}_0$ is antipolynomial of degree $< i$ in $n$. In addition, we prove that the postulation numbers of the components $H^i_{R_+} (M)_n$ with respect to ${\mathfrak q}_0$ have a common upper bound.
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Additional Information
  • M. Brodmann
  • Affiliation: Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
  • MR Author ID: 41830
  • Email: brodmann@math.unizh.ch
  • F. Rohrer
  • Affiliation: Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
  • Email: fred@math.unizh.ch
  • Received by editor(s): December 1, 2003
  • Published electronically: November 19, 2004
  • Communicated by: Bernd Ulrich
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 987-993
  • MSC (2000): Primary 13D45, 13E10
  • DOI: https://doi.org/10.1090/S0002-9939-04-07779-2
  • MathSciNet review: 2117198