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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules


Authors: M. Brodmann and F. Rohrer
Journal: Proc. Amer. Math. Soc. 133 (2005), 987-993
MSC (2000): Primary 13D45, 13E10
Published electronically: November 19, 2004
MathSciNet review: 2117198
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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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07779-2
PII: S 0002-9939(04)07779-2
Keywords: Local cohomology modules, graded components, Hilbert-Samuel polynomials
Received by editor(s): December 1, 2003
Published electronically: November 19, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society