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

DOI:
https://doi.org/10.1090/S0002-9939-04-07779-2

Published electronically:
November 19, 2004

MathSciNet review:
2117198

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Abstract | References | Similar Articles | Additional Information

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

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