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

Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-21680

Brodmann, M; Rohrer, F (2005). Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules. Proceedings of the American Mathematical Society, 133(4):987-993 (electronic).

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Abstract

Let $R=\bigoplus_{n\ge 0}R_n$ be a positively graded commutative Noetherian ring with one-dimensional local base ring $(R_0,\germ{m}_0)$ . Let $\germ{q}_0\subseteq R_0$ be an $\germ{m}_0$ -primary ideal. Let $M$ be a finitely generated graded $R$ -module and let $i\in\Bbb N_0$ . 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 $R$ . The main results in this paper are: $\roster$ $\item$ "(a)" Assume that $\dim_{R_0}(H^i_{R_+}(M)_n)=1$ for all $n\ll 0$ . Then there is a polynomial $S(x)\in\Bbb Q[x]$ of degree $<i$ such that for all $n\ll 0$ , $$$$ e_1( $\germ{q}$ _0,H^i_ ${R_+}$ (M)_n)=S(n), $$$$ where $e_1(\germ{q}_0,H^i_{R_+}(M)_n)$ is the so-called first Hilbert-Samuel coefficient of $H^i_{R_+}(M)_n$ with respect to $\germ{q}_0$ . $\item$ "(b)" There is some $c\in\Bbb N_0$ such that $\mu(\germ{q}_0,H^i_{R_+}(M)_n)\le c$ for all $n\in\Bbb Z$ , where $\mu(\germ{q}_0,H^i_{R_+}(M)_n)$ is the postulation number of the $R_0$ -module $H^i_{R_+}(M)_n$ with respect to $\germ{q}_0$ . $\endroster$

Item Type:	Journal Article, refereed, original work
Communities & Collections:	07 Faculty of Science > Institute of Mathematics
DDC:	510 Mathematics
Uncontrolled Keywords:	Local cohomology modules, graded components, Hilbert-Samuel polynomials
Language:	English
Date:	2005
Deposited On:	03 Feb 2010 09:42
Last Modified:	24 Nov 2012 23:08
Publisher:	American Mathematical Society
ISSN:	0002-9939
Additional Information:	First published in [Proc. Amer. Math. Soc. 133 (2005), no. 4, 987--993 (electronic)], published by the American Mathematical Society
Publisher DOI:	10.1090/S0002-9939-04-07779-2
Related URLs:	http://www.ams.org/mathscinet-getitem?mr=2117198

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