UZH-Logo

Maintenance Infos

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


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).

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

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

Citations

1 citation in Web of Science®
1 citation in Scopus®
Google Scholar™

Altmetrics

Downloads

25 downloads since deposited on 03 Feb 2010
11 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Local cohomology modules, graded components, Hilbert-Samuel polynomials
Language:English
Date:2005
Deposited On:03 Feb 2010 08:42
Last Modified:05 Apr 2016 13:24
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
Permanent URL: http://doi.org/10.5167/uzh-21680

Download

[img]
Preview
Filetype: PDF
Size: 169kB
View at publisher

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations