Multiplicities of graded components of local cohomology modules
type: Item
instance of: scholarly article
title: Multiplicities of graded components of local cohomology modules
zbMATH Open document ID: 1071.13005
author: Fred Rohrer
author: Reza Sazeedeh
author: M. P. Brodmann
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-18T00:00:00Z
review text: Let \(M\) be a finitely generated graded module over a Noetherian homogeneous ring \(R\) with local base ring \((R_0,m_0).\) Let \(H^i_{R_+}(M)_n\) be the \(i\)-th local cohomology module of \(M\) with respect to the irrelevant ideal \(R_+\) of \(R.\) The authors study the multiplicity of \(H^i_{R_+}(M)_n\) relative to an \(m_0\)-primary ideal of \(R_0.\) They prove that it is antipolynomial in \(n\) of degree less than \(i\) when \(i\) is not greater than the cohomological finite length dimension of \(M\) or \(\dim (R_0) = 2.\) A similar property of the lengths of the graded components of various graded submodules of \(H^i_{R_+}(M)\) is also described. Among other things the cases \(\dim (R_0) = 0\) and \(\dim (R_0) = 1\) are analyzed following \textit{M. P. Brodmann} and \textit{R. Y. Sharp} [``Local cohomology. An algebraic introduction with geometric applications.'' Cambridge: Cambridge University Press (1998; Zbl 0903.13006)] and \textit{M. Brodmann, S. Fumasoli} and \textit{R. Tajarod} [Proc. Am. Math. Soc. 131, No. 10, 2977--2985 (2003; Zbl 1041.13012)], respectively.
reviewed by: Aleksandr G. Aleksandrov
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 13A02
zbMATH DE Number: 2157528
zbMATH Keywords: homogeneous rings
zbMATH Keywords: graded modules
zbMATH Keywords: local cohomology
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Hilbert-Samuel multiplicity
zbMATH Keywords: antipolynomial function
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.034
OpenAlex ID: W2028436124
cites work: Cohomological invariants of coherent sheaves over projective schemes: A survey.
cites work: Low-codimensional associated primes of graded components of local cohomology modules
cites work: Local cohomology over homogeneous rings with one-dimensional local base ring
cites work: Cohomological patterns of coherent sheaves over projective schemes
cites work: Associated primes of graded components of local cohomology modules
cites work: Bounds for cohomological deficiency functions of projective schemes over Artinian rings
cites work: Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules
cites work: Q5441001
cites work: Q3843140
cites work: Some properties of top graded local cohomology modules.
cites work: ARTINIAN MODULES AND HILBERT POLYNOMIALS
cites work: Some properties of graded local cohomology modules
cites work: Baire's category theorem and prime avoidance in complete local rings
cites work: Q5693341
DOI: 10.1016/J.JPAA.2004.08.034
description: scientific article; zbMATH DE number 2157528