Multiplicity for ideals of maximal analytic spread and intersection theory
type: Item
instance of: scholarly article
title: Multiplicity for ideals of maximal analytic spread and intersection theory
zbMATH Open document ID: 0816.13019
author: Rüdiger Achilles
author: Mirella Manaresi
published in: Journal of Mathematics of Kyoto University
publication date: 1995-07-09T00:00:00Z
review text: Let \((A, {\mathfrak m})\) be a local ring and \(I\) an ideal in \(A\) having maximal analytic spread. The authors define a multiplicity \(\mu (I,A)\) for \(I\) which coincides with the usual multiplicity in case \(I\) is \({\mathfrak m}\)-primary. If \(K\) is an algebraically closed field, the Stückrad-Vogel intersection number for a \(K\)-rational component of an intersection in a projective space over \(K\) can be interpreted as such a multiplicity. Denote by \(G\) the associated graded ring \(\text{gr}_ I (A)\) of \(A\) with respect to \(I\). Then \(\mu (I,A) : = \sum e (G/{\mathfrak p})\) length\((G_{\mathfrak p})\), where \({\mathfrak p}\) runs over all associated prime ideals of \(G/{\mathfrak m} G\) in \(G\) having dimension \(\dim (G/{\mathfrak m} G)\), and \(e (G/{\mathfrak p})\) is the (usual) multiplicity of the graded ring \(G/{\mathfrak p}\). \(\mu (I,A)\) can be computed by means of `super-reductions' for \(I\) which generalize superficial systems of parameters for \(I\) in case \(I\) is \({\mathfrak m}\)-primary.
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 584580
zbMATH Keywords: local ring
zbMATH Keywords: maximal analytic spread
zbMATH Keywords: multiplicity
zbMATH Keywords: Stückrad-Vogel intersection number
zbMATH Keywords: associated graded ring
reviewed by: Udo Vetter
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519127
OpenAlex ID: W2117652825
DOI: 10.1215/KJM/1250519127
description: scientific article; zbMATH DE number 584580