Multiplicities of a bigraded ring and intersection theory
type: Item
instance of: scholarly article
title: Multiplicities of a bigraded ring and intersection theory
zbMATH Open document ID: 0894.14005
published in: Mathematische Annalen
publication date: 1998-08-13T00:00:00Z
review text: Let \(X\) and \(Y\) be projective varieties in \({\mathbb P}^n_K\). The Samuel multiplicity of an \(\mathfrak m\)-primary ideal in a local ring \((A,{\mathfrak m})\) can be used to define the intersection number of an irreducible component of the intersection of \(X\) and \(Y\). In this paper the authors define a multiplicity sequence \(c_0 (I,A),\dots,c_d (I,A)\) for an arbitrary ideal \(I\) of a \(d\)-dimensional local ring \((A,{\mathfrak m})\) which is closely related to the Stückrad-Vogel intersection cycle. It is defined by means of a certain bigraded ring \(G_{\mathfrak m} (G_I (A))\). The main result of this paper implies that each number of the multiplicity sequence equals the (local) degree of the part of the cycle in a certain dimension. Applications include an interpretation of the Segre classes of a subscheme as multiplicities in a bigraded ring, and a local version of Bezout's theorem. In the case where \(I\) is \(\mathfrak m\)-primary, \(c_0 (I,A)\) is the Samuel multiplicity of \(I\) and it is the only element of the sequence which is non-zero. If the embedded join of \(X\) and \(Y\) has minimal dimension, then again the sequence reduces to only one element different from zero. Interesting connections are made to earlier papers by the authors.
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 1121712
zbMATH Keywords: intersection theory
zbMATH Keywords: Segre classes
zbMATH Keywords: Samuel multiplicity
zbMATH Keywords: Stückrad-Vogel intersection cycle
zbMATH Keywords: embedded join
zbMATH Keywords: Bezout's theorem
zbMATH Keywords: analytic spread
author: Rüdiger Achilles
author: Mirella Manaresi
reviewed by: Juan C. Migliore
MaRDI profile type: Publication
full work available at URL: s002080050128
OpenAlex ID: W2057812107
DOI: 10.1007/S002080050128
Recommended article: Intersection multiplicities over Gorenstein rings
Recommended article: On the index of contact and multiplicities for bigraded rings
Recommended article: Q4295725
Recommended article: Q3728113
Recommended article: Generalized Intersection Multiplicities of Modules
Recommended article: Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity
Recommended article: An intersection multiplicity in terms of \(\text{Ext}\)-modules
Recommended article: Intersection multiplicity over noncommutative algebras
Recommended article: Intersection multiplicities on abelian varieties
description: scientific article; zbMATH DE number 1121712