Gabrielov's rank condition is equivalent to an inequality of reduced orders
type: Item
instance of: scholarly article
title: Gabrielov's rank condition is equivalent to an inequality of reduced orders
zbMATH Open document ID: 0612.32013
DOI: 10.1007/BF01450926
author: Shuzo Izumi
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164185
review text: Let \(\Phi: (Y,\eta)\to (X,\xi)\) be a morphism of germs of complex spaces. The paper under review relates the order of vanishing of the germ of holomorphic function f on x at \(\xi\), to the order of vanishing of the germ \(f\circ \Phi\) on Y at \(\eta\). Let \(\phi\) denote the naturally induced homorphism on the function rings \((\phi (f)=f\circ \Phi)\), and suppose \(\phi\) is injective. The author derives necessary and sufficient conditions for the existence of a number \(n\geq 1\) such that \({\bar \nu}\)(\(\phi\) (f))\(\leq n{\bar \nu}(f)\) for arbitrary f. Here \({\bar \nu}\) is the reduced order. The author defines the componentwise geometric rank of \(\phi\), and shows such n exists if and only if this rank is full.
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 13J05
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 3989771
zbMATH Keywords: analytic local ring
zbMATH Keywords: germs of complex spaces
zbMATH Keywords: order of vanishing
reviewed by: G. A. Harris
MaRDI profile type: Publication
cites work: When \(\varphi(f)\) convergent implies \(f\) is convergent
cites work: FORMAL RELATIONS BETWEEN ANALYTIC FUNCTIONS
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Linear complementary inequalities for orders of germs of analytic functions
cites work: A measure of integrity for local analytic algebras
cites work: Note on a paper of Samuel concerning asymptotic properties of ideals
cites work: Valuations Associated with a Local Ring (I)
cites work: Valuations Associated with a Local Ring (II)
cites work: Q3973367
description: scientific article; zbMATH DE number 3989771