On the irreducible components of globally defined semianalytic sets
type: Item
instance of: scholarly article
title: On the irreducible components of globally defined semianalytic sets
zbMATH Open document ID: 1370.14050
author: José F. Fernando
published in: Mathematische Zeitschrift
publication date: 2016-08-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1503.00990
full work available at URL: Fernando107.pdf
review text: Let \({\mathcal A}\) be a class of geometric objects, whatever it may be. The naive way to define a nonempty set \(X\in{\mathcal A}\) to be \({\mathcal A}\)-reducible is to say that there exist \(Y\subsetneq X\) and \(Z\subsetneq X\) in \({\mathcal A}\) such that \(X=Y\cup Z\). Of course, \(X\) is \({\mathcal A}\)-irreducible if it is not \({\mathcal A}\)-reducible. The \({\mathcal A}\)-irreducible components of \(X\in{\mathcal A}\) are the members of a family of \({\mathcal A}\)-irreducible sets \(X_i\in{\mathcal A}\) such that \(X_i\neq X_j\) if \(i\neq j\), \(X=\bigcup_{i}X_i\) and, whenever \(Y\in{\mathcal A}\) is \({\mathcal A}\)-irreducible, it contains a component \(X_i\) and \(Y\subset X\), then \(Y=X_i\). In a honest irreducibility theory, the family of irreducible components of a given object is unique. In such a case \(X\) is \({\mathcal A}\)-irreducible if it is its unique \({\mathcal A}\)-irreducible component. On the other hand in most situations we can attach to every \(X\in{\mathcal A}\) a ring \({\mathcal F}(X)\) of either \(\mathbb C\)-valued or \(\mathbb R\)-valued functions such that \(X\) is irreducible if and only if \({\mathcal F}(X)\) is an integral domain. The first example is given by the class \({\mathcal A}\) of algebraic subsets of \(\mathbb C^n\). As the ring \(\mathbb C[{ x}]\) of polynomial functions on \(\mathbb C^n\) is noetherian, the ideal \({\mathcal J}(X)\) of those polynomial functions vanishing on \(X\) is a finite intersection of prime ideals \({\mathfrak p}_1,\dots,{\mathfrak p}_r\). Let \(X_i:=\bigcap_{f\in{\mathfrak p}_i}f^{-1}(0)\). Then \(\{X_1,\dots,X_r\}\) is the family of irreducible components of \(X\). This interplay between algebra and geometry was succesfully employed to develop satisfactory theories of irreducible components for complex analytic sets and Stein spaces by H. Cartan, O. Forster, R. Remmert and K. Stein, and global real analytic sets, introduced by H. Cartan and studied by F. Bruhat and H. Whitney, i…
reviewed by: Jose Manuel Gamboa
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 58A07
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 32C20
zbMATH DE Number: 6618703
zbMATH Keywords: \(C\)-analytic and \(C\)-semianalytic sets
zbMATH Keywords: amenable \(C\)-semianalytic set
zbMATH Keywords: Zariski closure
zbMATH Keywords: Irreducibility, Irreducible components
zbMATH Keywords: analytic normalization.
MaRDI profile type: Publication
OpenAlex ID: W2963359140
arXiv ID: 1503.00990
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DOI: 10.1007/S00209-016-1634-9
description: scientific article; zbMATH DE number 6618703