irreducible component

Let $G\subset {ℂ}^N$ be an open set.

Definition.

A locally analytic set (or an analytic variety) $V\subset G$ is said to be irreducible if whenever we have two locally analytic sets $V_1$ and $V_2$ such that $V=V_1\cup V_2$, then either $V=V_1$ or $V=V_2$. Otherwise $V$ is said to be . A maximal irreducible subvariety of $V$ is said to be an irreducible component of $V$. Sometimes irreducible components are called ircomps.

Note that if $V$ is an analytic variety in $G$, then a subvariety $W$ is an irreducible component of $V$ if and only if $W^{*}$ (the set of regular points of $W$) is a connected complex analytic manifold. This means that the irreducible components of $V$ are the closures of the connected components of $V^{*}$.

Title	irreducible component
Canonical name	IrreducibleComponent1
Date of creation	2013-03-22 15:04:58
Last modified on	2013-03-22 15:04:58
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Definition
Classification	msc 32C25
Classification	msc 32A60
Synonym	ircomp
Related topic	AnalyticSet
Defines	irreducible analytic variety
Defines	irreducible locally analytic set
Defines	irreducible analytic variety
Defines	reducible locally analytic set
Defines	reducible analytic variety