View source for Unibranch local ring

In [[algebraic geometry]], a [[local ring]] ''A'' is said to be '''unibranch''' if the [[reduced ring]] ''A''<sub>red</sub> (obtained by quotienting ''A'' by its [[nilradical of a ring|nilradical]]) is an [[integral domain]], and the [[integral closure]] ''B'' of ''A''<sub>red</sub> is also a local ring.{{fact|date=August 2012}} A unibranch local ring is said to be '''geometrically unibranch''' if the [[residue field]] of ''B'' is a [[purely inseparable extension]] of the residue field of ''A''<sub>red</sub>. A complex variety ''X'' is called '''topologically unibranch''' at a point ''x'' if for all complements ''Y'' of closed algebraic subsets of ''X'' there is a fundamental system of neighborhoods (in the classical topology) of ''x'' whose intersection with ''Y'' is connected. 

In particular, a [[normal ring]] is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:

'''Theorem''' <ref>{{EGA |book=III-1|loc=III.4.3.7}}</ref> Let ''X'' and ''Y'' be two integral locally noetherian schemes and <math>f \colon X \to Y</math> a [[proper morphism|proper]] [[dominant morphism]]. Denote their function fields by ''K(X)'' and ''K(Y)'', respectively. Suppose that the algebraic closure of ''K(Y)'' in ''K(X)'' has separable degree ''n'' and that <math>y \in Y</math> is unibranch. Then the fiber <math>f^{-1}(y)</math> has at most ''n'' connected components. In particular, if ''f'' is [[birational]], then the fibers of unibranch points are connected.

In EGA, the theorem is obtained as a corollary of [[Zariski's main theorem]].

==References==
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[[Category:Algebraic geometry]]
[[Category:Commutative algebra]]


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