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A [[Scheme|scheme]] all local rings (cf. [[Local ring|Local ring]]) of which are normal (that is, reduced and integrally closed in their ring of fractions). A normal scheme is locally irreducible; for such a scheme the concepts of a connected component and an irreducible component are the same. The set of singular points of a Noetherian normal scheme has codimension greater than 1. The following normality criterion holds [[#References|[1]]]: A [[Noetherian scheme|Noetherian scheme]]  $  X $
is normal if and only if two conditions are satisfied: 1) for any point  $  x \in X $
of codimension  $  \leq  1 $
the local ring  $  {\mathcal O} _ {X,x} $
is regular (cf. [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]); and 2) for any point  $  x \in X $
of codimension  $  > 1 $
the depth of the ring (cf. [[Depth of a module|Depth of a module]])  $  {\mathcal O} _ {X,x} $
is greater than 1. Every [[Reduced scheme|reduced scheme]]  $  X $
has a normal scheme  $  X  ^  \nu  $
canonically connected with it (normalization). The  $  X $-
scheme  $  X  ^  \nu  $
is integral, but not always finite over  $  X $. 
However, if  $  X $
is excellent (see [[Excellent ring|Excellent ring]]), for example, if  $  X $
is a scheme of finite type over a field, then  $  X  ^  \nu  $
is finite over  $  X $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1975) {{MR|0201468}} {{ZBL|0296.13018}} </TD></TR></table>

====Comments====
A normalization of an irreducible algebraic variety  $  X $
is an irreducible normal variety  $  X  ^  \nu  $
together with a regular mapping  $  \nu :  X  ^  \nu  \rightarrow X $
that is finite and a birational isomorphism.

For an affine irreducible algebraic variety,  $  X  ^  \nu  $
is the integral closure of the ring  $  A ( X) $
of regular functions on  $  X $
in its field of fractions. The normalization has the following universality properties. Let  $  X $
be an integral scheme (i.e.  $  X $
is both reduced and irreducible, or, equivalently,  $  {\mathcal O} _ {X} ( U) $
is an integral domain for all open  $  U $
in  $  X $). 
For every normal integral scheme  $  Z $
and every dominant morphism  $  f :  Z \rightarrow X $(
i.e.  $  f ( Z) $
is dense in  $  X $), 
$  f $
factors uniquely through the normalization  $  X  ^  \nu  \rightarrow X $. 
So also [[Normal analytic space|Normal analytic space]].

Let  $  X $
be a curve and  $  x $
a, possibly singular, point on  $  X $. 
Let  $  X  ^  \nu  \rightarrow X $
be the normalization of  $  X $
and  $  \overline{x}\; _ {1} \dots \overline{x}\; _ {n} $
the inverse images of  $  x $
in  $  X  ^  \nu  $. 
These points are called the branches of  $  X $
passing through  $  x $. 
The terminology derives from the fact that the  $  \overline{x}\; _ {i} $
can be identified (in the case of varieties over  $  \mathbf R $
or  $  \mathbf C $) 
with the "branches" of  $  X $
passing through  $  x $. 
More precisely, if the  $  U _ {i} $
are sufficiently small complex or real neighbourhoods of the  $  x _ {i} $, 
then some neighbourhood of  $  x $
is the union of the branches  $  \nu ( U _ {i} ) $. 
Let  $  T _ {i} $
be the tangent space at  $  \overline{x}\; _ {i} $
to  $  X  ^  \nu  $. 
Then  $  ( d \nu ) ( \overline{x}\; _ {i} ) ( T _ {i} ) $
is some linear subspace of the tangent space to  $  X $
at  $  x $. 
It will be either a line or a point. In the first case the branch  $  \overline{x}\; _ {i} $
is called linear. The point  $  ( 0 , 0 ) $
on  $  y  ^ {2} = x  ^ {3} + x  ^ {2} $
is an example of a point with two linear branches (with tangents  $  y = x $, 
$  y = - x $), 
and the point  $  ( 0 , 0 ) $
on  $  y  ^ {2} = x  ^ {3} $
gives an example of a two-fold non-linear branch.

$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Sect. II.5 (Translated from Russian) {{MR|0366917}} {{ZBL|0284.14001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Matsumura, "Commutative algebra" , Benjamin (1970) {{MR|0266911}} {{ZBL|0211.06501}} </TD></TR></table>