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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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676101.png" /> is normal if and only if two conditions are satisfied: 1) for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676102.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676103.png" /> the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676104.png" /> is regular (cf. [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]); and 2) for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676105.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676106.png" /> the depth of the ring (cf. [[Depth of a module|Depth of a module]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676107.png" /> is greater than 1. Every [[Reduced scheme|reduced scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676108.png" /> has a normal scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n0676109.png" /> canonically connected with it (normalization). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761010.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761011.png" /> is integral, but not always finite over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761012.png" />. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761013.png" /> is excellent (see [[Excellent ring|Excellent ring]]), for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761014.png" /> is a scheme of finite type over a field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761015.png" /> is finite over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761016.png" />.
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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====
 
====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>
 
<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====
 
====Comments====
A normalization of an irreducible algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761017.png" /> is an irreducible normal variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761018.png" /> together with a regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761019.png" /> that is finite and a birational isomorphism.
+
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761020.png" /> is the integral closure of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761021.png" /> of regular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761022.png" /> in its field of fractions. The normalization has the following universality properties. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761023.png" /> be an integral scheme (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761024.png" /> is both reduced and irreducible, or, equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761025.png" /> is an integral domain for all open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761027.png" />). For every normal integral scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761028.png" /> and every dominant morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761029.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761030.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761031.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761032.png" /> factors uniquely through the normalization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761033.png" />. So also [[Normal analytic space|Normal analytic space]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761034.png" /> be a curve and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761035.png" /> a, possibly singular, point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761036.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761037.png" /> be the normalization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761039.png" /> the inverse images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761041.png" />. These points are called the branches of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761042.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761043.png" />. The terminology derives from the fact that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761044.png" /> can be identified (in the case of varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761045.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761046.png" />) with the "branches" of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761047.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761048.png" />. More precisely, if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761049.png" /> are sufficiently small complex or real neighbourhoods of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761050.png" />, then some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761051.png" /> is the union of the branches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761052.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761053.png" /> be the tangent space at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761054.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761055.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761056.png" /> is some linear subspace of the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761057.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761058.png" />. It will be either a line or a point. In the first case the branch <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761059.png" /> is called linear. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761060.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761061.png" /> is an example of a point with two linear branches (with tangents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761063.png" />), and the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761065.png" /> gives an example of a two-fold non-linear branch.
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067610/n06761066.png" /></td> </tr></table>
+
$$
  
 
====References====
 
====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>
 
<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>

Revision as of 08:03, 6 June 2020


A scheme all local rings (cf. 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 [1]: A 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)); and 2) for any point $ x \in X $ of codimension $ > 1 $ the depth of the ring (cf. Depth of a module) $ {\mathcal O} _ {X,x} $ is greater than 1. Every 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), for example, if $ X $ is a scheme of finite type over a field, then $ X ^ \nu $ is finite over $ X $.

References

[1] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1975) MR0201468 Zbl 0296.13018

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.

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

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001
[a2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Sect. II.5 (Translated from Russian) MR0366917 Zbl 0284.14001
[a3] H. Matsumura, "Commutative algebra" , Benjamin (1970) MR0266911 Zbl 0211.06501
How to Cite This Entry:
Normal scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_scheme&oldid=23916
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article