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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+1 = 66 n = 0
 
$#C+1 = 66 : ~/encyclopedia/old_files/data/N067/N.0607610 Normal scheme
 
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====References====
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<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>
  
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====
 
====Comments====
A normalization of an irreducible algebraic variety $  X $
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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.
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  $
+
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]].
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 $
+
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.
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 14:52, 7 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 is normal if and only if two conditions are satisfied: 1) for any point of codimension the local ring is regular (cf. Regular ring (in commutative algebra)); and 2) for any point of codimension the depth of the ring (cf. Depth of a module) is greater than 1. Every reduced scheme has a normal scheme canonically connected with it (normalization). The -scheme is integral, but not always finite over . However, if is excellent (see Excellent ring), for example, if is a scheme of finite type over a field, then is finite over .

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 is an irreducible normal variety together with a regular mapping that is finite and a birational isomorphism.

For an affine irreducible algebraic variety, is the integral closure of the ring of regular functions on in its field of fractions. The normalization has the following universality properties. Let be an integral scheme (i.e. is both reduced and irreducible, or, equivalently, is an integral domain for all open in ). For every normal integral scheme and every dominant morphism (i.e. is dense in ), factors uniquely through the normalization . So also Normal analytic space.

Let be a curve and a, possibly singular, point on . Let be the normalization of and the inverse images of in . These points are called the branches of passing through . The terminology derives from the fact that the can be identified (in the case of varieties over or ) with the "branches" of passing through . More precisely, if the are sufficiently small complex or real neighbourhoods of the , then some neighbourhood of is the union of the branches . Let be the tangent space at to . Then is some linear subspace of the tangent space to at . It will be either a line or a point. In the first case the branch is called linear. The point on is an example of a point with two linear branches (with tangents , ), and the point on 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=48017
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article