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''to an algebraic variety or scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991201.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991202.png" />''
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{{TEX|done}}
  
The vector space over the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991203.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991204.png" /> that is dual to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991206.png" /> is the maximal ideal of the [[Local ring|local ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991207.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991208.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991209.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912010.png" /> is defined by a system of equations
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''to an algebraic variety or scheme  $  X $
 +
at a point  $  x $''
  
<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/z/z099/z099120/z09912011.png" /></td> </tr></table>
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The vector space over the residue field  $  k ( x ) $
 +
of the point  $  x $
 +
that is dual to the space  $  \mathfrak M _ {x} / \mathfrak M _ {x}  ^ {2} $,
 +
where  $  \mathfrak M $
 +
is the maximal ideal of the [[Local ring|local ring]]  $  {\mathcal O} _ {X ,x }  $
 +
of  $  x $
 +
on  $  X $.  
 +
If  $  X \subset  A _ {k}  ^ {n} $
 +
is defined by a system of equations
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912012.png" />, then the Zariski tangent space at a rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912013.png" /> is defined by the system of linear equations
+
$$
 +
F _  \alpha  = 0 ,
 +
$$
  
<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/z/z099/z099120/z09912014.png" /></td> </tr></table>
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where  $  F _  \alpha  \in k [ X _ {1} \dots X _ {n} ] $,
 +
then the Zariski tangent space at a rational point  $  x = ( x _ {1} \dots x _ {n} ) $
 +
is defined by the system of linear equations
  
A variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912015.png" /> is non-singular at a rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912016.png" /> if and only if the dimension of the Zariski tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912017.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912018.png" /> is equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912019.png" />. For a rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912020.png" />, the Zariski tangent space is dual to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912021.png" /> — the stalk at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912022.png" /> of the cotangent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912023.png" />. An irreducible variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912024.png" /> over a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912025.png" /> is smooth if and only if the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912026.png" /> is locally free. The vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912027.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912028.png" /> is called the tangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912029.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912030.png" />; it is functorially related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912031.png" />. Its sheaf of sections is called the tangent sheaf to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912032.png" />. The Zariski tangent space was introduced by O. Zariski [[#References|[1]]].
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$$
 +
\sum _ { i=1 } ^ { n } 
 +
\frac{\partial  F _  \alpha  }{\partial  X _ {i} }
 +
( x ) ( X _ {i} - x _ {i} )  = 0 .
 +
$$
 +
 
 +
A variety  $  X $
 +
is non-singular at a rational point $  x $
 +
if and only if the dimension of the Zariski tangent space to $  X $
 +
at $  x $
 +
is equal to the dimension of $  X $.  
 +
For a rational point $  x \in X $,  
 +
the Zariski tangent space is dual to the space $  \Omega _ {X / k }  ^ {1} \otimes k ( x ) $
 +
— the stalk at $  x $
 +
of the cotangent sheaf $  \Omega _ {X / k }  ^ {1} $.  
 +
An irreducible variety $  X $
 +
over a perfect field $  k $
 +
is smooth if and only if the sheaf $  \Omega _ {X / k }  ^ {1} $
 +
is locally free. The vector bundle $  T _ {X} = V ( \Omega _ {X /k }  ^ {1} ) $
 +
associated with $  \Omega _ {X / k }  ^ {1} $
 +
is called the tangent bundle of $  X $
 +
over $  k $;  
 +
it is functorially related to $  X $.  
 +
Its sheaf of sections is called the tangent sheaf to $  X $.  
 +
The Zariski tangent space was introduced by O. Zariski [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, "The concept of a simple point of an abstract algebraic variety" ''Trans. Amer. Math. Soc.'' , '''62''' (1947) pp. 1–52 {{MR|0021694}} {{ZBL|0031.26101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) {{MR|0072531}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, "The concept of a simple point of an abstract algebraic variety" ''Trans. Amer. Math. Soc.'' , '''62''' (1947) pp. 1–52 {{MR|0021694}} {{ZBL|0031.26101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) {{MR|0072531}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 16:33, 10 February 2020


to an algebraic variety or scheme $ X $ at a point $ x $

The vector space over the residue field $ k ( x ) $ of the point $ x $ that is dual to the space $ \mathfrak M _ {x} / \mathfrak M _ {x} ^ {2} $, where $ \mathfrak M $ is the maximal ideal of the local ring $ {\mathcal O} _ {X ,x } $ of $ x $ on $ X $. If $ X \subset A _ {k} ^ {n} $ is defined by a system of equations

$$ F _ \alpha = 0 , $$

where $ F _ \alpha \in k [ X _ {1} \dots X _ {n} ] $, then the Zariski tangent space at a rational point $ x = ( x _ {1} \dots x _ {n} ) $ is defined by the system of linear equations

$$ \sum _ { i=1 } ^ { n } \frac{\partial F _ \alpha }{\partial X _ {i} } ( x ) ( X _ {i} - x _ {i} ) = 0 . $$

A variety $ X $ is non-singular at a rational point $ x $ if and only if the dimension of the Zariski tangent space to $ X $ at $ x $ is equal to the dimension of $ X $. For a rational point $ x \in X $, the Zariski tangent space is dual to the space $ \Omega _ {X / k } ^ {1} \otimes k ( x ) $ — the stalk at $ x $ of the cotangent sheaf $ \Omega _ {X / k } ^ {1} $. An irreducible variety $ X $ over a perfect field $ k $ is smooth if and only if the sheaf $ \Omega _ {X / k } ^ {1} $ is locally free. The vector bundle $ T _ {X} = V ( \Omega _ {X /k } ^ {1} ) $ associated with $ \Omega _ {X / k } ^ {1} $ is called the tangent bundle of $ X $ over $ k $; it is functorially related to $ X $. Its sheaf of sections is called the tangent sheaf to $ X $. The Zariski tangent space was introduced by O. Zariski [1].

References

[1] O. Zariski, "The concept of a simple point of an abstract algebraic variety" Trans. Amer. Math. Soc. , 62 (1947) pp. 1–52 MR0021694 Zbl 0031.26101
[2] P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) MR0072531
[3] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Zariski tangent space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zariski_tangent_space&oldid=44409
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article