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''in algebraic geometry''
 
''in algebraic geometry''
  
The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921901.png" /> on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921902.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921903.png" />, whose sections over an open affine subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921904.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921905.png" />-modules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921906.png" />-derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921907.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921908.png" />. An equivalent definition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921909.png" /> be the sheaf of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219010.png" /> of the sheaf of differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219011.png" /> into the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219012.png" /> (see [[Derivations, module of|Derivations, module of]]).
+
The sheaf $  \theta _ {X} $
 +
on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] $  X $
 +
over a field $  k $,  
 +
whose sections over an open affine subspace $  U = \mathop{\rm Spec} ( A) $
 +
are the $  A $-
 +
modules of $  k $-
 +
derivations $  \mathop{\rm Der} _ {k} ( A, A) $
 +
of the ring $  A $.  
 +
An equivalent definition is that $  \theta _ {X} $
 +
be the sheaf of homomorphisms $  \mathop{\rm Hom} ( \Omega _ {X/k}  ^ {1} , {\mathcal O} _ {X} ) $
 +
of the sheaf of differentials $  \Omega _ {X/k}  ^ {1} $
 +
into the structure sheaf $  {\mathcal O} _ {X} $(
 +
see [[Derivations, module of|Derivations, module of]]).
  
For any rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219013.png" />-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219014.png" />, the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219015.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219016.png" /> is identical to the [[Zariski tangent space|Zariski tangent space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219017.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219018.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219019.png" />, that is, to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219020.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219022.png" /> is the maximal ideal of the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219023.png" />. Instead of the tangent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219024.png" /> one can use the sheaf of germs of sections of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219025.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219026.png" /> (or the tangent bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219027.png" />). In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219028.png" /> is a smooth connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219029.png" />-scheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219030.png" /> is a locally free sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219031.png" /> of rank equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219032.png" />.
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For any rational $  k $-
 +
point $  x \in X $,  
 +
the stalk $  \theta _ {X} ( x) $
 +
of the sheaf $  \theta _ {X} $
 +
is identical to the [[Zariski tangent space|Zariski tangent space]] $  T _ {K,x} $
 +
to $  X $
 +
at $  x $,  
 +
that is, to the vector $  k $-
 +
space $  \mathop{\rm Hom} _ {k} ( \mathfrak M _ {x} / \mathfrak M _ {x}  ^ {2} , k) $,  
 +
where $  \mathfrak M _ {x} $
 +
is the maximal ideal of the local ring $  {\mathcal O} _ {K,x} $.  
 +
Instead of the tangent sheaf $  \theta _ {X} $
 +
one can use the sheaf of germs of sections of the vector bundle $  V ( \Omega _ {X/k}  ^ {1} ) $
 +
dual to $  \Omega _ {X}  ^ {1} $(
 +
or the tangent bundle to $  X $).  
 +
In the case when $  X $
 +
is a smooth connected $  k $-
 +
scheme, $  \theta _ {X} $
 +
is a locally free sheaf on $  X $
 +
of rank equal to the dimension of $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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"> 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>

Revision as of 08:25, 6 June 2020


in algebraic geometry

The sheaf $ \theta _ {X} $ on an algebraic variety or scheme $ X $ over a field $ k $, whose sections over an open affine subspace $ U = \mathop{\rm Spec} ( A) $ are the $ A $- modules of $ k $- derivations $ \mathop{\rm Der} _ {k} ( A, A) $ of the ring $ A $. An equivalent definition is that $ \theta _ {X} $ be the sheaf of homomorphisms $ \mathop{\rm Hom} ( \Omega _ {X/k} ^ {1} , {\mathcal O} _ {X} ) $ of the sheaf of differentials $ \Omega _ {X/k} ^ {1} $ into the structure sheaf $ {\mathcal O} _ {X} $( see Derivations, module of).

For any rational $ k $- point $ x \in X $, the stalk $ \theta _ {X} ( x) $ of the sheaf $ \theta _ {X} $ is identical to the Zariski tangent space $ T _ {K,x} $ to $ X $ at $ x $, that is, to the vector $ k $- space $ \mathop{\rm Hom} _ {k} ( \mathfrak M _ {x} / \mathfrak M _ {x} ^ {2} , k) $, where $ \mathfrak M _ {x} $ is the maximal ideal of the local ring $ {\mathcal O} _ {K,x} $. Instead of the tangent sheaf $ \theta _ {X} $ one can use the sheaf of germs of sections of the vector bundle $ V ( \Omega _ {X/k} ^ {1} ) $ dual to $ \Omega _ {X} ^ {1} $( or the tangent bundle to $ X $). In the case when $ X $ is a smooth connected $ k $- scheme, $ \theta _ {X} $ is a locally free sheaf on $ X $ of rank equal to the dimension of $ X $.

References

[1] 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:
Tangent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_sheaf&oldid=48948
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article