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''in algebraic geometry''
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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]]).
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$#A+1 = 32 n = 0
 
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$#C+1 = 32 : ~/encyclopedia/old_files/data/T092/T.0902190 Tangent sheaf
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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====References====
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<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>
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''in algebraic geometry''
  
====Comments====
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The sheaf  $  \theta _ {X} $
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on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]]  $  X $
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over a field  $  k $,
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whose sections over an open affine subspace  $  U = \mathop{\rm Spec} ( A) $
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are the  $  A $-
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modules of  $  k $-
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derivations  $  \mathop{\rm Der} _ {k} ( A, A) $
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of the ring  $  A $.
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An equivalent definition is that  $  \theta _ {X} $
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be the sheaf of homomorphisms  $  \mathop{\rm Hom} ( \Omega _ {X/k}  ^ {1} , {\mathcal O} _ {X} ) $
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of the sheaf of differentials  $  \Omega _ {X/k}  ^ {1} $
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into the structure sheaf  $  {\mathcal O} _ {X} $(
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see [[Derivations, module of|Derivations, module of]]).
  
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For any rational  $  k $-
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point  $  x \in X $,
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the stalk  $  \theta _ {X} ( x) $
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of the sheaf  $  \theta _ {X} $
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is identical to the [[Zariski tangent space|Zariski tangent space]]  $  T _ {K,x} $
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to  $  X $
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at  $  x $,
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that is, to the vector  $  k $-
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space  $  \mathop{\rm Hom} _ {k} ( \mathfrak M _ {x} / \mathfrak M _ {x}  ^ {2} , k) $,
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where  $  \mathfrak M _ {x} $
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is the maximal ideal of the local ring  $  {\mathcal O} _ {K,x} $.
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Instead of the tangent sheaf  $  \theta _ {X} $
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one can use the sheaf of germs of sections of the vector bundle  $  V ( \Omega _ {X/k}  ^ {1} ) $
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dual to  $  \Omega _ {X}  ^ {1} $(
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or the tangent bundle to  $  X $).
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In the case when  $  X $
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is a smooth connected  $  k $-
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scheme,  $  \theta _ {X} $
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is a locally free sheaf on  $  X $
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of rank equal to the dimension of  $  X $.
  
 
====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>
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<table>
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<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>
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<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>
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</table>

Latest revision as of 15:53, 10 April 2023


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
[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=23990
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article