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A generalization of the concept of a non-singular [[Algebraic variety|algebraic variety]]. A [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859001.png" /> of (locally) finite type over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859002.png" /> is called a smooth scheme (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859003.png" />) if the scheme obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859004.png" /> by replacing the field of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859005.png" /> with its algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859006.png" /> is a [[Regular scheme|regular scheme]], i.e. if all its local rings are regular. For a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859007.png" /> the concepts of a smooth scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859008.png" /> and a regular scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s0859009.png" /> are identical. In particular, a smooth scheme of finite type over an algebraically closed field is a non-singular algebraic variety. In the case of the field of complex numbers a non-singular algebraic variety has the structure of a complex [[Analytic manifold|analytic manifold]].
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A scheme is smooth if and only if it can be covered by smooth neighbourhoods. A point of a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590010.png" /> is called a simple point of the scheme if in a certain neighbourhood of it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590011.png" /> is smooth; otherwise the point is called a singular point. A connected smooth scheme is irreducible. A product of smooth schemes is itself a smooth scheme. In general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590012.png" /> is a smooth scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590014.png" /> is a [[Smooth morphism|smooth morphism]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590015.png" /> is a smooth scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590016.png" />.
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A generalization of the concept of a non-singular [[Algebraic variety|algebraic variety]]. A [[Scheme|scheme]]  $  X $
 +
of (locally) finite type over a field  $  k $
 +
is called a smooth scheme (over  $  k $)
 +
if the scheme obtained from  $  X $
 +
by replacing the field of constants  $  k $
 +
with its algebraic closure  $  \overline{k}  $
 +
is a [[Regular scheme|regular scheme]], i.e. if all its local rings are regular. For a perfect field  $  k $
 +
the concepts of a smooth scheme over  $  k $
 +
and a regular scheme over  $  k $
 +
are identical. In particular, a smooth scheme of finite type over an algebraically closed field is a non-singular algebraic variety. In the case of the field of complex numbers a non-singular algebraic variety has the structure of a complex [[Analytic manifold|analytic manifold]].
  
An affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590017.png" /> and a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590018.png" /> are smooth schemes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590019.png" />; any algebraic group (i.e. a reduced algebraic group scheme) over a perfect field is a smooth scheme. A reduced scheme over an algebraically closed field is smooth in an everywhere-dense open set.
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A scheme is smooth if and only if it can be covered by smooth neighbourhoods. A point of a scheme  $  X $
 +
is called a simple point of the scheme if in a certain neighbourhood of it  $  X $
 +
is smooth; otherwise the point is called a singular point. A connected smooth scheme is irreducible. A product of smooth schemes is itself a smooth scheme. In general, if  $  Y $
 +
is a smooth scheme over $  k $
 +
and  $  f: \  X \rightarrow Y $
 +
is a [[Smooth morphism|smooth morphism]], then  $  X $
 +
is a smooth scheme over $  k $.
  
If a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590020.png" /> is defined by the equations
 
  
<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/s/s085/s085900/s08590021.png" /></td> </tr></table>
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An affine space  $  A _{k} ^{n} $
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and a projective space  $  \mathbf P _{k} ^{n} $
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are smooth schemes over  $  k $;  
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any algebraic group (i.e. a reduced algebraic group scheme) over a perfect field is a smooth scheme. A reduced scheme over an algebraically closed field is smooth in an everywhere-dense open set.
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If a scheme  $  X $
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is defined by the equations $$
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F _{i} (X _{1} \dots X _{m} )  =   0
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i = 1 \dots n,
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$$
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in an affine space  $  A _{k} ^{m} $,
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then a point  $  x \in X $
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is simple if and only if the rank of the Jacobi matrix  $  \| {\partial F _{i} / \partial X _ j} (x) \| $
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is equal to  $  m - d $,
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where  $  d $
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is the dimension of  $  X $
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at  $  x $(
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Jacobi's criterion). In a more general case, a closed subscheme  $  X $
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of a smooth scheme  $  Y $
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defined by a sheaf of ideals  $  I $
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is smooth in a neighbourhood of a point  $  x $
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if and only if there exists a system of generators  $  g _{1} \dots g _{n} $
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of the ideal  $  I _{x} $
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in the ring  $  {\mathcal O} _{X,x} $
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for which  $  dg _{1} \dots dg _{n} $
 +
form part of a basis of a free  $  O _{X,x} $-
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module of the differential sheaf  $  \Omega _{X/k,x} $.
  
in an affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590022.png" />, then a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590023.png" /> is simple if and only if the rank of the Jacobi matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590024.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590026.png" /> is the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590027.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590028.png" /> (Jacobi's criterion). In a more general case, a closed subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590029.png" /> of a smooth scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590030.png" /> defined by a sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590031.png" /> is smooth in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590032.png" /> if and only if there exists a system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590033.png" /> of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590034.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590035.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590036.png" /> form part of a basis of a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590037.png" />-module of the differential sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085900/s08590038.png" />.
 
  
 
====References====
 
====References====

Latest revision as of 00:26, 22 December 2019


A generalization of the concept of a non-singular algebraic variety. A scheme $ X $ of (locally) finite type over a field $ k $ is called a smooth scheme (over $ k $) if the scheme obtained from $ X $ by replacing the field of constants $ k $ with its algebraic closure $ \overline{k} $ is a regular scheme, i.e. if all its local rings are regular. For a perfect field $ k $ the concepts of a smooth scheme over $ k $ and a regular scheme over $ k $ are identical. In particular, a smooth scheme of finite type over an algebraically closed field is a non-singular algebraic variety. In the case of the field of complex numbers a non-singular algebraic variety has the structure of a complex analytic manifold.

A scheme is smooth if and only if it can be covered by smooth neighbourhoods. A point of a scheme $ X $ is called a simple point of the scheme if in a certain neighbourhood of it $ X $ is smooth; otherwise the point is called a singular point. A connected smooth scheme is irreducible. A product of smooth schemes is itself a smooth scheme. In general, if $ Y $ is a smooth scheme over $ k $ and $ f: \ X \rightarrow Y $ is a smooth morphism, then $ X $ is a smooth scheme over $ k $.


An affine space $ A _{k} ^{n} $ and a projective space $ \mathbf P _{k} ^{n} $ are smooth schemes over $ k $; any algebraic group (i.e. a reduced algebraic group scheme) over a perfect field is a smooth scheme. A reduced scheme over an algebraically closed field is smooth in an everywhere-dense open set.

If a scheme $ X $ is defined by the equations $$ F _{i} (X _{1} \dots X _{m} ) = 0, i = 1 \dots n, $$ in an affine space $ A _{k} ^{m} $, then a point $ x \in X $ is simple if and only if the rank of the Jacobi matrix $ \| {\partial F _{i} / \partial X _ j} (x) \| $ is equal to $ m - d $, where $ d $ is the dimension of $ X $ at $ x $( Jacobi's criterion). In a more general case, a closed subscheme $ X $ of a smooth scheme $ Y $ defined by a sheaf of ideals $ I $ is smooth in a neighbourhood of a point $ x $ if and only if there exists a system of generators $ g _{1} \dots g _{n} $ of the ideal $ I _{x} $ in the ring $ {\mathcal O} _{X,x} $ for which $ dg _{1} \dots dg _{n} $ form part of a basis of a free $ O _{X,x} $- module of the differential sheaf $ \Omega _{X/k,x} $.


References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" Publ. Math. IHES : 32 (1967) MR0238860 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901
[3] 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


Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Smooth scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Smooth_scheme&oldid=44319
This article was adapted from an original article by V.I. DanilovI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article