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''of schemes''
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The concept of a family of non-singular algebraic varieties (cf. [[Algebraic variety|Algebraic variety]]) generalized to the case of schemes. In the classical case of a morphism of complex algebraic varieties this concept reduces to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858801.png" /> is called a smooth morphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858802.png" /> is a [[Flat morphism|flat morphism]] and if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858803.png" /> the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858804.png" /> is a [[Smooth scheme|smooth scheme]] (over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858805.png" />). A scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858806.png" /> is called a smooth scheme over a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858807.png" />, or a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s0858809.png" />-scheme, if the structure morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588010.png" /> is a smooth morphism.
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An example of a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588011.png" />-scheme is the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588012.png" />. A special case of the concept of a smooth morphism is that of an [[Etale morphism|étale morphism]]. Conversely, any smooth morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588013.png" /> can be locally factored with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588014.png" /> into a composition of an étale morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588015.png" /> and a projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588016.png" />.
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''of schemes''
 
 
A composite of smooth morphisms is again a smooth morphism; this is also true for any base change. A smooth morphism is distinguished by its differential property: A flat finitely-represented morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588017.png" /> is a smooth morphism if and only if the sheaf of relative differentials is a locally free sheaf of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588018.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588019.png" />.
 
 
 
The concept of a smooth morphism is analogous to the concept of a [[Serre fibration|Serre fibration]] in topology. E.g., a smooth morphism of complex algebraic varieties is a locally trivial differentiable fibration. In the general case the following analogue of the covering homotopy axiom is valid: For any [[Affine scheme|affine scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588020.png" />, any closed subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588021.png" /> of it which is definable by a nilpotent ideal and any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588022.png" />, the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588023.png" /> is surjective.
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588024.png" /> is a smooth morphism and if the [[Local ring|local ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588025.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588026.png" /> is regular (respectively, normal or reduced), then the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588027.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085880/s08588029.png" /> will also have this property.
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The concept of a family of non-singular algebraic varieties (cf. [[Algebraic variety|Algebraic variety]]) generalized to the case of schemes. In the classical case of a morphism of complex algebraic varieties this concept reduces to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes  $  f: X \rightarrow Y $
 
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is called a smooth morphism if  $  f $
====References====
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is a [[Flat morphism|flat morphism]] and if for any point  $  y \in Y $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" ''Publ. Math. IHES'' : 32 (1967) {{MR|0238860}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR></table>
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the fibre  $  f ^ { - 1 } ( y) $
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is a [[Smooth scheme|smooth scheme]] (over the field  $  k( y) $).  
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A scheme  $  X $
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is called a smooth scheme over a scheme  $  Y $,  
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or a smooth  $  Y $-
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scheme, if the structure morphism  $  f:  X \rightarrow Y $
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is a smooth morphism.
  
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An example of a smooth  $  Y $-
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scheme is the affine space  $  A _ {Y}  ^ {n} $.
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A special case of the concept of a smooth morphism is that of an [[Etale morphism|étale morphism]]. Conversely, any smooth morphism  $  f:  X \rightarrow Y $
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can be locally factored with respect to  $  X $
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into a composition of an étale morphism  $  X \rightarrow A _ {Y}  ^ {n} $
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and a projection  $  A _ {Y}  ^ {n} \rightarrow Y $.
  
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A composite of smooth morphisms is again a smooth morphism; this is also true for any base change. A smooth morphism is distinguished by its differential property: A flat finitely-represented morphism  $  f:  X \rightarrow Y $
 +
is a smooth morphism if and only if the sheaf of relative differentials is a locally free sheaf of rank  $  \mathop{\rm dim} _ {x}  f $
 +
at a point  $  x $.
  
====Comments====
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The concept of a smooth morphism is analogous to the concept of a [[Serre fibration|Serre fibration]] in topology. E.g., a smooth morphism of complex algebraic varieties is a locally trivial differentiable fibration. In the general case the following analogue of the covering homotopy axiom is valid: For any [[Affine scheme|affine scheme]]  $  Y  ^  \prime  $,
 +
any closed subscheme  $  Y _ {0}  ^  \prime  $
 +
of it which is definable by a nilpotent ideal and any morphism  $  Y  ^  \prime  \rightarrow Y $,
 +
the canonical mapping  $  \mathop{\rm Hom} _ {Y} ( Y  ^  \prime  , X) \rightarrow  \mathop{\rm Hom} _ {Y} ( Y _ {0}  ^  \prime  , X) $
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is surjective.
  
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If  $  f:  X \rightarrow Y $
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is a smooth morphism and if the [[Local ring|local ring]]  $  {\mathcal O} _ {Y,y} $
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at the point  $  y \in Y $
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is regular (respectively, normal or reduced), then the local ring  $  {\mathcal O} _ {X,x} $
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of any point  $  x \in X $
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with  $  f( x) = y $
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will also have this property.
  
 
====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>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géométrie algébrique IV. Etude locale des schémas et des morphismes des schémas" ''Publ. Math. IHES'' : 32 (1967) {{MR|0238860}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR>
 +
<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 07:00, 21 July 2024


of schemes

The concept of a family of non-singular algebraic varieties (cf. Algebraic variety) generalized to the case of schemes. In the classical case of a morphism of complex algebraic varieties this concept reduces to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes $ f: X \rightarrow Y $ is called a smooth morphism if $ f $ is a flat morphism and if for any point $ y \in Y $ the fibre $ f ^ { - 1 } ( y) $ is a smooth scheme (over the field $ k( y) $). A scheme $ X $ is called a smooth scheme over a scheme $ Y $, or a smooth $ Y $- scheme, if the structure morphism $ f: X \rightarrow Y $ is a smooth morphism.

An example of a smooth $ Y $- scheme is the affine space $ A _ {Y} ^ {n} $. A special case of the concept of a smooth morphism is that of an étale morphism. Conversely, any smooth morphism $ f: X \rightarrow Y $ can be locally factored with respect to $ X $ into a composition of an étale morphism $ X \rightarrow A _ {Y} ^ {n} $ and a projection $ A _ {Y} ^ {n} \rightarrow Y $.

A composite of smooth morphisms is again a smooth morphism; this is also true for any base change. A smooth morphism is distinguished by its differential property: A flat finitely-represented morphism $ f: X \rightarrow Y $ is a smooth morphism if and only if the sheaf of relative differentials is a locally free sheaf of rank $ \mathop{\rm dim} _ {x} f $ at a point $ x $.

The concept of a smooth morphism is analogous to the concept of a Serre fibration in topology. E.g., a smooth morphism of complex algebraic varieties is a locally trivial differentiable fibration. In the general case the following analogue of the covering homotopy axiom is valid: For any affine scheme $ Y ^ \prime $, any closed subscheme $ Y _ {0} ^ \prime $ of it which is definable by a nilpotent ideal and any morphism $ Y ^ \prime \rightarrow Y $, the canonical mapping $ \mathop{\rm Hom} _ {Y} ( Y ^ \prime , X) \rightarrow \mathop{\rm Hom} _ {Y} ( Y _ {0} ^ \prime , X) $ is surjective.

If $ f: X \rightarrow Y $ is a smooth morphism and if the local ring $ {\mathcal O} _ {Y,y} $ at the point $ y \in Y $ is regular (respectively, normal or reduced), then the local ring $ {\mathcal O} _ {X,x} $ of any point $ x \in X $ with $ f( x) = y $ will also have this property.

References

[1] A. Grothendieck, "Eléments de géométrie 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
[2] A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971) MR0354651 Zbl 1039.14001
[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Smooth morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smooth_morphism&oldid=23980
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