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Difference between revisions of "Smooth morphism"

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<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)</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)</TD></TR></table>
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<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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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. Sect. IV.2</TD></TR></table>
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<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 21:56, 30 March 2012

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 is called a smooth morphism if is a flat morphism and if for any point the fibre is a smooth scheme (over the field ). A scheme is called a smooth scheme over a scheme , or a smooth -scheme, if the structure morphism is a smooth morphism.

An example of a smooth -scheme is the affine space . A special case of the concept of a smooth morphism is that of an étale morphism. Conversely, any smooth morphism can be locally factored with respect to into a composition of an étale morphism and a projection .

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 is a smooth morphism if and only if the sheaf of relative differentials is a locally free sheaf of rank at a point .

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 , any closed subscheme of it which is definable by a nilpotent ideal and any morphism , the canonical mapping is surjective.

If is a smooth morphism and if the local ring at the point is regular (respectively, normal or reduced), then the local ring of any point with will also have this property.

References

[1] 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
[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


Comments

References

[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