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Difference between revisions of "Smooth morphism"

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<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>
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<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">[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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<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>
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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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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=55735
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