Smooth morphism

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 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

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001