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.
|||A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" Publ. Math. IHES : 32 (1967)|
|||A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971)|
|[a1]||R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2|
Smooth morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smooth_morphism&oldid=15097