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 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) |
[2] | A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971) |
Comments
References
[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