# Smooth morphism

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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.

How to Cite This Entry:
Smooth morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smooth_morphism&oldid=15097
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