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Concept
Smooth morphism
Summary
In algebraic geometry, a morphism between schemes is said to be smooth if
(i) it is locally of finite presentation
(ii) it is flat, and
(iii) for every geometric point the fiber is regular.
(iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.
If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety.
There are many equivalent definitions of a smooth morphism. Let be locally of finite presentation. Then the following are equivalent.
f is smooth.
f is formally smooth (see below).
f is flat and the sheaf of relative differentials is locally free of rank equal to the relative dimension of .
For any , there exists a neighborhood of x and a neighborhood of such that and the ideal generated by the m-by-m minors of is B.
Locally, f factors into where g is étale.
Locally, f factors into where g is étale.
A morphism of finite type is étale if and only if it is smooth and quasi-finite.
A smooth morphism is stable under base change and composition.
A smooth morphism is universally locally acyclic.
Smooth morphisms are supposed to geometrically correspond to smooth submersions in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by Ehresmann's theorem).
Let be the morphism of schemes
It is smooth because of the Jacobian condition: the Jacobian matrix
vanishes at the points which has an empty intersection with the polynomial, since
which are both non-zero.
Given a smooth scheme the projection morphism
is smooth.
Every vector bundle over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of over is the weighted projective space minus a point
sending
Notice that the direct sum bundles can be constructed using the fiber product
Recall that a field extension is called separable iff given a presentation
we have that .
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