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Wikipedia / Smooth morphism

In algebraic geometry, a morphism between schemes is said to be smooth if

(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.

A singular variety is called smoothable if it can be put in a flat family so that the nearby fibers are all smooth. Such a family is called a smoothning of the variety.

Equivalent definitions

be locally of finite presentation. Then the following are equivalent.

  1. f is smooth.

  2. f is formally smooth (see below).

  3. f is flat and the sheaf of relative differentials is locally free of rank equal to the relative dimension.

  4. and the ideal generated by the m-by-m is B.

  5. Locally, f 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.

Examples

Smooth morphisms are supposed to geometrically correspond to smooth submersionss in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by Ehresmann's theorem).

Smooth morphism to a point

be the morphism of schemes

It is smooth because of the Jacobian condition: the Jacobian matrix

which has an empty intersection with the polynomial, since

which are both non-zero.

Trivial fibrations

Given a smooth scheme the projection morphism

is smooth.

Vector bundles

Every vector bundle is the weighted projective space minus a point

sending

can be constructed using the fiber product

Separable field extensions

Recall that a field extension is called separable iff given a presentation

. We can reinterpret this definition in terms of Kähler differentials as follows: the field extension is separable iff

Notice that this includes every perfect field: finite fields and fields of characteristic 0.

Non-examples

Singular varieties

, then the point at the origin is always singular. For example, consider the affine cone-fold given by

Then the Jacobian matrix is given by

which vanishes at the origin, hence the cone is singular. Affine hypersurfaces like these are popular in singularity theory because of their relatively simple algebra but rich underlying structures.

Another example of a singular variety is the projective cone. For example, the projective cone of the points

is the scheme

chart this is the scheme

, this is a family of four points degenerating at the origin. The non-singularity of this scheme can also be checked using the Jacobian condition.

Degenerating families

Consider the flat family

Then the fibers are all smooth except for the point at the origin. Since smoothness is stable under base-change, this family is not smooth.

Non-separable field extensions

is non-separable, hence the associated morphism of schemes is not smooth. If we look at the minimal polynomial of the field extension,

, hence the Kähler differentials will be non-zero.

Formally smooth morphism

One can define smoothness without reference to geometry. We say that an S-scheme X is formally smooth if for any affine S-scheme T of T given by a nilpotent ideal. Then a morphism locally of finite presentation is smooth if and only if it is formally smooth.

In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition of formally étale (resp. formally unramified).

Smooth base change

Let S. The smooth base change theorem be a quasi-compact morphism a torsion sheaf is injective, then the base change morphism is an isomorphism.

See also

References