Smooth morphism

(Redirected from Formally smooth)

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.

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

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There are many equivalent definitions of a smooth morphism. Let   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 of  .
  4. 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.
  5. 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.

Examples

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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).

Smooth Morphism to a Point

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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.

Trivial Fibrations

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Given a smooth scheme   the projection morphism

 

is smooth.

Vector Bundles

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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

 

Separable Field Extensions

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

 

we have that  . 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

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Singular Varieties

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If we consider   of the underlying algebra   for a projective variety  , called the affine cone of  , then the point at the origin is always singular. For example, consider the affine cone of a quintic  -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 of a smooth variety: given a smooth projective variety   its projective cone is the union of all lines in   intersecting  . For example, the projective cone of the points

 

is the scheme

 

If we look in the   chart this is the scheme

 

and project it down to the affine line  , 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

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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

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

 

then  , hence the Kähler differentials will be non-zero.

Formally smooth morphism

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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 and a subscheme   of T given by a nilpotent ideal,   is surjective where we wrote  . 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

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Let S be a scheme and   denote the image of the structure map  . The smooth base change theorem states the following: let   be a quasi-compact morphism,   a smooth morphism and   a torsion sheaf on  . If for every   in  ,   is injective, then the base change morphism   is an isomorphism.

See also

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References

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