Smooth morphism

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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.

  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 s 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.
  6. 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 locally of finite presentation.

A smooth morphism is universally locally acyclic.

Formally smooth morphism[edit]

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 type 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[edit]

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.

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