In algebraic geometry, a morphism
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(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
- f is smooth.
- f is formally smooth (see below).
- f is flat and the sheaf of relative differentials
Ω X / S X / S . - For any
x ∈ X , there exists a neighborhoodSpec B of x and a neighborhoodSpec A off ( x ) such thatB = A [ t 1 , … , t n ] / ( P 1 , … , P m ) and the ideal generated by the m-by-m minors of( ∂ P i / ∂ t j ) is B. - Locally, f factors into
X → g A S n → S where g is étale. - Locally, f factors into
X → g A S n → A S n − 1 → ⋯ → A S 1 → S 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
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
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 be a scheme and