Smooth morphism

In algebraic geometry, a morphism ${\displaystyle f:X\to S}$ 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.

There are many equivalent definitions of a smooth morphism. Let ${\displaystyle f:X\to S}$ be locally of finite presentation. Then the following are equivalent.

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.

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 ${\displaystyle T_{0}}$ of T given by a nilpotent ideal, ${\displaystyle X(T)\to X(T_{0})}$ is surjective where we wrote ${\displaystyle X(T)=\operatorname {Hom} _{S}(T,X)}$. Then a morphism locally of finite type is smooth if and only if it is formally smooth.

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