Smooth morphism

In algebraic geometry, a morphism $f:X \to S$ 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 $\overline{s} \to S$ the fiber $X_{\overline{s}} = X \times_S {\overline{s}}$ 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 $f: X \to S$ 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 $\Omega_{X/S}$ is locally free of rank equal to the relative dimension of $X/S$.
4. For any $s \in S$, there exists a neighborhood $\operatorname{Spec}B$ of s and a neighborhood $\operatorname{Spec}A$ of $f(s)$ such that $B = A[t_1, \dots, t_n]/(P_1, \dots, P_m)$ and the ideal generated by the m-by-m minors of $(\partial P_i/\partial t_j)$ is B.
5. Locally, f factors into $X \overset{g}\to \mathbb{A}^n_S \to S$ where g is étale.
6. Locally, f factors into $X \overset{g}\to \mathbb{A}^n_S \to \mathbb{A}^{n-1}_S \to \cdots \to \mathbb{A}^1_S \to S$ 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 $T_0$ of T given by a nilpotent ideal, $X(T) \to X(T_0)$ is surjective where we wrote $X(T) = \operatorname{Hom}_S(T, X)$. 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

Let S be a scheme and $\operatorname{char}(S)$ denote the image of the structure map $S \to \operatorname{Spec}\mathbb{Z}$. The smooth base change theorem states the following: let $f: X \to S$ be a quasi-compact morphism, $g: S' \to S$ a smooth morphism and $\mathcal{F}$ a torsion sheaf on $X_\text{et}$. If for every $0 \ne p$ in $\operatorname{char}(S)$, $p:\mathcal{F} \to \mathcal{F}$ is injective, then the base change morphism $g^*(R^if_*\mathcal{F}) \to R^if'_*(g'^*\mathcal{F})$ is an isomorphism.

See also

• smooth algebra
• regular embedding

References

• J. S. Milne (2012). "Lectures on Étale Cohomology"
• J. S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series . Princeton University Press, Princeton, N.J., 1980.