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.

f is smooth.
f is formally smooth (see below).
f is flat and the sheaf of relative differentials $\Omega _{{X/S}}$ is locally free of rank equal to the relative dimension of $X/S$ .
For any $x\in X$ , there exists a neighborhood $\operatorname {Spec}B$ of x and a neighborhood $\operatorname {Spec} A$ of $f(x)$ 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.
Locally, f factors into $X{\overset {g}\to }{\mathbb {A}}_{S}^{n}\to S$ where g is étale.
Locally, f factors into $X{\overset {g}\to }{\mathbb {A}}_{S}^{n}\to {\mathbb {A}}_{S}^{{n-1}}\to \cdots \to {\mathbb {A}}_{S}^{1}\to S$ 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

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 $0\neq p$ in $\operatorname {char}(S)$ , $p:{\mathcal {F}}\to {\mathcal {F}}$ is injective, then the base change morphism $g^{*}(R^{i}f_{*}{\mathcal {F}})\to R^{i}f'_{*}(g'^{*}{\mathcal {F}})$ is an isomorphism.

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.

This article is issued from Wikipedia - version of the 10/10/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Smooth morphism

Formally smooth morphism

Smooth base change

See also

References