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Smooth morphism

For any non negative integer $ n$ and for any scheme $ S$ , we put

$\displaystyle \mathbb{A}^n_S=\operatorname{Spec}\mathbb{Z}[x_1,x_2,\dots,x_n] \times_{\operatorname{Spec}\mathbb{Z}} S
$

and $ \pi_S:\mathbb{A}^n_S\to S$ the standard projection.

A smooth scheme over $ S$ is a scheme which ''étale locally look like'' $ \mathbb{A}^n_S$ .

DEFINITION 10.7   A separated morphism $ \phi:X\to S$ of finite type is smooth of relative dimension $ n$ if for any point $ x$ on $ X$ , there exists an open neighborhood $ U$ of $ X$ and an 'etale morphism $ \psi: U\to \mathbb{A}^n_S$ such that

$\displaystyle \phi =\psi \circ \pi_S
$

holds.

Let us close this section by quoting the following fundamental result.

THEOREM 10.8 (SGA1,Éxpose II,Corollary4.6)   Let $ f:X\to Y$ be a morphism of smooth $ S$ -schemes. then $ f$ is étale at $ x\in X$ if and only if $ f^*(\Omega^1_{Y/S}) \to \Omega^1_{X/S}$ is isomorphism at $ x$ .



2007-12-11