Definition 29.34.1. Let $f : X \to S$ be a morphism of schemes.

We say that $f$ is

*smooth at $x \in X$*if there exist an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is smooth.We say that $f$ is

*smooth*if it is smooth at every point of $X$.A morphism of affine schemes $f : X \to S$ is called

*standard smooth*if there exists a standard smooth ring map $R \to R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ (see Algebra, Definition 10.137.6) such that $X \to S$ is isomorphic to\[ \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)) \to \mathop{\mathrm{Spec}}(R). \]

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Comment #116 by michele serra on

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