The Stacks project

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

  1. 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.

  2. We say that $f$ is smooth if it is smooth at every point of $X$.

  3. 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

    \[ \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)) \to \mathop{\mathrm{Spec}}(R). \]

Comments (2)

Comment #116 by michele serra on

This comment is just about a typing mistake: in point 3. of the definition there is an "if" missing, is should be:

A morphism of affine schemes is called {\it standard smooth} if there exists a standard smooth...Best wishes, michele

There are also:

  • 2 comment(s) on Section 29.34: Smooth morphisms

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View Definition 29.34.1 as pdf