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

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