Solitons: Differential equations, symmetries, and infinite-dimensional algebras
《Solitons: Differential equations, symmetries, and infinite-dimensional algebras》 · 第 156 页 · Definition 6.14
Let $f:X\to Y$ be a morphism of schemes and $d\ge0$ be an integer. We say that $f$ is smooth of relative dimension $d$ at $x\in X$ if there exist affine open neighborhoods $U$ of $x$ and $V=\operatorname{Spec}R$ of $f(x)$, an open immersion $U\hookrightarrow\operatorname{Spec}R[T_1,\ldots,T_n]/(f_1,\ldots,f_{n-d})$, and the Jacobian matrix $J_{f_1,\ldots,f_{n-d}}(x)$ has rank $n-d$. We say that $f$ is smooth if it is smooth at all points $x\in X$.