Example 2

Let $A$ be a noetherian ring, $T = (T_i)_{i \in I}$ a finite family of indeterminates. The $A$-algebra $A[T]$ is smooth. More generally, let $F_1, \ldots, F_m$ be elements of $A[T]$, and let $B$ be the $A$-algebra $A[T]/(F_1, \ldots, F_m)$. If at every maximal ideal $n$ of $B$ the class (mod. $n$) of the matrix $\left( \frac{\partial F_j}{\partial T_i} \right)$ is of rank $m$, the $A$-algebra $B$ is smooth (Theorem 3 of No. 9).