The Stacks project

Example 16.2.6. The set of points where a finitely presented ring map is smooth needn't be a quasi-compact open. For example, let $R = k[x, y_1, y_2, y_3, \ldots ]/(xy_ i)$ and $A = R/(x)$. Then the smooth locus of $R \to A$ is $\bigcup D(y_ i)$ which is not quasi-compact.