The Stacks project

Example 26.21.4. Here is an example of a non-quasi-separated morphism. Suppose $X = X_1 \cup X_2 \to S = \mathop{\mathrm{Spec}}(k)$ with $X_1 = X_2 = \mathop{\mathrm{Spec}}(k[t_1, t_2, t_3, \ldots ])$ glued along the complement of $\{ 0\} = \{ (t_1, t_2, t_3, \ldots )\} $ (glued as in Example 26.14.3). In this case the inverse image of the affine scheme $X_1 \times _ S X_2$ under $\Delta _{X/S}$ is the scheme $\mathop{\mathrm{Spec}}(k[t_1, t_2, t_3, \ldots ]) \setminus \{ 0\} $ which is not quasi-compact.