The Stacks project

Remark 28.29.2. Lemma 28.29.1 above is false if $X$ is not quasi-separated. Here is an example. Take $R = \mathbf{Q}[x, y_1, y_2, \ldots ]/((x-i)y_ i)$. Consider the minimal prime ideal $\mathfrak p = (y_1, y_2, \ldots )$ of $R$. Glue two copies of $\mathop{\mathrm{Spec}}(R)$ along the (not quasi-compact) open $\mathop{\mathrm{Spec}}(R) \setminus V(\mathfrak p)$ to get a scheme $X$ (glueing as in Schemes, Example 26.14.3). Then the two maximal points of $X$ corresponding to $\mathfrak p$ are not contained in a common affine open. The reason is that any open of $\mathop{\mathrm{Spec}}(R)$ containing $\mathfrak p$ contains infinitely many of the “lines” $x = i$, $y_ j = 0$, $j \not= i$ with parameter $y_ i$. Details omitted.

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