The Stacks project

Example 34.9.3. Let $X = X_1 \cup X_2$ where $X_1 = X_2 = \mathop{\mathrm{Spec}}(k[t_1,t_2,\dots ])$ be the non-quasi-separated scheme from Schemes, Example 26.21.4, and $Y = X_1 \amalg \mathop{\mathrm{Spec}}(\mathcal{O}_{X_2, 0})$. Then the obvious morphism $f : Y \to X$ is flat and surjective, and (since $Y$ is quasi-compact) trivially satisfies (4) of Lemma 34.9.2. But $\{ f : Y \to X\} $ isn't an fpqc covering. Namely, we claim there is no quasi-compact open $W$ of $Y$ whose image by $f$ is $X_2$. Namely, $f^{-1}(X_2) = X_1 \setminus \{ 0\} \amalg \mathop{\mathrm{Spec}}(\mathcal{O}_{X_2, 0})$. Hence if $W$ exists, then $W = U \amalg \mathop{\mathrm{Spec}}(\mathcal{O}_{X_2, 0})$ for some quasi-compact open $U$ of $X_1 \setminus \{ 0\} = X_2 \setminus \{ 0\} $. But then we may write $U = D(f_1) \cup \ldots \cup D(f_ n)$ for some $f_ i \in k[t_1, t_2, \ldots ]$ vanishing at $0$. Suppose that $f_1, \ldots , f_ n$ use only the variables $x_1, \ldots , x_ m$. Then $p = (0, \ldots , 0, 1, 0, \ldots )$ with $1$ in the $(m + 1)$st spot is a closed point of $X_2$ not in the image of $W$.