The Stacks project

Lemma 69.17.4. Let $\varphi : X \to \mathop{\mathrm{Spec}}(A)$ be a quasi-compact and quasi-separated morphism from an algebraic space to an affine scheme. If $X$ is not a scheme, then there exists an ideal $I \subset A$ such that the base change $X_{A/I}$ is not a scheme, but for every $I \subset I'$, $I \not= I'$ the base change $X_{A/I'}$ is a scheme.

Proof. We prove this by Zorn's lemma. Let $\mathcal{I}$ be the set of ideals $I$ such that $X_{A/I}$ is not a scheme. By assumption $\mathcal{I}$ contains $(0)$. If $I_\alpha $ is a chain of ideals in $\mathcal{I}$, then $I = \bigcup I_\alpha $ is in $\mathcal{I}$. Namely, $A/I = \mathop{\mathrm{colim}}\nolimits A/I_\alpha $, hence

\[ X_{A/I} = \mathop{\mathrm{lim}}\nolimits X_{A/I_\alpha } \]

Thus we may apply Lemma 69.5.11 to see that if $X_{A/I}$ were a scheme, then so would be one of the $X_{A/I_\alpha }$. Thus $\mathcal{I}$ has maximal elements by Zorn's lemma. $\square$

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