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The Stacks project

Lemma 70.17.4. 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 70.5.11 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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