The Stacks project

Lemma 87.16.1. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent

  1. the reduction of $X$ (Lemma 87.12.1) is a quasi-separated algebraic space,

  2. for $U \to X$, $V \to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \times _ X V$ is quasi-compact,

  3. for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times _ X V$ is quasi-compact.

Proof. Observe that $U \times _ X V$ is a scheme by Lemma 87.11.2. Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$. Then

\[ U_{red} \times _{X_{red}} V_{red} = U_{red} \times _ X V_{red} \to U \times _ X V \]

is a thickening of schemes. From this the equivalence of (1) and (2) is clear, keeping in mind the analogous lemma for algebraic spaces, see Properties of Spaces, Lemma 66.3.3. We omit the proof of the equivalence of (2) and (3). $\square$

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