Lemma 38.33.1. Let $X \to S$ be a morphism of schemes. If $X = U \cup V$ is an open cover such that $U \to S$ and $V \to S$ are separated and $U \cap V \to U \times _ S V$ is closed, then $X \to S$ is separated.

Proof. Omitted. Hint: check that $\Delta : X \to X \times _ S X$ is closed by using the open covering of $X \times _ S X$ given by $U \times _ S U$, $U \times _ S V$, $V \times _ S U$, and $V \times _ S V$. $\square$

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