Lemma 30.2.5. Let $X$ be a scheme. The following are equivalent

$X$ has affine diagonal $\Delta : X \to X \times X$,

for $U, V \subset X$ affine open, the intersection $U \cap V$ is affine, and

there exists an open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ such that $U_{i_0 \ldots i_ p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots , i_ p \in I$.

In particular this holds if $X$ is separated.

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