The Stacks project

Lemma 65.4.4. All of the separation axioms listed in Definition 65.4.2 are stable under base change.

Proof. Let $f : X \to Y$ and $Y' \to Y$ be morphisms of algebraic spaces. Let $f' : X' \to Y'$ be the base change of $f$ by $Y' \to Y$. Then $\Delta _{X'/Y'}$ is the base change of $\Delta _{X/Y}$ by the morphism $X' \times _{Y'} X' \to X \times _ Y X$. By the results of Section 65.3 each of the properties of the diagonal used in Definition 65.4.2 is stable under base change. Hence the lemma is true. $\square$

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