The Stacks project

Lemma 65.4.8. All of the separation axioms listed in Definition 65.4.2 are stable under composition of morphisms.

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces to which the axiom in question applies. The diagonal $\Delta _{X/Z}$ is the composition

\[ X \longrightarrow X \times _ Y X \longrightarrow X \times _ Z X. \]

Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 65.4.5 above we see that the second arrow also has this property. Hence the lemma follows since the composition of (representable) morphisms with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$, see Section 65.3. $\square$


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