The Stacks project

Lemma 101.4.10. All of the separation axioms listed in Definition 101.4.1 are stable under composition of morphisms.

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

\[ \mathcal{X} \longrightarrow \mathcal{X} \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{X} \times _\mathcal {Z} \mathcal{X}. \]

Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 101.4.7 above we see that the second arrow also has this property. Hence the lemma follows since the composition of morphisms which are representable by algebraic spaces with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$, see our general discussion in Properties of Stacks, Section 100.3 and Morphisms of Spaces, Lemmas 67.38.3, 67.27.3, 67.40.4, 67.8.5, and 67.4.8. $\square$

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