The Stacks project

Lemma 100.23.5. A composition of a locally quasi-finite morphisms is locally quasi-finite.

Proof. We have seen this for quasi-DM morphisms in Lemma 100.4.10 and for locally finite type morphisms in Lemma 100.17.2. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be locally quasi-finite. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ be a morphism. It suffices to show that $|\mathcal{X}_ k|$ is discrete. By Lemma 100.23.3 the morphisms $\mathcal{X}_ k \to \mathcal{Y}_ k$ and $\mathcal{Y}_ k \to \mathop{\mathrm{Spec}}(k)$ are locally quasi-finite. In particular we see that $\mathcal{Y}_ k$ is a quasi-DM algebraic stack, see Lemma 100.4.13. By Theorem 100.21.3 we can find a scheme $V$ and a surjective, flat, locally finitely presented, locally quasi-finite morphism $V \to \mathcal{Y}_ k$. By Lemma 100.23.4 we see that $V$ is locally quasi-finite over $k$, in particular $|V|$ is discrete. The morphism $V \times _{\mathcal{Y}_ k} \mathcal{X}_ k \to \mathcal{X}_ k$ is surjective, flat, and locally of finite presentation hence $|V \times _{\mathcal{Y}_ k} \mathcal{X}_ k| \to |\mathcal{X}_ k|$ is surjective and open. Thus it suffices to show that $|V \times _{\mathcal{Y}_ k} \mathcal{X}_ k|$ is discrete. Note that $V$ is a disjoint union of spectra of Artinian local $k$-algebras $A_ i$ with residue fields $k_ i$, see Varieties, Lemma 33.20.2. Thus it suffices to show that each

\[ |\mathop{\mathrm{Spec}}(A_ i) \times _{\mathcal{Y}_ k} \mathcal{X}_ k| = |\mathop{\mathrm{Spec}}(k_ i) \times _{\mathcal{Y}_ k} \mathcal{X}_ k| = |\mathop{\mathrm{Spec}}(k_ i) \times _\mathcal {Y} \mathcal{X}| \]

is discrete, which follows from the assumption that $\mathcal{X} \to \mathcal{Y}$ is locally quasi-finite. $\square$

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