Lemma 101.23.5. A composition of a locally quasi-finite morphisms is locally quasi-finite.

**Proof.**
We have seen this for quasi-DM morphisms in Lemma 101.4.10 and for locally finite type morphisms in Lemma 101.17.2. 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 101.23.3 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 101.4.13. By Theorem 101.21.3 we can find a scheme $V$ and a surjective, flat, locally finitely presented, locally quasi-finite morphism $V \to \mathcal{Y}_ k$. By Lemma 101.23.4 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

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

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: