## 100.23 Locally quasi-finite morphisms

The property “locally quasi-finite” of morphisms of algebraic spaces is not smooth local on the source-and-target so we cannot use the material in Section 100.16 to define locally quasi-finite morphisms of algebraic stacks. We do already know what it means for a morphism of algebraic stacks representable by algebraic spaces to be locally quasi-finite, see Properties of Stacks, Section 99.3. To find a condition suitable for general morphisms we make the following observation.

Lemma 100.23.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is representable by algebraic spaces. The following are equivalent

1. $f$ is locally quasi-finite (as in Properties of Stacks, Section 99.3), and

2. $f$ is locally of finite type and for every morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Y}$ where $k$ is a field the space $|\mathop{\mathrm{Spec}}(k) \times _\mathcal {Y} \mathcal{X}|$ is discrete.

Proof. Assume (1). In this case the morphism of algebraic spaces $\mathcal{X}_ k \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite as a base change of $f$. Hence $|\mathcal{X}_ k|$ is discrete by Morphisms of Spaces, Lemma 66.27.5. Conversely, assume (2). Pick a surjective smooth morphism $V \to \mathcal{Y}$ where $V$ is a scheme. It suffices to show that the morphism of algebraic spaces $V \times _\mathcal {Y} \mathcal{X} \to V$ is locally quasi-finite, see Properties of Stacks, Lemma 99.3.3. The morphism $V \times _\mathcal {Y} \mathcal{X} \to V$ is locally of finite type by assumption. For any morphism $\mathop{\mathrm{Spec}}(k) \to V$ where $k$ is a field

$\mathop{\mathrm{Spec}}(k) \times _ V (V \times _\mathcal {Y} \mathcal{X}) = \mathop{\mathrm{Spec}}(k) \times _\mathcal {Y} \mathcal{X}$

has a discrete space of points by assumption. Hence we conclude that $V \times _\mathcal {Y} \mathcal{X} \to V$ is locally quasi-finite by Morphisms of Spaces, Lemma 66.27.5. $\square$

A morphism of algebraic stacks which is representable by algebraic spaces is quasi-DM, see Lemma 100.4.3. Combined with the lemma above we see that the following definition does not conflict with the already existing notion in the case of morphisms representable by algebraic spaces.

Definition 100.23.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is locally quasi-finite if $f$ is quasi-DM, locally of finite type, and for every morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Y}$ where $k$ is a field the space $|\mathcal{X}_ k|$ is discrete.

The condition that $f$ be quasi-DM is natural. For example, let $k$ be a field and consider the morphism $\pi : [\mathop{\mathrm{Spec}}(k)/\mathbf{G}_ m] \to \mathop{\mathrm{Spec}}(k)$ which has singleton fibres and is locally of finite type. As we will see later this morphism is smooth of relative dimension $-1$, and we'd like our locally quasi-finite morphisms to have relative dimension $0$. Also, note that the section $\mathop{\mathrm{Spec}}(k) \to [\mathop{\mathrm{Spec}}(k)/\mathbf{G}_ m]$ does not have discrete fibres, hence is not locally quasi-finite, and we'd like to have the following permanence property for locally quasi-finite morphisms: If $f : \mathcal{X} \to \mathcal{X}'$ is a morphism of algebraic stacks locally quasi-finite over the algebraic stack $\mathcal{Y}$, then $f$ is locally quasi-finite (in fact something a bit stronger holds, see Lemma 100.23.8).

Another justification for the definition above is Lemma 100.23.7 below which characterizes being locally quasi-finite in terms of the existence of suitable “presentations” or “coverings” of $\mathcal{X}$ and $\mathcal{Y}$.

Lemma 100.23.3. A base change of a locally quasi-finite morphism is locally quasi-finite.

Proof. We have seen this for quasi-DM morphisms in Lemma 100.4.4 and for locally finite type morphisms in Lemma 100.17.3. It is immediate that the condition on fibres is inherited by a base change. $\square$

Lemma 100.23.4. Let $\mathcal{X} \to \mathop{\mathrm{Spec}}(k)$ be a locally quasi-finite morphism where $\mathcal{X}$ is an algebraic stack and $k$ is a field. Let $f : V \to \mathcal{X}$ be a locally quasi-finite morphism where $V$ is a scheme. Then $V \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite.

Proof. By Lemma 100.17.2 we see that $V \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type. Assume, to get a contradiction, that $V \to \mathop{\mathrm{Spec}}(k)$ is not locally quasi-finite. Then there exists a nontrivial specialization $v \leadsto v'$ of points of $V$, see Morphisms, Lemma 29.20.6. In particular $\text{trdeg}_ k(\kappa (v)) > \text{trdeg}_ k(\kappa (v'))$, see Morphisms, Lemma 29.28.7. Because $|\mathcal{X}|$ is discrete we see that $|f|(v) = |f|(v')$. Consider $R = V \times _\mathcal {X} V$. Then $R$ is an algebraic space and the projections $s, t : R \to V$ are locally quasi-finite as base changes of $V \to \mathcal{X}$ (which is representable by algebraic spaces so this follows from the discussion in Properties of Stacks, Section 99.3). By Properties of Stacks, Lemma 99.4.3 we see that there exists an $r \in |R|$ such that $s(r) = v$ and $t(r) = v'$. By Morphisms of Spaces, Lemma 66.33.3 we see that the transcendence degree of $v/k$ is equal to the transcendence degree of $r/k$ is equal to the transcendence degree of $v'/k$. This contradiction proves the lemma. $\square$

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. $\square$

Before we characterize locally quasi-finite morphisms in terms of coverings we do it for quasi-DM morphisms.

Lemma 100.23.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

1. $f$ is quasi-DM,

2. for any morphism $V \to \mathcal{Y}$ with $V$ an algebraic space there exists a surjective, flat, locally finitely presented, locally quasi-finite morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ where $U$ is an algebraic space, and

3. there exist algebraic spaces $U$, $V$ and a morphism $V \to \mathcal{Y}$ which is surjective, flat, and locally of finite presentation, and a morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite.

Proof. The implication (2) $\Rightarrow$ (3) is immediate.

Assume (1) and let $V \to \mathcal{Y}$ be as in (2). Then $\mathcal{X} \times _\mathcal {Y} V \to V$ is quasi-DM, see Lemma 100.4.4. By Lemma 100.4.3 the algebraic space $V$ is DM, hence quasi-DM. Thus $\mathcal{X} \times _\mathcal {Y} V$ is quasi-DM by Lemma 100.4.11. Hence we may apply Theorem 100.21.3 to get the morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ as in (2).

Assume (3). Let $V \to \mathcal{Y}$ and $U \to \mathcal{X} \times _\mathcal {Y} V$ be as in (3). To prove that $f$ is quasi-DM it suffices to show that $\mathcal{X} \times _\mathcal {Y} V \to V$ is quasi-DM, see Lemma 100.4.5. By Lemma 100.4.14 we see that $\mathcal{X} \times _\mathcal {Y} V$ is quasi-DM. Hence $\mathcal{X} \times _\mathcal {Y} V \to V$ is quasi-DM by Lemma 100.4.13 and (1) holds. This finishes the proof of the lemma. $\square$

Lemma 100.23.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

1. $f$ is locally quasi-finite,

2. $f$ is quasi-DM and for any morphism $V \to \mathcal{Y}$ with $V$ an algebraic space and any locally quasi-finite morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ where $U$ is an algebraic space the morphism $U \to V$ is locally quasi-finite,

3. for any morphism $V \to \mathcal{Y}$ from an algebraic space $V$ there exists a surjective, flat, locally finitely presented, and locally quasi-finite morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ where $U$ is an algebraic space such that $U \to V$ is locally quasi-finite,

4. there exists algebraic spaces $U$, $V$, a surjective, flat, and locally of finite presentation morphism $V \to \mathcal{Y}$, and a morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite such that $U \to V$ is locally quasi-finite.

Proof. Assume (1). Then $f$ is quasi-DM by assumption. Let $V \to \mathcal{Y}$ and $U \to \mathcal{X} \times _\mathcal {Y} V$ be as in (2). By Lemma 100.23.5 the composition $U \to \mathcal{X} \times _\mathcal {Y} V \to V$ is locally quasi-finite. Thus (1) implies (2).

Assume (2). Let $V \to \mathcal{Y}$ be as in (3). By Lemma 100.23.6 we can find an algebraic space $U$ and a surjective, flat, locally finitely presented, locally quasi-finite morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. By (2) the composition $U \to V$ is locally quasi-finite. Thus (2) implies (3).

It is immediate that (3) implies (4).

Assume (4). We will prove (1) holds, which finishes the proof. By Lemma 100.23.6 we see that $f$ is quasi-DM. To prove that $f$ is locally of finite type it suffices to prove that $g : \mathcal{X} \times _\mathcal {Y} V \to V$ is locally of finite type, see Lemma 100.17.6. Then it suffices to check that $g$ precomposed with $h : U \to \mathcal{X} \times _\mathcal {Y} V$ is locally of finite type, see Lemma 100.17.7. Since $g \circ h : U \to V$ was assumed to be locally quasi-finite this holds, hence $f$ is locally of finite type. Finally, let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to \mathcal{Y}$ be a morphism. Then $V \times _\mathcal {Y} \mathop{\mathrm{Spec}}(k)$ is a nonempty algebraic space which is locally of finite presentation over $k$. Hence we can find a finite extension $k'/k$ and a morphism $\mathop{\mathrm{Spec}}(k') \to V$ such that

$\xymatrix{ \mathop{\mathrm{Spec}}(k') \ar[r] \ar[d] & V \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & \mathcal{Y} }$

commutes (details omitted). Then $\mathcal{X}_{k'} \to \mathcal{X}_ k$ is representable (by schemes), surjective, and finite locally free. In particular $|\mathcal{X}_{k'}| \to |\mathcal{X}_ k|$ is surjective and open. Thus it suffices to prove that $|\mathcal{X}_{k'}|$ is discrete. Since

$U \times _ V \mathop{\mathrm{Spec}}(k') = U \times _{\mathcal{X} \times _\mathcal {Y} V} \mathcal{X}_{k'}$

we see that $U \times _ V \mathop{\mathrm{Spec}}(k') \to \mathcal{X}_{k'}$ is surjective, flat, and locally of finite presentation (as a base change of $U \to \mathcal{X} \times _\mathcal {Y} V$). Hence $|U \times _ V \mathop{\mathrm{Spec}}(k')| \to |\mathcal{X}_{k'}|$ is surjective and open. Thus it suffices to show that $|U \times _ V \mathop{\mathrm{Spec}}(k')|$ is discrete. This follows from the fact that $U \to V$ is locally quasi-finite (either by our definition above or from the original definition for morphisms of algebraic spaces, via Morphisms of Spaces, Lemma 66.27.5). $\square$

Lemma 100.23.8. Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. Assume that $\mathcal{X} \to \mathcal{Z}$ is locally quasi-finite and $\mathcal{Y} \to \mathcal{Z}$ is quasi-DM. Then $\mathcal{X} \to \mathcal{Y}$ is locally quasi-finite.

Proof. Write $\mathcal{X} \to \mathcal{Y}$ as the composition

$\mathcal{X} \longrightarrow \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \longrightarrow \mathcal{Y}$

The second arrow is locally quasi-finite as a base change of $\mathcal{X} \to \mathcal{Z}$, see Lemma 100.23.3. The first arrow is locally quasi-finite by Lemma 100.4.8 as $\mathcal{Y} \to \mathcal{Z}$ is quasi-DM. Hence $\mathcal{X} \to \mathcal{Y}$ is locally quasi-finite by Lemma 100.23.5. $\square$

Comment #1786 by Matthew Emerton on

Unless I missed it, this section discusses locally q.f. morphisms, but doesn't actually define quasi-finite morphisms, despite its title. I guess these should be loc. q.f. plus fin. type?

Comment #1822 by on

OK, I put this on the todo list. Right now I do not have the time to think through completely the following: should we also put some condition on the diagonal?

Comment #3177 by Ariyan Javanpeykar on

Typo: "with all of the already existing notion" --> "with the already existing notion"

Comment #5479 by on

OK, I have now (finally) added a section on quasi-finite morphisms of algebraic stacks which are defined as quasi-compact, locally quasi-finite morphisms. See this commit.

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