67.27 Quasi-finite morphisms
The property “locally quasi-finite” of morphisms of schemes is étale local on the source-and-target, see Descent, Remark 35.32.7. It is also stable under base change and fpqc local on the target, see Morphisms, Lemma 29.20.13, and Descent, Lemma 35.23.24. Hence, by Lemma 67.22.1 above, we may define what it means for a morphism of algebraic spaces to be locally quasi-finite as follows and it agrees with the already existing notion defined in Section 67.3 when the morphism is representable.
Definition 67.27.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
We say $f$ is locally quasi-finite if the equivalent conditions of Lemma 67.22.1 hold with $\mathcal{P} = \text{locally quasi-finite}$.
Let $x \in |X|$. We say $f$ is quasi-finite at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally quasi-finite.
A morphism of algebraic spaces $f : X \to Y$ is quasi-finite if it is locally quasi-finite and quasi-compact.
The last part is compatible with the notion of quasi-finiteness for morphisms of schemes by Morphisms, Lemma 29.20.9.
Lemma 67.27.2. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be morphisms of algebraic spaces over $S$. Denote $f' : X' \to Y'$ the base change of $f$ by $g$. Denote $g' : X' \to X$ the projection. Assume $f$ is locally of finite type. Let $W \subset |X|$, resp. $W' \subset |X'|$ be the set of points where $f$, resp. $f'$ is quasi-finite.
$W \subset |X|$ and $W' \subset |X'|$ are open,
$W' = (g')^{-1}(W)$, i.e., formation of the locus where $f$ is quasi-finite commutes with base change,
the base change of a locally quasi-finite morphism is locally quasi-finite, and
the base change of a quasi-finite morphism is quasi-finite.
Proof.
Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a scheme $V'$ and a surjective étale morphism $V' \to Y' \times _ Y V$. Set $U' = V' \times _ V U$ so that $U' \to X'$ is a surjective étale morphism as well. Picture
\[ \vcenter { \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ V' \ar[r] & V } } \quad \text{lying over}\quad \vcenter { \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } } \]
Choose $u \in |U|$ with image $x \in |X|$. The property of being "locally quasi-finite" is étale local on the source-and-target, see Descent, Remark 35.32.7. Hence Lemmas 67.22.5 and 67.22.7 apply and we see that $f : X \to Y$ is quasi-finite at $x$ if and only if $U \to V$ is quasi-finite at $u$. Similarly for $f' : X' \to Y'$ and the morphism $U' \to V'$. Hence parts (1), (2), and (3) reduce to Morphisms, Lemmas 29.20.13 and 29.56.2. Part (4) follows from (3) and Lemma 67.8.4.
$\square$
Lemma 67.27.3. The composition of quasi-finite morphisms is quasi-finite. The same holds for locally quasi-finite.
Proof.
See Remark 67.22.3 and Morphisms, Lemma 29.20.12.
$\square$
Lemma 67.27.4. A base change of a quasi-finite morphism is quasi-finite. The same holds for locally quasi-finite.
Proof.
Immediate consequence of Lemma 67.27.2.
$\square$
The following lemma characterizes locally quasi-finite morphisms as those morphisms which are locally of finite type and have “discrete fibres”. However, this is not the same thing as asking $|X| \to |Y|$ to have discrete fibres as the discussion in Examples, Section 110.50 shows.
Lemma 67.27.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces. Assume $f$ is locally of finite type. The following are equivalent
$f$ is locally quasi-finite,
for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field the space $|X_ k|$ is discrete. Here $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$.
Proof.
Assume $f$ is locally quasi-finite. Let $\mathop{\mathrm{Spec}}(k) \to Y$ be as in (2). Choose a surjective étale morphism $U \to X$ where $U$ is a scheme. Then $U_ k = \mathop{\mathrm{Spec}}(k) \times _ Y U \to X_ k$ is an étale morphism of algebraic spaces by Properties of Spaces, Lemma 66.16.5. By Lemma 67.27.4 we see that $X_ k \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite. By definition this means that $U_ k \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite. Hence $|U_ k|$ is discrete by Morphisms, Lemma 29.20.8. Since $|U_ k| \to |X_ k|$ is surjective and open we conclude that $|X_ k|$ is discrete.
Conversely, assume (2). Choose a surjective étale morphism $V \to Y$ where $V$ is a scheme. Choose a surjective étale morphism $U \to V \times _ Y X$ where $U$ is a scheme. Note that $U \to V$ is locally of finite type as $f$ is locally of finite type. Picture
\[ \xymatrix{ U \ar[r] \ar[rd] & X \times _ Y V \ar[d] \ar[r] & V \ar[d] \\ & X \ar[r] & Y } \]
If $f$ is not locally quasi-finite then $U \to V$ is not locally quasi-finite. Hence there exists a specialization $u \leadsto u'$ for some $u, u' \in U$ lying over the same point $v \in V$, see Morphisms, Lemma 29.20.6. We claim that $u, u'$ do not have the same image in $X_ v = \mathop{\mathrm{Spec}}(\kappa (v)) \times _ Y X$ which will contradict the assumption that $|X_ v|$ is discrete as desired. Let $d = \text{trdeg}_{\kappa (v)}(\kappa (u))$ and $d' = \text{trdeg}_{\kappa (v)}(\kappa (u'))$. Then we see that $d > d'$ by Morphisms, Lemma 29.28.7. Note that $U_ v$ (the fibre of $U \to V$ over $v$) is the fibre product of $U$ and $X_ v$ over $X \times _ Y V$, hence $U_ v \to X_ v$ is étale (as a base change of the étale morphism $U \to X \times _ Y V$). If $u, u' \in U_ v$ map to the same element of $|X_ v|$ then there exists a point $r \in R_ v = U_ v \times _{X_ v} U_ v$ with $t(r) = u$ and $s(r) = u'$, see Properties of Spaces, Lemma 66.4.3. Note that $s, t : R_ v \to U_ v$ are étale morphisms of schemes over $\kappa (v)$, hence $\kappa (u) \subset \kappa (r) \supset \kappa (u')$ are finite separable extensions of fields over $\kappa (v)$ (see Morphisms, Lemma 29.36.7). We conclude that the transcendence degrees are equal. This contradiction finishes the proof.
$\square$
Lemma 67.27.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:
$f$ is locally quasi-finite,
for every $x \in |X|$ the morphism $f$ is quasi-finite at $x$,
for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is locally quasi-finite,
for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is locally quasi-finite,
there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is locally quasi-finite,
there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi $ is locally quasi-finite,
for every commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is locally quasi-finite,
there exists a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ is surjective such that the top horizontal arrow is locally quasi-finite, and
there exist Zariski coverings $Y = \bigcup _{i \in I} Y_ i$, and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is locally quasi-finite.
Proof.
Omitted.
$\square$
Lemma 67.27.7. An immersion is locally quasi-finite.
Proof.
Omitted.
$\square$
Lemma 67.27.8. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. If $X \to Z$ is locally quasi-finite, then $X \to Y$ is locally quasi-finite.
Proof.
Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } \]
with vertical arrows étale and surjective. (See Spaces, Lemma 65.11.6.) Apply Morphisms, Lemma 29.20.17 to the top row.
$\square$
Lemma 67.27.9. Let $S$ be a scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $y \in |Y|$. There are at most finitely many points of $|X|$ lying over $y$ at which $f$ is quasi-finite.
Proof.
Choose a field $k$ and a morphism $\mathop{\mathrm{Spec}}(k) \to Y$ in the equivalence class determined by $y$. The fibre $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$ is an algebraic space of finite type over a field, in particular quasi-compact. The map $|X_ k| \to |X|$ surjects onto the fibre of $|X| \to |Y|$ over $y$ (Properties of Spaces, Lemma 66.4.3). Moreover, the set of points where $X_ k \to \mathop{\mathrm{Spec}}(k)$ is quasi-finite maps onto the set of points lying over $y$ where $f$ is quasi-finite by Lemma 67.27.2. Choose an affine scheme $U$ and a surjective étale morphism $U \to X_ k$ (Properties of Spaces, Lemma 66.6.3). Then $U \to \mathop{\mathrm{Spec}}(k)$ is a morphism of finite type and there are at most a finite number of points where this morphism is quasi-finite, see Morphisms, Lemma 29.20.14. Since $X_ k \to \mathop{\mathrm{Spec}}(k)$ is quasi-finite at a point $x'$ if and only if it is the image of a point of $U$ where $U \to \mathop{\mathrm{Spec}}(k)$ is quasi-finite, we conclude.
$\square$
Lemma 67.27.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite.
Proof.
A monomorphism is separated, see Lemma 67.10.3. By Lemma 67.27.6 it suffices to prove the lemma after performing a base change by $Z \to Y$ with $Z$ affine. Hence we may assume that $Y$ is an affine scheme. Choose an affine scheme $U$ and an étale morphism $U \to X$. Since $X \to Y$ is locally of finite type the morphism of affine schemes $U \to Y$ is of finite type. Since $X \to Y$ is a monomorphism we have $U \times _ X U = U \times _ Y U$. In particular the maps $U \times _ Y U \to U$ are étale. Let $y \in Y$. Then either $U_ y$ is empty, or $\mathop{\mathrm{Spec}}(\kappa (u)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} U_ y$ is isomorphic to the fibre of $U \times _ Y U \to U$ over $u$ for some $u \in U$ lying over $y$. This implies that the fibres of $U \to Y$ are finite discrete sets (as $U \times _ Y U \to U$ is an étale morphism of affine schemes, see Morphisms, Lemma 29.36.7). Hence $U \to Y$ is quasi-finite, see Morphisms, Lemma 29.20.6. As $U \to X$ was an arbitrary étale morphism with $U$ affine this implies that $X \to Y$ is locally quasi-finite.
$\square$
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