The Stacks project

63.50 Separated, locally quasi-finite morphisms

In this section we prove that an algebraic space which is locally quasi-finite and separated over a scheme, is representable. This implies that a separated and locally quasi-finite morphism is representable (see Lemma 63.51.1). But first... a lemma (which will be obsoleted by Proposition 63.50.2).

Lemma 63.50.1. Let $S$ be a scheme. Consider a commutative diagram

\[ \xymatrix{ V' \ar[r] \ar[rd] & T' \times _ T X \ar[r] \ar[d] & X \ar[d] \\ & T' \ar[r] & T } \]

of algebraic spaces over $S$. Assume

  1. $T' \to T$ is an étale morphism of affine schemes,

  2. $X \to T$ is a separated, locally quasi-finite morphism,

  3. $V'$ is an open subspace of $T' \times _ T X$, and

  4. $V' \to T'$ is quasi-affine.

In this situation the image $U$ of $V'$ in $X$ is a quasi-compact open subspace of $X$ which is representable.

Proof. We first make some trivial observations. Note that $V'$ is representable by Lemma 63.21.3. It is also quasi-compact (as a quasi-affine scheme over an affine scheme, see Morphisms, Lemma 28.12.2). Since $T' \times _ T X \to X$ is étale (Properties of Spaces, Lemma 62.16.5) the map $|T' \times _ T X| \to |X|$ is open, see Properties of Spaces, Lemma 62.16.7. Let $U \subset X$ be the open subspace corresponding to the image of $|V'|$, see Properties of Spaces, Lemma 62.4.8. As $|V'|$ is quasi-compact we see that $|U|$ is quasi-compact, hence $U$ is a quasi-compact algebraic space, by Properties of Spaces, Lemma 62.5.2.

By Morphisms, Lemma 28.54.10 the morphism $T' \to T$ is universally bounded. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $T' \to T$, see Morphisms, Lemma 28.54.9 for a description of this integer in the case of an étale morphism. If $n = 1$, then $T' \to T$ is an open immersion (see Étale Morphisms, Theorem 40.14.1), and the result is clear. Assume $n > 1$.

Consider the affine scheme $T'' = T' \times _ T T'$. As $T' \to T$ is étale we have a decomposition (into open and closed affine subschemes) $T'' = \Delta (T') \amalg T^*$. Namely $\Delta = \Delta _{T'/T}$ is open by Morphisms, Lemma 28.33.13 and closed because $T' \to T$ is separated as a morphism of affines. As a base change the degrees of the fibres of the second projection $\text{pr}_1 : T' \times _ T T' \to T'$ are bounded by $n$, see Morphisms, Lemma 28.54.6. On the other hand, $\text{pr}_1|_{\Delta (T')} : \Delta (T') \to T'$ is an isomorphism and every fibre has exactly one point. Thus, on applying Morphisms, Lemma 28.54.9 we conclude the degrees of the fibres of the restriction $\text{pr}_1|_{T^*} : T^* \to T'$ are bounded by $n - 1$. Hence the induction hypothesis applied to the diagram

\[ \xymatrix{ p_0^{-1}(V') \cap X^* \ar[r] \ar[rd] & X^* \ar[r]_{p_1|_{X^*}} \ar[d] & X' \ar[d] \\ & T^* \ar[r]^{\text{pr}_1|_{T^*}} & T' } \]

gives that $p_1(p_0^{-1}(V') \cap X^*)$ is a quasi-compact scheme. Here we set $X'' = T'' \times _ T X$, $X^* = T^* \times _ T X$, and $X' = T' \times _ T X$, and $p_0, p_1 : X'' \to X'$ are the base changes of $\text{pr}_0, \text{pr}_1$. Most of the hypotheses of the lemma imply by base change the corresponding hypothesis for the diagram above. For example $p_0^{-1}(V') = T'' \times _{T'} V'$ is a scheme quasi-affine over $T''$ as a base change. Some verifications omitted.

By Properties of Spaces, Lemma 62.13.1 we conclude that

\[ p_1(p_0^{-1}(V')) = V' \cup p_1(p_0^{-1}(V') \cap X^*) \]

is a quasi-compact scheme. Moreover, it is clear that $p_1(p_0^{-1}(V'))$ is the inverse image of the quasi-compact open subspace $U \subset X$ discussed in the first paragraph of the proof. In other words, $T' \times _ T U$ is a scheme! Note that $T' \times _ T U$ is quasi-compact and separated and locally quasi-finite over $T'$, as $T' \times _ T X \to T'$ is locally quasi-finite and separated being a base change of the original morphism $X \to T$ (see Lemmas 63.4.4 and 63.27.4). This implies by More on Morphisms, Lemma 36.38.2 that $T' \times _ T U \to T'$ is quasi-affine.

By Descent, Lemma 34.36.1 this gives a descent datum on $T' \times _ T U / T'$ relative to the étale covering $\{ T' \to W\} $, where $W \subset T$ is the image of the morphism $T' \to T$. Because $U'$ is quasi-affine over $T'$ we see from Descent, Lemma 34.35.1 that this datum is effective, and by the last part of Descent, Lemma 34.36.1 this implies that $U$ is a scheme as desired. Some minor details omitted. $\square$

Proposition 63.50.2. Let $S$ be a scheme. Let $f : X \to T$ be a morphism of algebraic spaces over $S$. Assume

  1. $T$ is representable,

  2. $f$ is locally quasi-finite, and

  3. $f$ is separated.

Then $X$ is representable.

Proof. Let $T = \bigcup T_ i$ be an affine open covering of the scheme $T$. If we can show that the open subspaces $X_ i = f^{-1}(T_ i)$ are representable, then $X$ is representable, see Properties of Spaces, Lemma 62.13.1. Note that $X_ i = T_ i \times _ T X$ and that locally quasi-finite and separated are both stable under base change, see Lemmas 63.4.4 and 63.27.4. Hence we may assume $T$ is an affine scheme.

By Properties of Spaces, Lemma 62.6.2 there exists a Zariski covering $X = \bigcup X_ i$ such that each $X_ i$ has a surjective étale covering by an affine scheme. By Properties of Spaces, Lemma 62.13.1 again it suffices to prove the proposition for each $X_ i$. Hence we may assume there exists an affine scheme $U$ and a surjective étale morphism $U \to X$. This reduces us to the situation in the next paragraph.

Assume we have

\[ U \longrightarrow X \longrightarrow T \]

where $U$ and $T$ are affine schemes, $U \to X$ is étale surjective, and $X \to T$ is separated and locally quasi-finite. By Lemmas 63.39.5 and 63.27.3 the morphism $U \to T$ is locally quasi-finite. Since $U$ and $T$ are affine it is quasi-finite. Set $R = U \times _ X U$. Then $X = U/R$, see Spaces, Lemma 61.9.1. As $X \to T$ is separated the morphism $R \to U \times _ T U$ is a closed immersion, see Lemma 63.4.5. In particular $R$ is an affine scheme also. As $U \to X$ is étale the projection morphisms $t, s : R \to U$ are étale as well. In particular $s$ and $t$ are quasi-finite, flat and of finite presentation (see Morphisms, Lemmas 28.34.6, 28.34.12 and 28.34.11).

Let $(U, R, s, t, c)$ be the groupoid associated to the étale equivalence relation $R$ on $U$. Let $u \in U$ be a point, and denote $p \in T$ its image. We are going to use More on Groupoids, Lemma 39.13.2 for the groupoid $(U, R, s, t, c)$ over the scheme $T$ with points $p$ and $u$ as above. By the discussion in the previous paragraph all the assumptions (1) – (7) of that lemma are satisfied. Hence we get an étale neighbourhood $(T', p') \to (T, p)$ and disjoint union decompositions

\[ U_{T'} = U' \amalg W, \quad R_{T'} = R' \amalg W' \]

and $u' \in U'$ satisfying conclusions (a), (b), (c), (d), (e), (f), (g), and (h) of the aforementioned More on Groupoids, Lemma 39.13.2. We may and do assume that $T'$ is affine (after possibly shrinking $T'$). Conclusion (h) implies that $R' = U' \times _{X_{T'}} U'$ with projection mappings identified with the restrictions of $s'$ and $t'$. Thus $(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \times _{t', U', s'} R'})$ of conclusion (g) is an étale equivalence relation. By Spaces, Lemma 61.10.2 we conclude that $U'/R'$ is an open subspace of $X_{T'}$. By conclusion (d) the schemes $U'$, $R'$ are affine and the morphisms $s'|_{R'}, t'|_{R'}$ are finite étale. Hence Groupoids, Proposition 38.23.9 kicks in and we see that $U'/R'$ is an affine scheme.

We conclude that for every pair of points $(u, p)$ as above we can find an étale neighbourhood $(T', p') \to (T, p)$ with $\kappa (p) = \kappa (p')$ and a point $u' \in U_{T'}$ mapping to $u$ such that the image $x'$ of $u'$ in $|X_{T'}|$ has an open neighbourhood $V'$ in $X_{T'}$ which is an affine scheme. We apply Lemma 63.50.1 to obtain an open subspace $W \subset X$ which is a scheme, and which contains $x$ (the image of $u$ in $|X|$). Since this works for every $x$ we see that $X$ is a scheme by Properties of Spaces, Lemma 62.13.1. This ends the proof. $\square$

Comments (0)

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0417. Beware of the difference between the letter 'O' and the digit '0'.