The Stacks project

Proof. We first make some trivial observations. Note that $V'$ is representable by Lemma 67.21.3. It is also quasi-compact (as a quasi-affine scheme over an affine scheme, see Morphisms, Lemma 29.13.2). Since $T' \times _ T X \to X$ is étale (Properties of Spaces, Lemma 66.16.5) the map $|T' \times _ T X| \to |X|$ is open, see Properties of Spaces, Lemma 66.16.7. Let $U \subset X$ be the open subspace corresponding to the image of $|V'|$, see Properties of Spaces, Lemma 66.4.8. 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 66.5.2.

By Morphisms, Lemma 29.57.9 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 29.57.8 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 41.14.1), 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 29.35.13 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 29.57.5. 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 29.57.8 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

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 66.13.1 we conclude that

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 67.4.4 and 67.27.4). This implies by More on Morphisms, Lemma 37.43.2 that $T' \times _ T U \to T'$ is quasi-affine.

By Descent, Lemma 35.39.1 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 35.38.1 that this datum is effective, and by the last part of Descent, Lemma 35.39.1 this implies that $U$ is a scheme as desired. Some minor details omitted. $\square$

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 66.13.1. Note that $X_ i = T_ i \times _ T X$ and that locally quasi-finite and separated are both stable under base change, see Lemmas 67.4.4 and 67.27.4. Hence we may assume $T$ is an affine scheme.

By Properties of Spaces, Lemma 66.6.2 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 66.13.1 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

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 67.39.5 and 67.27.3 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 65.9.1. As $X \to T$ is separated the morphism $R \to U \times _ T U$ is a closed immersion, see Lemma 67.4.5. 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 29.36.6, 29.36.12 and 29.36.11).

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 40.13.2 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

and $u' \in U'$ satisfying conclusions (a), (b), (c), (d), (e), (f), (g), and (h) of the aforementioned More on Groupoids, Lemma 40.13.2. 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 65.10.2 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 39.23.9 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 67.50.1 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 66.13.1. This ends the proof. $\square$