The Stacks project

Proof. We may assume that each $S_ i \to S$ is locally quasi-finite, see Lemma 37.23.6.

Fix a point $s \in S$. Pick an $i \in I$ and a point $s_ i \in S_ i$ mapping to $s$. Choose an elementary étale neighbourhood $(S', s) \to (S, s)$ such that there exists an open

which contains a unique point $v \in V$ mapping to $s \in S'$ and such that $V \to S'$ is finite, see Lemma 37.41.1. Then $V \to S'$ is finite locally free, because it is finite and because $S_ i \times _ S S' \to S'$ is flat and locally of finite presentation as a base change of the morphism $S_ i \to S$, see Morphisms, Lemmas 29.21.4, 29.25.8, and 29.48.2. Hence $V \to S'$ is open, and after shrinking $S'$ we may assume that $V \to S'$ is surjective finite locally free. Since we can do this for every point of $S$ we conclude that $\{ S_ i \to S\} $ can be refined by a covering of the form $\{ V_ a \to S\} _{a \in A}$ where each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ étale and $V_ a \to S'_ a$ surjective finite locally free. $\square$

Proof. Let $\{ V_ a \to S\} _{a \in A}$ be the fppf covering found in Lemma 37.48.1. In other words, this covering refines $\{ S_ i \to S\} $ and each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ étale and $V_ a \to S'_ a$ surjective finite locally free.

By Remark 37.40.3 there exists a Zariski open covering $S = \bigcup U_ j$, for each $j$ a finite locally free, surjective morphism $W_ j \to U_ j$, and for each $j$ a Zariski open covering $\{ W_{j, k} \to W_ j\} $ such that the family $\{ W_{j, k} \to S\} $ refines the étale covering $\{ S'_ a \to S\} $, i.e., for each pair $j, k$ there exists an $a(j, k)$ and a factorization $W_{j, k} \to S'_ a \to S$ of the morphism $W_{j, k} \to S$. Set $T_{j, k} = W_{j, k} \times _{S'_ a} V_ a$ and everything is clear. $\square$

Proof. Let $V' \to S$ be the normalization of $S$ in $U$, see Morphisms, Section 29.53. By construction $V' \to S$ is integral. By Morphisms, Lemmas 29.53.6 and 29.53.12 we see that the inverse image of $U$ in $V'$ is $V$. Let $Z$ be the reduced induced scheme structure on $S \setminus U$. Then $\overline{V} = V' \amalg Z$ works. $\square$

Proof. By Zariski's Main Theorem (Lemma 37.43.3) we can assume $V$ is a quasi-compact open in a scheme $V'$ finite over $S$. After replacing $V'$ by the scheme theoretic image of $V$ we may assume that $V$ is dense in $V'$. It follows that $V' \times _ S U = V$ because $V \to V' \times _ S U$ is closed as $V$ is finite over $U$. Let $Z$ be the reduced induced scheme structure on $S \setminus U$. Then $\overline{V} = V' \amalg Z$ works. $\square$

Proof. Choose $S = \bigcup U_ j$, $W_ j \to U_ j$, $W_ j = \bigcup W_{j, k}$, and $T_{j, k} \to W_{j, k}$ as in Lemma 37.48.2. By Lemma 37.48.3 we can extend $W_ j \to U_ j$ to a surjective integral morphism $\overline{W}_ j \to S$. After this we can extend $T_{j, k} \to W_{j, k}$ to a surjective integral morphism $\overline{T}_{j, k} \to \overline{W}_ j$. We set $\overline{T}_ j$ equal to the product of all the schemes $\overline{T}_{j, k}$ over $\overline{W}_ j$ (Limits, Lemma 32.3.1). Then we set $S'$ equal to the product of all the schemes $\overline{T}_ j$ over $S$. If $x \in S'$, then there is a $j$ such that the image of $x$ in $S$ lies in $U_ j$. Hence there is a $k$ such that the image of $x$ under the projection $S' \to \overline{W}_ j$ lies in $W_{j, k}$. Hence under the projection $S' \to \overline{T}_ j \to \overline{T}_{j, k}$ the point $x$ ends up in $T_{j, k}$. And $T_{j, k} \to S$ factors through $S_ i$ for some $i$. Finally, the morphism $S' \to S$ is integral and surjective by Limits, Lemmas 32.3.3 and 32.3.2. $\square$

Proof. Let $Y \to X$ be the integral surjective morphism found in Lemma 37.48.5. Choose a finite affine open covering $Y = \bigcup V_ j$ such that $V_ j \to X$ factors through $S_{i(j)}$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_\lambda $ with $Y_\lambda \to X$ finite and of finite presentation, see Limits, Lemma 32.7.3. For large enough $\lambda $ we can find affine opens $V_{\lambda , j} \subset Y_\lambda $ whose inverse image in $Y$ recovers $V_ j$, see Limits, Lemma 32.4.11. For even larger $\lambda $ the morphisms $V_ j \to S_{i(j)}$ over $X$ come from morphisms $V_{\lambda , j} \to S_{i(j)}$ over $X$, see Limits, Proposition 32.6.1. Setting $S' = Y_\lambda $ for this $\lambda $ finishes the proof. $\square$

Proof. Let $\{ T_ i \to T\} $ be an fppf covering of schemes, see Topologies, Definition 34.7.1. Observe that $T_ i \to T$ is locally of finite type. Let $U \subset T$ be an affine open. It suffices to show that $\{ T_ i \times _ T U \to U\} $ can be refined by a standard ph covering, see Topologies, Definition 34.8.4. This follows immediately from Lemma 37.48.6 and the fact that a finite morphism is proper (Morphisms, Lemma 29.44.11). $\square$