Lemma 37.74.4. Let $S = \mathop{\mathrm{lim}}\nolimits S_ i$ be a limit of a directed system of schemes with affine transition morphisms. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_ i : X_ i \to Y_ i$ be the base change of $f_0$ to $S_ i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$. If
$f$ is locally quasi-finite and universally open, and
$f_0$ is locally of finite presentation,
then there exists an $i \geq 0$ such that $f_ i$ is locally quasi-finite and universally open.
Proof.
By Limits, Lemma 32.18.2 after increasing $0$ we may assume $f_0$ is locally quasi-finite. Let $x \in X$. By étale localization of quasi-finite morphisms we can find a diagram
\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \\ Y & V \ar[l] } \]
where $V \to Y$ is étale, $U \subset X_ V$ is open, $U \to V$ is finite, and $x$ is in the image of $U \to X$, see Lemma 37.41.1. After shrinking $V$ we may assume $V$ and $U$ are affine. Since $X$ is quasi-compact, it follows, by taking a finite disjoint union of such $V$ and $U$, that we can make a diagram as above such that $U \to X$ is surjective. By Limits, Lemmas 32.10.1, 32.4.11, 32.8.15, 32.8.3, 32.8.10, and 32.4.13 after possibly increasing $0$ we may assume we have a diagram
\[ \xymatrix{ X_0 \ar[d] & U_0 \ar[l] \ar[d] \\ Y_0 & V_0 \ar[l] } \]
where $V_0$ is affine, $V_0 \to Y_0$ is étale, $U_0 \subset (X_0)_{V_0}$ is open, $U_0 \to V_0$ is finite, and $U_0 \to X_0$ is surjective. Since $V_ i \to Y_ i$ is étale and hence universally open, follows that it suffices to prove that $U_ i \to V_ i$ is universally open for large enough $i$. This reduces us to the case discussed in the next paragraph.
Let $A = \mathop{\mathrm{colim}}\nolimits A_ i$ be a filtered colimit of rings. Let $A_0 \to B_0$ be a ring map. Set $B = A \otimes _{A_0} B_0$ and $B_ i = A_ i \otimes _{A_0} B_0$. Assume $A_0 \to B_0$ is finite, of finite presentation, and $A \to B$ is universally open. We have to show that $A_ i \to B_ i$ is universally open for $i$ large enough. Pick $b_{0, 1}, \ldots , b_{0, d} \in B_0$ which generate $B_0$ as an $A_0$-module. Set $h_0 = \sum _{j = 1, \ldots , d} x_ jb_{0, j}$ in $B_0[x_1, \ldots , x_ d]$. Denote $h$, resp. $h_ i$ the image of $h_0$ in $B[x_1, \ldots , x_ d]$, resp. $B_ i[x_1, \ldots , x_ d]$. The image $U$ of $D(h)$ in $\mathop{\mathrm{Spec}}(A[x_1, \ldots , x_ d])$ is open as $A \to B$ is universally open. Of course $U$ is quasi-compact as the image of an affine scheme. For $i$ large enough there is a quasi-compact open $U_ i \subset \mathop{\mathrm{Spec}}(A_ i[x_1, \ldots , x_ d])$ whose inverse image in $\mathop{\mathrm{Spec}}(A[x_1, \ldots , x_ d])$ is $U$, see Limits, Lemma 32.4.11. After increasing $i$ we may assume that $D(h_ i)$ maps into $U_ i$; this follows from the same lemma by considering the pullback of $U_ i$ in $D(h_ i)$. Finally, for $i$ even larger the morphism of schemes $D(h_ i) \to U_ i$ will be surjective by an application of the already used Limits, Lemma 32.8.15. We conclude $A_ i \to B_ i$ is universally open by Lemma 37.74.3.
$\square$
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