The Stacks project

Lemma 37.72.4. 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

  1. $f$ is locally quasi-finite and universally open, and

  2. $f_0$ 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 37.72.3. $\square$

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 0F34. Beware of the difference between the letter 'O' and the digit '0'.