The Stacks project

Lemma 32.4.19. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume

  1. $S$ quasi-compact and quasi-separated,

  2. $X_ i$ quasi-compact and quasi-separated,

  3. the transition morphisms $X_{i'} \to X_ i$ are finite,

  4. $X_ i \to S$ locally of finite type

  5. $X \to S$ integral.

Then $X_ i \to S$ is finite for $i$ large enough.

Proof. By Lemma 32.4.18 we may assume $X_ i \to S$ is affine for all $i$. Choose a finite affine open covering $S = \bigcup _{j = 1, \ldots , n} V_ j$. Denote $f : X \to S$ and $f_ i : X_ i \to S$ the structure morphisms. It suffices to show that there exists an $i$ such that $f_ i^{-1}(V_ j)$ is finite over $V_ j$ for $j = 1, \ldots , m$ (Morphisms, Lemma 29.44.3). Namely, for $i' \geq i$ the composition $X_{i'} \to X_ i \to S$ will be finite as a composition of finite morphisms (Morphisms, Lemma 29.44.5). This reduces us to the affine case: Let $R$ be a ring and $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $R \to A$ integral and $A_ i \to A_{i'}$ finite for all $i \leq i'$. Moreover $R \to A_ i$ is of finite type for all $i$. Goal: Show that $A_ i$ is finite over $R$ for some $i$. To prove this choose an $i \in I$ and pick generators $x_1, \ldots , x_ m \in A_ i$ of $A_ i$ as an $R$-algebra. Since $A$ is integral over $R$ we can find monic polynomials $P_ j \in R[T]$ such that $P_ j(x_ j) = 0$ in $A$. Thus there exists an $i' \geq i$ such that $P_ j(x_ j) = 0$ in $A_{i'}$ for $j = 1, \ldots , m$. Then the image $A'_ i$ of $A_ i$ in $A_{i'}$ is finite over $R$ by Algebra, Lemma 10.36.5. Since $A'_ i \subset A_{i'}$ is finite too we conclude that $A_{i'}$ is finite over $R$ by Algebra, Lemma 10.7.3. $\square$

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