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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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