Lemma 70.22.3 (0CMF)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.22: Refined Noetherian valuative criteria
Lemma 70.22.3 (cite)

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Lemma 70.22.3. Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$ . Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is quasi-separated, and that $h(U)$ is dense in $X$ . If given any commutative solid diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & Y }$

where $A$ is a discrete valuation ring with field of fractions $K$ , there exists a unique dotted arrow making the diagram commute, then $f$ is proper.

Proof. Combine Lemmas 70.22.2 and 70.22.1. $\square$

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