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The Stacks project

Lemma 67.42.3. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Assume f is quasi-compact and separated. Then the following are equivalent

  1. f is universally closed,

  2. the existence part of the valuative criterion holds as in Definition 67.41.1, and

  3. given any commutative solid diagram

    \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & Y }

    where A is a valuation ring with field of fractions K, there exists a dotted arrow, i.e., f satisfies the existence part of the valuative criterion as in Schemes, Definition 26.20.3.

Proof. Since f is separated parts (2) and (3) are equivalent by Lemma 67.41.5. The equivalence of (3) and (1) follows from Lemma 67.42.2. \square

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