The Stacks project

Lemma 69.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-separated and of finite type and $Y$ is locally Noetherian. The following are equivalent:

  1. $f$ is proper,

  2. $f$ satisfies the valuative criterion, see Morphisms of Spaces, Definition 66.41.1,

  3. for any diagram (69.21.1.1) there exists exactly one dotted arrow,

  4. for all diagrams (69.21.1.1) with $A$ a discrete valuation ring there exists exactly one dotted arrow, and

  5. for all diagrams (69.21.1.1) where $A$ is a discrete valuation ring and where the image of $\mathop{\mathrm{Spec}}(K) \to X$ is a point of codimension $0$ on $X$ there exists exactly one dotted arrow1.

Proof. We have (1) $\Leftrightarrow $ (2) $\Leftrightarrow $ (3) by Morphisms of Spaces, Lemma 66.44.1. It is clear that (3) $\Rightarrow $ (4) $\Rightarrow $ (5). To finish the proof we will now show (5) implies (1).

Assume (5). By Lemma 69.21.2 we see that $f$ is separated. To finish the proof it suffices to show that $f$ is universally closed. Let $V \to Y$ be an ├ętale morphism where $V$ is an affine scheme. It suffices to show that the base change $V \times _ Y X \to V$ is universally closed, see Morphisms of Spaces, Lemma 66.9.5. Let

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

of algebraic spaces over $S$ be a commutative diagram where $A$ is a discrete valuation ring with fraction field $K$ and where $\mathop{\mathrm{Spec}}(K) \to V \times _ Y X$ maps to a point of codimension $0$ of the algebraic space $V \times _ Y X$. Since $V \times _ Y X \to X$ is ├ętale it follows that the image of $\mathop{\mathrm{Spec}}(K) \to X$ is a point of codimension $0$ of $X$. Thus by (5) we obtain the longer of the two dotted arrows fitting into the diagram. Then of course we obtain the shorter one as well. It follows that our assumptions hold for the morphism $V \times _ Y X \to V$ and we reduce to the case discussed in the next paragraph.

Aassume $Y$ is a Noetherian affine scheme. In this case $X$ is a separated Noetherian algebraic space (we already know $f$ is separated) of finite type over $Y$. (In particular, the algebraic space $X$ has a dense open subspace which is a scheme by Properties of Spaces, Proposition 65.13.3 although strictly speaking we will not need this.) Choose a quasi-projective scheme $X'$ over $Y$ and a proper surjective morphism $X' \to X$ as in the weak form of Chow's lemma (Cohomology of Spaces, Lemma 68.18.1). We may replace $X'$ by the disjoint union of the irreducible components which dominate an irreducible component of $X$; details omitted. In particular, we may assume that generic points of the scheme $X'$ map to points of codimension $0$ of $X$ (in this case these are exactly the generic points of $X$). We claim that $X' \to Y$ is proper. The claim implies $X$ is proper over $Y$ by Morphisms of Spaces, Lemma 66.40.7. To prove this, according to Limits, Lemma 32.15.3 it suffices to prove that in every solid commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X' \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[ru]^ a \ar@{-->}[rru]_ b & & Y } \]

where $A$ is a dvr with fraction field $K$ and where $K$ is the residue field of a generic point of $X'$ we can find the dotted arrow $a$ (we already know uniqueness as $X'$ is separated). By assumption (5) we can find the dotted arrow $b$. Then the morphism $X' \times _{X, b} \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A)$ is a proper morphism of schemes and by the valuative criterion for morphisms of schemes we can lift $b$ to the desired morphism $a$. $\square$

[1] There is a sharper formulation where in the existence part one only requires the dotted arrow exists after an extension of discrete valuation rings.

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