Lemma 69.19.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume
$Y$ is locally Noetherian,
$f$ is of finite type and quasi-separated,
for every commutative diagram
where $A$ is a discrete valuation ring and $K$ its fraction field, there is a unique dotted arrow making the diagram commute.
Then $f$ is proper.
Proof. It suffices to prove $f$ is universally closed because $f$ is separated by Lemma 69.19.1. To do this we may work étale locally on $Y$ (Morphisms of Spaces, Lemma 67.9.5). Hence we may assume $Y = \mathop{\mathrm{Spec}}(A)$ is a Noetherian affine scheme. Choose $X' \to X$ as in the weak form of Chow's lemma (Lemma 69.18.1). We claim that $X' \to \mathop{\mathrm{Spec}}(A)$ is universally closed. The claim implies the lemma by Morphisms of Spaces, Lemma 67.40.7. To prove this, according to Limits, Lemma 32.15.4 it suffices to prove that in every solid commutative diagram
where $A$ is a dvr with fraction field $K$ we can find the dotted arrow $a$. By assumption 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 $a$. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)