Properness and completeness of curves

In this post, I want to show an application of fpqc descent (specifically, pro-Zariski descent) to a classical lemma about properness. Recall (EGA II, Thm 7.3.8) the valuative criterion of properness:

Theorem. Let f \colon X \ra Y be a finite type morphism of locally Noetherian schemes. Then f is proper if and only if for every commutative diagram

Rendered by QuickLaTeX.com

where A is a discrete valuation ring with fraction field K, there exists a unique morphism \Spec A \ra X making commutative the diagram

Rendered by QuickLaTeX.com

Lemma. Suppose f \colon X \ra Y is a finite type morphism of locally Noetherian schemes. Then f is proper if and only if for every Dedekind scheme C \in \ob(\Sch/Y) and every closed point p \in C, every Y-morphism g \colon C\setminus\{p\} \ra X extends uniquely to C \ra X.

Proof. If A is a discrete valuation ring with fraction field K and maximal ideal \fr m, then \Spec A is a Dedekind scheme, and (\Spec A)\setminus\{\fr m\} = \Spec K. Thus, the condition of the lemma clearly implies properness, by the theorem above.

Conversely, suppose f is proper, and let C be a Dedekind scheme over Y, and p \in C a closed point. Write U = C\setminus\{p\}, and let V = \Spec \O_{C,p}. Let \eta be the generic point of C, and K = \O_{C,\eta}.

The valuative criterion shows that the the induced map g|_{\eta} \colon \Spec K \ra X extends uniquely to a Y-morphism \tilde{g} \colon V \ra X. Moreover, since U \sbq C is an open immersion, the fibre product U \times_C V is the open \Spec K \sbq V.

Now \{U, V\} is an fpqc cover of C (in fact, a pro-Zariski cover). The above shows that g and \tilde{g} have the same restriction to U \times_C V. Since representable presheaves are sheaves for the fpqc topology (Tag 03O3), we thus see that they glue to a unique map C \ra X. \qedsymbol

Remark. Of the course, the classical proof of the lemma goes by noting that the morphism V \ra X factors through some Zariski-open V' containing p, since X is of finite type over Y. The only thing that we changed is that we didn’t pass from the pro-Zariski to the Zariski covering, but instead argued directly using fpqc descent.

Leave a Reply

Your email address will not be published. Required fields are marked *