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 be a finite type morphism of locally Noetherian schemes. Then is proper if and only if for every commutative diagram
where is a discrete valuation ring with fraction field , there exists a unique morphism making commutative the diagram
Lemma. Suppose is a finite type morphism of locally Noetherian schemes. Then is proper if and only if for every Dedekind scheme and every closed point , every -morphism extends uniquely to .
Proof. If is a discrete valuation ring with fraction field and maximal ideal , then is a Dedekind scheme, and . Thus, the condition of the lemma clearly implies properness, by the theorem above.
Conversely, suppose is proper, and let be a Dedekind scheme over , and a closed point. Write , and let . Let be the generic point of , and .
The valuative criterion shows that the the induced map extends uniquely to a -morphism . Moreover, since is an open immersion, the fibre product is the open .
Now is an fpqc cover of (in fact, a pro-Zariski cover). The above shows that and have the same restriction to . Since representable presheaves are sheaves for the fpqc topology (Tag 03O3), we thus see that they glue to a unique map .
Remark. Of the course, the classical proof of the lemma goes by noting that the morphism factors through some Zariski-open containing , since is of finite type over . 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.
I want to ask why we have .
Because is the open , taking the fibre product has the effect of removing . Applying this to gives the punctured spectrum . Since is the spectrum of a DVR, it has two points, and removing the closed point gives the spectrum of the fraction field (exercise in local algebra).