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.