The Stacks project

Lemma 32.17.3. Let $S$ be a Nagata scheme (and in particular locally Noetherian). Let $f : X \to Y$ be a quasi-compact morphism of schemes locally of finite type over $S$. The following are equivalent

  1. $f$ proper,

  2. for any commutative diagram

    \[ \xymatrix{ U \ar[r] \ar[d] & X \ar[d]^ f \\ C \ar[r] \ar@{..>}[ru] & Y } \]

    of schemes over $S$ such that

    1. $C$ is a normal integral scheme of finite type over $S$,

    2. $U = C \setminus \{ c\} $ for some closed point $c \in C$,

    3. $A = \mathcal{O}_{C, c}$ has dimension $1$1

    then in the commutative diagram

    \[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & Y } \]

    where $K = \text{Frac}(A)$ there exists exactly one dotted arrow2 making the diagram commute.

Proof. This is formal from Lemmas 32.17.1 and 32.17.2 and the definition of proper morphisms as being finite type, separated, and universally closed. $\square$

[1] It follows that $A$ is a discrete valuation ring, see Algebra, Lemma 10.119.7. Moreover, $c$ maps to a finite type point $s \in S$ and $A$ is essentially of finite type over $\mathcal{O}_{S, s}$.
[2] By Lemma 32.6.4 this is equivalent to asking for the existence and uniqueness of the dotted arrow making the first commutative diagram commute.

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