Lemma 32.17.2 (0GWW)—The Stacks project

Lemma 32.17.2. Let $S$ be a Nagata scheme (and in particular locally Noetherian). Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. The following are equivalent

$f$ separated,
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 at most one dotted arrow ² making the diagram commute.

Proof. By Lemma 32.15.2 we see that (1) implies (2). Assume (2). In order to show that $f$ is separated, we have to show that $\Delta : X \to X \times _ Y X$ is closed. By Morphisms, Lemma 29.15.7 the morphism $\Delta $ is quasi-compact. By Lemma 32.17.1 it suffices to show: for any commutative diagram

\[ \xymatrix{ U \ar[rr] \ar[d] & & X \ar[d]^\Delta \\ C \ar[rr]^{(a_1, a_2)} \ar@{..>}[rru] & & X \times _ Y X } \]

of schemes over $S$ such that

$C$ is a normal integral scheme of finite type over $S$,
$U = C \setminus \{ c\} $ for some closed point $c \in C$,
$A = \mathcal{O}_{C, c}$ has dimension $1$.

then in the commutative diagram

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

where $K = \text{Frac}(A)$ there exists some dotted arrow making the diagram commute. By Lemma 32.6.4 the existence of the dotted arrow in the second diagram is equivalent to the existence of the dotted arrow in the first diagram. Moreover, the existence there is the same as asking $a_1 = a_2$. However $a_1|_ U = a_2|_ U$, so by the uniqueness assumption (2) we see that this is true and the proof is complete. $\square$

[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 there to be at most one dotted arrow making the first commutative diagram commute.

The Stacks project

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