Lemma 32.17.3 (0GWX)—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

$f$ proper,
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 arrow ² 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.

The Stacks project

Comments (0)

Post a comment