32.16 Refined Noetherian valuative criteria
One usually does not have to consider all possible diagrams with valuation rings when checking valuative criteria. An example is given by Morphisms, Lemma 29.42.2. In the Noetherian setting, we have also seen this in Lemmas 32.15.2 and 32.15.3. Here is another variant.
Lemma 32.16.1. Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that $h(U)$ is dense in $X$. If given any commutative solid diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & S } \]
where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute, then $f$ is proper.
Proof.
There is an immediate reduction to the case where $S$ is affine. Then $U$ is quasi-compact. Let $U = U_1 \cup \ldots \cup U_ n$ be an affine open covering. We may replace $U$ by $U_1 \amalg \ldots \amalg U_ n$ without changing the assumptions, hence we may assume $U$ is affine. Thus we can find an open immersion $U \to Y$ over $X$ with $Y$ proper over $X$. (First put $U$ inside $\mathbf{A}^ n_ X$ using Morphisms, Lemma 29.39.2 and then take the closure inside $\mathbf{P}^ n_ X$, or you can directly use Morphisms, Lemma 29.43.12.) We can assume $U$ is dense in $Y$ (replace $Y$ by the scheme theoretic closure of $U$ if necessary, see Morphisms, Section 29.7). Note that $g : Y \to X$ is surjective as the image is closed and contains the dense subset $h(U)$. We will show that $Y \to S$ is proper. This will imply that $X \to S$ is proper by Morphisms, Lemma 29.41.9 thereby finishing the proof. To show that $Y \to S$ is proper we will use part (4) of Lemma 32.15.3. To do this consider a diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_ y \ar[d] & Y \ar[d]^{f \circ g} \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{..>}[ru] & S } \]
where $A$ is a discrete valuation ring with fraction field $K$ and where $y : \mathop{\mathrm{Spec}}(K) \to Y$ is the inclusion of a generic point. We have to show there exists a unique dotted arrow. Uniqueness holds by the converse to the valuative criterion for separatedness (Schemes, Lemma 26.22.1) since $Y \to S$ is separated as the composition of the separated morphisms $Y \to X$ and $X \to S$ (Schemes, Lemma 26.21.12). Existence can be seen as follows. As $y$ is a generic point of $Y$, it is contained in $U$. By assumption of the lemma there exists a morphism $a : \mathop{\mathrm{Spec}}(A) \to X$ such that
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_ y \ar[d] & U \ar[r] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar[rru]^ a & & S } \]
is commutative. Then since $Y \to X$ is proper, we can apply the valuative criterion for properness (Morphisms, Lemma 29.42.1) to find a morphism $b : \mathop{\mathrm{Spec}}(A) \to Y$ such that
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_ y \ar[d] & Y \ar[d]^ g \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ a \ar[ru]^ b & X } \]
is commutative. This finishes the proof since $b$ can serve as the dotted arrow above.
$\square$
Lemma 32.16.2. Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ is locally of finite type, that $h$ is of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & S } \]
where $A$ is a discrete valuation ring with field of fractions $K$, there exists at most one dotted arrow making the diagram commute, then $f$ is separated.
Proof.
We will apply Lemma 32.16.1 to the morphisms $U \to X$ and $\Delta : X \to X \times _ S X$. We check the conditions. Observe that $\Delta $ is quasi-compact by Properties, Lemma 28.5.4 (and Schemes, Lemma 26.21.13). Of course $\Delta $ is locally of finite type and separated (true for any diagonal morphism). Finally, suppose given a commutative solid diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^{(a, b)} \ar@{-->}[rru] & & X \times _ S X } \]
where $A$ is a discrete valuation ring with field of fractions $K$. Then $a$ and $b$ give two dotted arrows in the diagram of the lemma and have to be equal. Hence as dotted arrow we can use $a = b$ which gives existence. This finishes the proof.
$\square$
Lemma 32.16.3. Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & S } \]
where $A$ is a discrete valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is proper.
Proof.
Combine Lemmas 32.16.2 and 32.16.1.
$\square$
Comments (0)