# Wednesday, November 03

:::{.remark}
Let $R\in \DVR, k = \ff(R)$, then a $k\dash$point of $X$ is a morphism $\spec k \to X$ and is given by the data of an inclusion $p_1\in \realize{X}$ and an inclusion $\kappa(p_1) \injects k$.
:::

:::{.example title="?"}
Why these are called $k\dash$points?
Given $\spec \QQ\to \spec S\da \spec \ZZ[x,y,z]/ \gens{x^5 +y^5-z^5}$, then there is a ring map $S\to \QQ$ where $x,y,z \mapsto x_0,y_0, z_0$ satisfying $x_0^2 +y_0^2=z_0^2$.
So these are rational solutions to the defining equations.
:::

:::{.remark}
Lifting a $k\dash$point to an $R\dash$point $\spec R \to X$ requires $p_0 \in \cl(\ts{p_1})$ and a domination $\OO_{\cl(\ts{p_1}), p_0} \to R$ inducing the $k\dash$point in the sense that the generic point of $\spec R$ maps to the generic point of $\spec \OO_{\cl\ts{p_1}, p_0}$ corresponding to an inclusion of fields.
So we get a morphism of local rings.

:::

:::{.remark}
We saw that $f:X\to Y$ is separated iff $\Delta(X) \injects X\fiberpower{Y}{2}$ is closed iff any $k\dash$point of $X$ has at most one specialization over a given $R\dash$point of $Y$.
Idea: rules out two lifts.

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\end{tikzpicture}

Note that this needs $X$ to be Noetherian.

> Is separatedness local? Perhaps $X$ only locally Noetherian would suffice.

:::

:::{.remark}
Review the difference between "of finite type" and "*locally* of finite type".
:::

:::{.definition title="Closed and universally closed"}
A morphism $f:X\to Y$ is **closed** if the underlying map $\realize{f}: \realize{X} \to\realize{Y} \in \Top$ is a closed continuous map (so images of closed sets are closed), and $f$ is **universally closed** if for all $Y' \to Y$ the change $f': X\fiberprod{Y} Y'\to Y'$ is closed.
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:::{.example title="?"}
Identity maps $\id_X:X\to X$ are closed, using that $Y\fiberprod{X} X \cong Y$ and pulling back $\id_X$ yields $\id_Y$.
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:::{.example title="A non-example"}
Consider $\AA^1\slice k\to \spec k$, which is closed since $\spec k = \pt$ and has the discrete topology.
This is not universally closed, since we have

\begin{tikzcd}
	{\AA^2\slice k} && {\AA^1\slice k} \\
	\\
	{\AA^1 \slice k} && {\spec k}
	\arrow["f", from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow["{f'}"', from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEFBXjJcXHNsaWNlIGsiXSxbMCwyLCJcXEFBXjEgXFxzbGljZSBrIl0sWzIsMiwiXFxzcGVjIGsiXSxbMiwwLCJcXEFBXjFcXHNsaWNlIGsiXSxbMywyLCJmIl0sWzEsMl0sWzAsMSwiZiciLDJdLFswLDNdXQ==)

Consider $Z\da V(xy-1) \subseteq \AA^2\slice k$, then $f'(Z) = \AA^2\slice k\smz$ is projection onto the $x\dash$axis and is not closed in $\AA^2\slice k$.
What this is a projection onto the $x\dash$axis: this comes from the map $f: k[x] \injects k[x, y] \cong k[x] \tensor_k k[y]$ where $f\inv(\gens{x-x_0, y-y_0}) = \gens{x-x_0}$, so geometrically this yields the $(x_0, y_0) \to x_0$.
:::

:::{.example title="?"}
Consider $\PP^1\slice k \da \proj k[x,y]$ and consider $\PP^1\slice k \to \spec k$.
Here $\PP^1\slice k$ is supposed to be "compact" in the sense that graphs of all functions are closed.
:::

:::{.exercise title="?"}
Are compact spaces universally closed in $\Top$?
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:::{.definition title="Proper"}
A morphism $f:X\to Y$ is **proper** if

1. $f$ is of finite type,
2. $f$ is separated,
3. $f$ is *universally closed*

:::

:::{.remark}
This ranges over all possible base changes, so it's quite hard to actually check!
The following result gives an easier way:
:::

:::{.theorem title="Valuative criterion of properness"}
Let $f:X\to Y$ be a finite type morphism with $X$ Noetherian.
Then $f$ is proper $\iff$ there exists unique lifts $\Theta$ of the following form:

\begin{tikzcd}
	{\spec k} && X \\
	\\
	{\spec R} && Y
	\arrow["f", from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["{!\Theta}"{description}, dashed, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHNwZWMgayJdLFswLDIsIlxcc3BlYyBSIl0sWzIsMiwiWSJdLFsyLDAsIlgiXSxbMywyLCJmIl0sWzEsMl0sWzAsM10sWzAsMV0sWzEsMywiIVxcVGhldGEiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)


:::

:::{.remark}
Most spaces in practice are separated and of finite type, unless you're working with moduli of K3 surfaces!
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:::{.proof title="$\implies$"}
Suppose $f$ is proper, then $f$ is separated and we have uniqueness for any lifts by the valuative criterion for separatedness.
This uses that $X$ is Noetherian.
It then suffices to show existence of $\Theta$, using that $f$ is universally closed.
Consider the base change $X_R \da X\fiberprod{Y} \spec R$, then using commutativity we get a morphism $s: \spec k \to X_R$.
Let $p_1 = s(\gens{0}) \subseteq X_R$, we'll then try to specialize $p_1$.
Let $Z \da \cl\ts{p_1} \subseteq X_R$, then since $f$ is proper and $Z$ is closed in $X_R$, $f_R(Z)$ is closed:

\begin{tikzcd}
	{X_R} && X \\
	\\
	{\spec R} && Y
	\arrow["f", from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow["{f_R}"', from=1-1, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYX1IiXSxbMCwyLCJcXHNwZWMgUiJdLFsyLDAsIlgiXSxbMiwyLCJZIl0sWzIsMywiZiJdLFsxLDNdLFswLDJdLFswLDEsImZfUiIsMl1d)

We can compose $\spec k \to Z \mapsvia{f_R} \spec R$ to get $\tilde f$, which is an inclusion of the generic point:

\begin{tikzcd}
	{\spec k} \\
	& {X_R} && X \\
	\\
	& {\spec R} && Y
	\arrow["f", from=2-4, to=4-4]
	\arrow[from=4-2, to=4-4]
	\arrow[from=1-1, to=2-4]
	\arrow["{\tilde f}"', from=1-1, to=4-2]
	\arrow["{!\Theta}"{description}, dashed, from=4-2, to=2-4]
	\arrow[from=1-1, to=2-2]
	\arrow[from=2-2, to=2-4]
	\arrow["{f_R}"{description}, from=2-2, to=4-2]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXHNwZWMgayJdLFsxLDMsIlxcc3BlYyBSIl0sWzMsMywiWSJdLFszLDEsIlgiXSxbMSwxLCJYX1IiXSxbMywyLCJmIl0sWzEsMl0sWzAsM10sWzAsMSwiXFx0aWxkZSBmIiwyXSxbMSwzLCIhXFxUaGV0YSIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDRdLFs0LDNdLFs0LDEsImZfUiIsMV1d)


Then $f_R(Z) = \spec R$ and so there exists $p_0 \in Z$ with $f_R(p_0) = \mfm$, the closed point in $\spec R$.
So we get
\[
g: Z &\to \spec R \\
\cl\ts{p_1}\ni p_0 &\mapsto \mfm \\
p_1 &\mapsto \gens{0}
.\]

Taking stalks yields a local ring morphism $g^\sharp_{p_0}: R\to \OO_{Z, p_0}$, and this completes to a diagram:

\begin{tikzcd}
	R && {\OO_{Z, p_0}} \\
	\\
	k && {\ff \OO_{Z, p_0} = \kappa(p_1)}
	\arrow[from=1-1, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJSIl0sWzIsMCwiXFxPT197WiwgcF8wfSJdLFswLDIsImsiXSxbMiwyLCJcXGZmIFxcT09fe1osIHBfMH0gPSBcXGthcHBhKHBfMSkiXSxbMCwyXSxbMiwzXSxbMSwzXSxbMCwxXV0=)

But $R$ is final with respect to domination for local rings $R'$ in $k$ with $\ff R' = k$, and if final objects admit morphisms to other objects, those objects must also be final, so $R = \OO_{Z, p_0}$.
This yields a domination $\OO_{Z, p_0} \to R$, which corresponds to a lift $\spec R \to X$.
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