# Friday, November 05

:::{.remark}
Last time: valuative criterion for properness.
A morphism $f:X\to Y\in \Sch$ is proper $\iff$

- $f$ is separated,
- $f$ is of finite type,
- $f$ is universally closed (closed to closed, and preserved under base change)

If $X$ is Noetherian and $f$ is of finite type, then $f$ is proper $\iff$ for $R \in \DVR, K = \ff(R)$, we have lifts:

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

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHNwZWMgSyJdLFswLDIsIlxcc3BlYyBSIl0sWzIsMiwiWSJdLFsyLDAsIlgiXSxbMywyLCJmIl0sWzAsM10sWzAsMV0sWzEsMl0sWzEsMywiXFxleGlzdHMgIVxcVGhldGEiLDFdXQ==)

We proved that $f$ proper implies $\exists \Theta$.
Erratum: we said $R \subseteq K$ is final with respect to local rings contained in $K$ with fraction field $K$, but rather it's maximal.
As an example, $\ZZ\plocal{2}, \ZZ\plocal{3} \embeds \QQ$ but there's no common ring they map to.
Proof of $\impliedby$: see Hartshorne.
:::

:::{.corollary title="?"}
Some applications/corollaries of the valuative criterion for properness:

| Separated | Proper | 
|-----------|--------|
| Open or closed immersions | Closed immersions[^rarely_proper] |
| Compositions | Compositions |
| Stable under base change | Stable under base change |
| Products | Products | 
| Local on base | Local on base |
| $X \mapsvia{f} Y \mapsvia{g} Z$ with $g \circ f$ separated $\implies f$ separated | $X \mapsvia{f} Y \mapsvia{g} Z$ with $g \circ f$ proper and $g$ separated $\implies f$ separated |

[^rarely_proper]: 
Very rarely proper! Only if inclusion of a connected component.

- "Stable under base change" means that whenever $X \mapsvia{f} Y$ has a property $P(f)$, any fiber product along $Y' \to Y$ yields the same property $P(f')$:

\begin{tikzcd}
	{X'} && X \\
	\\
	{Y'} && Y
	\arrow[from=3-1, to=3-3]
	\arrow["f", color={rgb,255:red,92;green,92;blue,214}, from=1-3, to=3-3]
	\arrow["{f'}", color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYJyJdLFswLDIsIlknIl0sWzIsMiwiWSJdLFsyLDAsIlgiXSxbMSwyXSxbMywyLCJmIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX0sWzI0MCw2MCw2MCwxXV0sWzAsMSwiZiciLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdfSxbMjQwLDYwLDYwLDFdXSxbMCwzXSxbMCwyLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)


- A *product* of morphisms in $\Sch\slice S$ is the product in $\Sch\slice S$, or equivalently the fiber product over $S$.
  So given $f:X\to Y$ and $f':X'\to Y'$, the product is $(f, f'): X\fiberprod{S} X' \to Y\fiberprod{S} Y'$.
  So here "Products" means that if $P(f), P(f')$ holds, then $P(f, f')$ holds.

- $P$ is *local on the base* if whenever $P$ holds for $X \mapsvia{f} Y$ then for all open $U \subseteq Y$, the restriction $\ro{f}{f\inv(U)}: f\inv(U) \to U$ also satisfies $P$

:::

:::{.proof title="stability under base change"}
Diagram chases involving the valuative criteria and universal properties of the fiber product.
For example, we'll do stability under base change: let $X \mapsvia{f} Y$ be separated and $Y'\to Y$, we'll show $X' \mapsvia{f'} Y'$ is separated where $X' \da X\fiberprod{Y} Y'$.
We need to show an extension $\Theta$ of the following form is unique if it exists:

\begin{tikzcd}
	{\spec K} && {X'} && X \\
	\\
	{\spec R} && {Y'} && Y
	\arrow["{f'}", from=1-3, to=3-3]
	\arrow["\alpha"', from=3-1, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
	\arrow["{\Theta_1}", shift left=1, dashed, from=3-1, to=1-3]
	\arrow[from=3-3, to=3-5]
	\arrow["f"{description}, from=1-5, to=3-5]
	\arrow["\beta"{description}, from=1-3, to=1-5]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=3-5]
	\arrow["{\Theta_2}"', shift right=1, dashed, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJcXHNwZWMgSyJdLFswLDIsIlxcc3BlYyBSIl0sWzIsMCwiWCciXSxbMiwyLCJZJyJdLFs0LDAsIlgiXSxbNCwyLCJZIl0sWzIsMywiZiciXSxbMSwzLCJcXGFscGhhIiwyXSxbMCwyXSxbMCwxXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbMSwyLCJcXFRoZXRhXzEiLDAseyJvZmZzZXQiOi0xLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMyw1XSxbNCw1LCJmIiwxXSxbMiw0LCJcXGJldGEiLDFdLFsyLDUsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFsxLDIsIlxcVGhldGFfMiIsMix7Im9mZnNldCI6MSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

Note that $\beta \circ \theta_1 = \beta \circ \Theta_1$, since $f$ is separated, using the valuative criterion for separatedness.
Since $X'$ is a fiber product, by the universal property there exists a unique product morphism $(\beta \circ \Theta_1, \alpha) = (\beta \circ \Theta_2, \alpha)$.
So $\Theta_1 = \Theta_2$ and $f'$ is separated by the valuative criterion of separatedness.
:::

:::{.proof title="of products"}
We want to show that if $f:X\to Y, f':X'\to Y'$ are proper then $(f, f'): X\fiberprod{S} X' \to Y\fiberprod{S} Y'$ is proper.
We can produce a diagram:

\begin{tikzcd}
	\textcolor{rgb,255:red,92;green,92;blue,214}{\spec K} && {X\fiberprod{S} X'} && {X'} \\
	& \textcolor{rgb,255:red,92;green,92;blue,214}{\spec R} && {Y\fiberprod{S} Y'} && {Y'} \\
	&& X && S \\
	&&& Y && S
	\arrow[from=2-4, to=4-4]
	\arrow[from=4-4, to=4-6]
	\arrow[from=2-6, to=4-6]
	\arrow[from=2-4, to=2-6]
	\arrow[from=1-5, to=3-5]
	\arrow[from=3-5, to=4-6]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=1-3, to=2-4]
	\arrow[from=3-3, to=4-4]
	\arrow[from=1-5, to=2-6]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=2-2, to=2-4]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=2-2]
	\arrow["{\text{WTS: } \exists ! \Theta}"{description}, dashed, from=2-2, to=1-3]
	\arrow[curve={height=30pt}, dotted, from=1-1, to=3-3]
	\arrow[curve={height=30pt}, dotted, from=2-2, to=4-4]
	\arrow["{\exists !}"{description}, color={rgb,255:red,214;green,92;blue,92}, squiggly, from=2-2, to=3-3]
	\arrow["{\exists !}"{description, pos=0.7}, color={rgb,255:red,214;green,92;blue,92}, squiggly, from=2-2, to=1-5]
\end{tikzcd}

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

Here we get existence of unique maps $\spec R\to X, \spec R\to X'$, which thus yields a unique map $\spec R \to X\fiberprod{S} X'$.
:::


:::{.proof title="locality on base"}
Suppose $X \mapsvia{f} Y$ is proper, we'll show that $\ro{f}{f\inv(U)}:f\inv(U) \to U$ is proper for $U \subseteq Y$.
We use that $f\inv(U)$ is a fiber product and apply the universal property:

\begin{tikzcd}
	{\spec K} && {f\inv(U)} && X \\
	\\
	{\spec R} && U && Y
	\arrow["f", from=1-5, to=3-5]
	\arrow[from=1-1, to=3-1]
	\arrow[from=3-3, to=3-5]
	\arrow[from=1-3, to=1-5]
	\arrow["{\ro{f}{f\inv(U)}}"{pos=0.8}, from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=3-5]
	\arrow["{\exists !}"{description, pos=0.4}, curve={height=-6pt}, from=3-1, to=1-5]
	\arrow["{\therefore \exists!}"{description}, dashed, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbNCwyLCJZIl0sWzQsMCwiWCJdLFswLDAsIlxcc3BlYyBLIl0sWzAsMiwiXFxzcGVjIFIiXSxbMiwyLCJVIl0sWzIsMCwiZlxcaW52KFUpIl0sWzEsMCwiZiJdLFsyLDNdLFs0LDBdLFs1LDFdLFs1LDQsIlxccm97Zn17ZlxcaW52KFUpfSIsMCx7ImxhYmVsX3Bvc2l0aW9uIjo4MH1dLFszLDRdLFsyLDVdLFs1LDAsIiIsMCx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFszLDEsIlxcZXhpc3RzICEiLDEseyJsYWJlbF9wb3NpdGlvbiI6NDAsImN1cnZlIjotMX1dLFszLDUsIlxcdGhlcmVmb3JlIFxcZXhpc3RzISIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)


For the converse, it suffices to check properness on an open cover $\mcu \covers Y$.
Why?
It follows if for any diagram of the following form, there exists an open $U_i \subseteq Y$ such that $\im \spec R \subseteq U_i$:

\begin{tikzcd}
	{\spec K} && X \\
	\\
	{\spec R} && Y
	\arrow[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]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHNwZWMgSyJdLFswLDIsIlxcc3BlYyBSIl0sWzIsMiwiWSJdLFsyLDAsIlgiXSxbMywyXSxbMSwyXSxbMCwzXSxbMCwxXV0=)

Note that $\spec R$ has two points, so this is not completely trivial.
Consider the closed point $\mfm \in \spec R$ and let $p_0 = \im(\mfm)$.
:::