# Wednesday, October 27


:::{.remark}
Recall that if $X\in \Sch\slice S$ with $f:X\to S$, there is an induced diagonal map 
\[
\Delta: X\to X\fiberpower{S}{2} \\
,\]
which is induced by $(\id_X, \id_X): X\to X\cartpower{2}$.
We said $f$ is **separated** if $\Delta$ is a closed immersion, which in particular is a homeomorphism onto a closed subset.
:::


:::{.example title="?"}
An example: any morphism of affine schemes $f\in \Sch(\spec B, \spec A)$.

A non-example: $\AA^1\slice k \Disjoint_{\AA^1\slice k\smz} \AA^1\slice k$, the line with the doubled origin.
We saw $(0, 0')\in \bd\Delta(X) = \bar{\Delta(X)}\sm \Delta(X)$, and in fact $X\fiberpower{\spec k}{2} = \union_{i\leq 4} \AA^2\slice k$.
:::


:::{.proposition title="?"}
$f\in \Sch(X, S)$ is separated $\iff \im(\Delta)$ is closed.
:::


:::{.proof title="?"}
$\implies$:
This is definitional.

$\impliedby$:
First show $\Delta$ is a homeomorphism onto its image.
Use the universal property to get $p_1 \circ \delta = \id_X$ where $p_i: X\fiberpower{S}{2}\to X$ are the projections.
Since both are continuous, $\Delta$ is a homeomorphism onto its image, which is closed.

It then suffices to show $\Delta^\#: \OO_{X\fiberpower{S}{2}} \to \Delta_* \OO_{X}$ is a surjective map in $\Sh(X\fiberpower{S}{2})$ to get a closed immersion.
For any $p\in X\fiberpower{S}{2}$, we need to show that there exists an open $N\ni p$ such that there is a surjection on sections
\[
\Delta^\#(N): \OO_{X\fiberpower{S}{2}}(N) \surjects \Delta_* \OO_X(N) \in \CRing
.\]

Observe that if $p\not\in \im \Delta$ and $N= (\im \Delta)^c$ is open, then $\Delta_* \OO_X(N) = \OO_X(\Delta\inv(N)) = \OO_X(\emptyset) = 0$ and anything surjects onto the zero ring.
So if $p\not\in \supp \Delta_* \OO_X$, this is surjective.
For $p\in \im\Delta$, write $p = \Delta(q)$ using the $\Delta$ is a homeomorphism onto its image and thus $q$ is unique, and choose $U\ni q$ an affine open in $X$.
Then $f(U) \subseteq V$ is contained in an affine open in $S$.
The gluing construction of fiber products yields that $U\fiberpower{V}{2} \subseteq X\fiberpower{S}{2}$ is again an affine open, and $N\ni p$.

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\end{tikzpicture}

Then $\Delta^*(N)$ is surjective since $U, V$ are affine and thus $\ro{f}{U}:U\to V$ is separated.
:::


:::{.example title="?"}
A morphism of schemes can be a homeomorphism onto a closed subset but *not* a closed immersion.
Consider $k\injects k[x]/\gens{x^2}$, inducing $\spec k[x]/\gens{x^2} \to \spec k$.
This is not a surjective map of rings, and on affines this is equivalent to being a closed immersion.

Note that even though $f$ is a closed immersion here, $\Delta$ is not: the fiber product is given by
\[
\spec k[x]/x^2 \fiberprod{\spec k} \spec k[y]/y^2 = \spec k[x, y] / \gens{x^2, y^2}
.\]

Note that taking $k[x]/x^2\to k$ where $x\to 0$ *is* a closed immersion. 
:::

> To google: if $f$ is a homeomorphism onto its image and satisfies some condition for the induced map on Zariski tangent spaces, is it necessarily a closed immersion?

## Valuative Criterion of Separatedness



:::{.remark}
A map from a punctured curve should extend uniquely!

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:::

:::{.theorem title="?"}
Let $f\in \Sch(X, Y)$ with $X$ Noetherian, then $f$ is separated iff for any $R\in \DVR$ and $K \da \ff(R)$, if the following lift exists, it is unique:

\begin{tikzcd}
	K && {\spec K} && X \\
	\\
	R && {\spec R} && Y
	\arrow[from=1-5, to=3-5]
	\arrow[from=3-3, to=3-5]
	\arrow["{!\theta}", dashed, from=3-3, to=1-5]
	\arrow[hook, from=3-1, to=1-1]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-3, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMiwwLCJcXHNwZWMgSyJdLFsyLDIsIlxcc3BlYyBSIl0sWzQsMCwiWCJdLFs0LDIsIlkiXSxbMCwyLCJSIl0sWzAsMCwiSyJdLFsyLDNdLFsxLDNdLFsxLDIsIiFcXHRoZXRhIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzQsNSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMCwxXSxbMCwyXV0=


:::

:::{.example title="?"}
For $\spec R = \ts{\mfm, 0}$, $\spec K = \ts{0}$ is just the generic point:

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![](figures/2021-10-27_12-23-53.png)

And moreover the generic points must be mapped to each other.
:::

:::{.example title="?"}
Consider $R \da \CC[t]\localize{t}$, which is the stalk $\OO_{\AA^1\slice \CC, 0}$, and $K = \CC(t)$.
:::

:::{.slogan}
$\spec R$ for $R\in \DVR$ is like a small piece of a curve.
:::