# Monday, November 01

:::{.remark}
Recall that $f:X\to Y$ is **separated** if $\Delta X\to X\fiberpower{Y}{2}$ is a closed immersion, or equivalently $\Delta(X) \subseteq X\fiberpower{Y}{2}$ is closed.
We discussed the valuative criterion of separatedness, which is slightly more useful when proving things, but only holds for Noetherian (quasicompact, admits a a finite cover of affines) schemes: $X$ is separated iff any diagram admitting a lift $\Theta$ of the following form admits a *unique* lift:

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

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJcXHNwZWMgUiwgUlxcaW4gXFxEVlIiXSxbMCwwLCJcXHNwZWMgSyJdLFsyLDAsIlgiXSxbMiwyLCJZIl0sWzIsMywiZiJdLFswLDNdLFsxLDBdLFsxLDJdLFswLDIsIiFcXFRoZXRhIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

Here $R\in \DVR$ and $K\in \ff(R)$.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-11-01_11-40.pdf_tex} };
\end{tikzpicture}



:::

:::{.example title="?"}
Consider mapping to the line with two origins:

\begin{tikzpicture}
\fontsize{42pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-11-01_11-42.pdf_tex} };
\end{tikzpicture}

Then given $f:X\to Y$ and $\spec R\to Y$ there is an induced map $k[t]\localize{t} \from k[t]$.
But note that there are two distinct extensions $\spec R\to X$, say $\Theta_1, \Theta_2$, and there is an extension of the following form:

\begin{tikzcd}
	{\spec k(t)} && {\spec k[t]} \\
	\\
	{\spec k[t]\localize{t}}
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow["{\exists \Theta_1}", shift left=1, from=3-1, to=1-3]
	\arrow["{\exists \Theta_2}"', shift right=1, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHNwZWMgayh0KSJdLFsyLDAsIlxcc3BlYyBrW3RdIl0sWzAsMiwiXFxzcGVjIGtbdF1cXGxvY2FsaXple3R9Il0sWzAsMl0sWzAsMV0sWzIsMSwiXFxleGlzdHMgXFxUaGV0YV8xIiwwLHsib2Zmc2V0IjotMX1dLFsyLDEsIlxcZXhpc3RzIFxcVGhldGFfMiIsMix7Im9mZnNldCI6MX1dXQ==)

:::

:::{.remark}
Taking fraction fields corresponds to throwing out everything but the generic point.
:::

:::{.proof title="of valuative criterion"}
Omitted, see Hartshorne.
We'll discuss one key idea: **specialization**.
:::

## Specialization

:::{.remark}
Consider the data of a morphism from $\spec K$:

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

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHNwZWMgSyJdLFsyLDAsIlgiXSxbMiwyLCJZIl0sWzAsMiwiXFxzcGVjIFIiXSxbMywyLCJcXHBzaSJdLFsxLDIsImYiLDJdLFswLDNdLFswLDEsIlxccGhpIl1d)

This is the data of a point $p\in X$, so $\phi( \gens{0}) = p\in \realize{X}$, and (it suffices to have) a pushforward $\phi_p^\sharp$ which is a morphism of local rings inducing a diagram:

\begin{tikzcd}
	{\OO_{X, p}} && K \\
	\\
	{\kappa(p)}
	\arrow["{\mod \mfm_p}"', from=1-1, to=3-1]
	\arrow["{\phi^\sharp_p}", from=1-1, to=1-3]
	\arrow[from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXE9PX3tYLCBwfSJdLFsyLDAsIksiXSxbMCwyLCJcXGthcHBhKHApIl0sWzAsMiwiXFxtb2QgXFxtZm1fcCIsMl0sWzAsMSwiXFxwaGleXFxzaGFycF9wIl0sWzIsMV1d)

Here $\phi^\sharp_p(\mfm_p) = \gens{0}$ and $\kappa(p)$ is the residue field at $p$.
:::

:::{.remark}
What data is needed to specify $\psi: \spec R \to Y$?
We need two points $p_0, p_1\in \realize{X}$ with $p_1 = \psi(\gens{0}) \in \psi(\spec K)$ and $p_0 = \psi(\mfm)$.
Since $\psi \inv \qty{\bar{\ts{p_1}}}$ is closed, we also need $p_0 \in \bar{\ts{p_1}}$, so $\kappa(p_1) \subseteq K$.

Consider $Z \da \bar{\ts{p_1}} \ni p_0$ with its structure as a reduced closed subscheme of $X$.
This yields a map $\spec R\to Z$, and we need an injective (dominant) ring map $\OO_{Z, p_0}\to R$.
Why does this produce a map $\spec R\to Y$?
We have a closed immersion $\spec R\embeds \spec \OO_{Z, p_0} \to Z \embeds X$
:::

:::{.definition title="Specialization"}
A point $p_0$ is a **specialization** of $p_1$ relative to $R\in \DVR$ and $K = \ff(R)$ if $p_1$ is a $K\dash$point, so $\kappa(p_1) \subseteq K$ such that $p_0 \in \bar{p_1}$ and $\OO_{\bar p_1, p} \surjects R$
:::

:::{.example title="?"}
Take $R\da k\formalpowerseries{t}$ and $\ff(R) = k((t))$ for $k\in \Field$.
Consider $\spec k\to \AA^1\slice k$ corresponding to $k[t] \injects k((t))$.
Setting $p_1 = \im \gens{0} = \gens{0} \in k[t]$ to be the generic point of $\AA^1\slice k$, we have $\bar{p_1} = \AA^1\slice k$.
Set $p_0 = \gens{t}$, and note that $p_0 \in \bar{p_1}$, we then want a ring map $\OO_{\bar{p_1}, p_0} \to R = k[[t]]$.
Note that $\OO_{\bar{p_1}, p_0} = k[t] \localize{t}$, and there is a ring map $\ts{f(t)/g(t) \st g(0) \neq 0} \to k[[t]]$.
This is injective, yielding a domination of rings.
:::