# Monday, September 20

:::{.remark}
Last time: we proved the following:
:::

:::{.proposition title="?"}
\envlist

a. If $A\in \Ring$, then $(\spec A, \OO_{\spec A})\in \LRS$.
b. If $f\in \CRing(A, B)$ is a ring morphism, it induces a morphism $(f, f^\#) \in \LRS(\spec B, \spec A)$.
c. Moreover, every $(f, f^\#) \in \LRS(\spec B, \spec A)$ is induced by some $f\in \Top(A, B)$.
:::

:::{.remark}
Recall that a *scheme* $(X, \OO_X)$ is a LRS which is locally isomorphic to some affine scheme $(\spec A, \OO_{\spec A})$:

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\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-09-20_11-35.pdf_tex} };
\end{tikzpicture}

:::

:::{.definition title="Complete Ring"}
A ring $R$ is **complete** with respect to $I\normal R$ if $R = \invlim_{\leftarrow} R/I^k$.
Elements can be written as sequence $(a_k)_{k\geq 0}$ such that $a_k \equiv a_{k-1} \mod I^{k-1}$.
:::

:::{.example title="?"}
A non-example: let $R = \CC\adjoint{t}$ and $I = \gens{t}$, then set

- $a_0 = 1$,
- $a_1 = 1 + t$,
- $a_2 = 1 + t + t^2$,
- $a_3 = 1 + t + t^2 + t^3$

This is an element in $\colim_n R/I^n$, but is not realized by any polynomial, since any polynomial is annihilated by quotienting by a large enough power of $t$.
Note that $\CC\powerseries{t} = \CC\adjoin[t] \complete{\gens{t}}$.
:::

:::{.example title="?"}
Part (c) of the proposition would be false if we considered all ringed spaces.
Let $R \in \DVR$, so $R$ is local with a principal maximal idea, or equivalently equipped with a valuation $v: R\smz \to \ZZ_{\geq 0}$ satisfying

- $v(x+y) = v(x) + v(y)$
- $v(x+y) \geq \min{v(x), v(y)}$

Then $v\inv(\ZZ_{\geq 0})$ is the maximal ideal.
Here we'll additionally require that $R$ is **complete** with respect to its maximal ideal $\mfm$.

An example is $R = \CC\powerseries{t}$, with $v$ the $t\dash$adic valuation.
Another is $R = \ZZpadic = \ZZ\complete{p}$, the completion of $\ZZ$ at the prime $p$, given by elements $a_n\cdots a_0$ with $a_i \in \ts{0, \cdots, p-1}$.
This has maximal ideal $\mfm = p\ZZpadic$.
:::

:::{.example title="A morphisms of ringed spaces that isn't a morphism of locally ringed spaces"}
Let $K = \ff(R)$ be the fraction field of $R$, then 
\[
\ff(\CC\powerseries{t}) = \CC\functionfield{t} = \ts{\sum_{k\geq -N} a_k t^k \st N\in \ZZ_{\geq 0} }
.\]
Also $\ff(\ZZpadic) = \QQpadic$, and these are both examples of complete DVRs.

Consider $(\spec R, \OO_{\spec R})$ and $(\spec K, \OO_{\spec K})$.
Then $X_1\da \spec R = \ts{ \gens 0, \mfm}$, and the closed sets are $\emptyset, X_1, V(\mfm) = \ts{\mfm}$.
For $X_2\da \spec K = \ts{\gens 0}$, there is one points since it's a field.
Use that $\iota: R \injects K$, so define a morphism that does not come from a ring morphism $R\to K$:
\[
(f, f^\#): (\spec K, \OO_{\spec K}) &\to (\spec R, \OO_{\spec R}) \\
\\
\\
f: \spec K &\to \spec R\\
0 &\mapsto \mfm 
\\
\\
f^\#: \OO_{\spec R} &\to f_* \OO_{\spec K} \\
\emptyset &\mapsto 0 \\
\spec R & \mapsto R \\
\ts{\gens 0} &\mapsto K
.\]
using that $f\inv(\gens 0) = \gens m$ and we can realize the last assignment as a distinguished open mapping to its stalk/localization.
Then check
\[
f_* \OO_{\spec K}(\emptyset) &= 0
f_* \OO_{\spec K}(\spec R) &= K \\
f_* \OO_{\spec K}(\ts{\gens 0}) &= \OO_{\spec K}(f\inv (\gens 0)) = \OO_{\spec K}(\emptyset) = 0 \\
.\]
This would induces a commutative diagram, showing this is a morphism of ringed spaces:

\begin{tikzcd}
	R && K \\
	\\
	K && 0 \\
	\\
	0 && 0
	\arrow["{f^\#(X)}", from=1-1, to=1-3]
	\arrow["{f^\#(\ts{\gens 0})}", from=3-1, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-3, to=5-3]
	\arrow[from=3-1, to=5-1]
	\arrow["{f^\#(\emptyset)}", from=5-1, to=5-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJSIl0sWzIsMCwiSyJdLFswLDIsIksiXSxbMiwyLCIwIl0sWzAsNCwiMCJdLFsyLDQsIjAiXSxbMCwxLCJmXlxcIyhYKSJdLFsyLDMsImZeXFwjKFxcdHN7XFxnZW5zIDB9KSJdLFswLDJdLFsxLDNdLFszLDVdLFsyLDRdLFs0LDUsImZeXFwjKFxcZW1wdHlzZXQpIl1d)


:::{.question}
Is this a morphism of *locally* ringed spaces?
:::


The answer is **no**, since the induced morphism on stalks won't be morphisms of *local* rings.
We can check
\[
f^\#_{\gens 0}: \OO_{\spec R, \mfm} &\to \OO_{\spec K, \gens 0} \\
f^\#_{\gens 0}: R = R\localize{\mfm} &\to K\localize{\gens 0} = K
,\]
and $(f^\#_{\gens 0})\inv(\gens 0) = \gens{0} \neq \mfm$, which is not the maximal ideal of $R$.

On the other hand, using part (b) of the proposition, any $\phi\in \Ring(R, K)$ induces a morphism $\tilde \phi: \Loc\RingedSpace(\spec K, \spec R)$.
So $(f, f^\#)$ is not induced by any such ring map $\phi$.
:::

:::{.remark}
So the functor
\[
\Ring &\to \RingedSpace \\
A &\mapsto (\spec A, \OO_{\spec A})
\]
is not fully faithful, but restricting the essential image to $\Loc\RingedSpace$.
:::

:::{.remark}
Consider the heuristic $\spec \CC\powerseries{t} \sim \DD \subseteq \CC$ and $\spec \CC\functionfield{t} \sim \DD\smz$.
:::