# Friday, September 17

:::{.remark}
Recall from last time: a *locally ringed space* $(X, \OO_X)$ is a ringed space (so $X\in \Top, \OO_X \in \Sh(X, \Ring))$ such that the stalks $\OO_{X, p} \in \Loc\Ring$ for all points $p\in X$.
Some examples:

- $(M, \OO_M)$ with $M$ a manifold and $\OO_M$ any sheaf of reasonable functions,
- $(\spec A, \OO_{\spec A})$

We defined a scheme as a locally ringed space which is locally isomorphic in $\Loc\Ringedspace$ to $(\spec A, \OO_{\spec A})$.
Recall that **locally** meant there exists an open cover $\mcu$ with the property holding for every element in the cover.
Note that most "local" conditions from commutative algebra (that can be checked at all localizations) will correspond to properties that hold on *all* open covers.

> There are generally two ways to define properties of schemes: either it holds for every affine open cover, or there exists an affine open cover.

:::

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

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

:::

:::{.remark}
Note that morphisms in $\Ringedspace$ don't necessarily restrict to morphisms in $\Loc\Ringedspace$, i.e. this is not a full subcategory, since morphisms of rings need not be morphisms of local rings (i.e. those preserving the maximal ideal).
:::

:::{.proof title="of (a) and (b)"}
Part (a):
This follows from the theorem last week that $\OO_{\spec A, p} = A\localize{p}$.

Part(b):
There's only ever one thing to do!
Define the set-theoretic map
\[
f: \spec B \to \spec A \\
p &\mapsto \phi\inv(p)
.\]
Why this is also continuous: we'll show preimages preserve closed sets.
We can write
\[
f\inv(V(I)) 
&= f\inv\qty{\ts{Q \st Q \contains I}} \\
&= \ts{\mfp \st \phi\inv(\mfp) \contains I} \\
&= \ts{\mfp \st \mfp \contains I} \\
&= V(\gens{ \phi(I)} )
,\]
using that $f\inv(Q) \da \ts{\mfp \st \phi\inv(\mfp) = Q}$.

Now localize to get $\phi_p: A\localize{\phi\inv(p)}\to B\localize{p}$.
We now need a sheaf map:
\[
f^\#: \OO_{\spec A} \to f_* \OO_{\spec B}
,\]
i.e. an assignment $f^\#(V): \OO_{\spec A}(V) \to \OO_{\spec B}(f\inv(V))$ for all $V \subseteq \spec A$ open.
We can write 
\[
\OO_{\spec A}(V) \da \ts{ s \in \Top(V, \prod_{p\in V} A\localize{p} ) \st s(p)\in A\localize{p}, s \text{ locally a fraction} }
&\too
\OO_{\spec B}(f\inv V) \da \ts{ t \in \Top(f\inv(V) , \prod_{q\in f\inv(V) } B\localize{q} ) \st t(q)\in B\localize{q}, t \text{ locally a fraction} } \\
(s_p)_{p\in V} &\mapsto (p_q(s_p))_{q\in f\inv(V)}
.\]
But then $q\in f\inv(p)$ for some $p\in V$ iff $p\in \phi\inv(q)$.
So using the map on stalks gives a map on sections, and it preserves the property of locally being a fraction, so this yields a morphism of sheaves of rings, and it remains to check that it's a local morphism.

> Note that you can get this by composing $f\inv(V) \mapsvia{f} V \mapsvia{s} \prod A\localize{p} \mapsvia{\prod \phi_p} \prod B\localize{\phi(p)}$.

We now claim $f^\#$ is a local homeomorphism.
This is clear: we can write $f^\#_p: A\localize{f(p)} \to B\localize{p}$, and $f_p^\# = \phi_p$ by construction, which is a local morphism of rings.
So $f^\#$ is a morphism in $\Loc\Ringedspace$.

:::

:::{.proof title="of (c)"}
Let $(f, f^\#): (\spec B, \OO_{\spec B} ) \to (\spec A, \OO_{\spec A})$ be a morphism in $\LRS$.
Goal: define $\phi \in \CRing(A, B)$, inducing $(f, f^\#)$ in the sense of part (b).
Note that by definition, $f^\#(\spec A): \OO_{\spec A}(\spec A) \to \OO_{\spec B}(\spec B)$.
By the previous theorem, global sections recovers rings on affines, so $f^\#(\spec A): A\to B$.

:::{.claim}
$\phi\inv(p) = f(p)$.
:::

For any $p\in \spec B$, we can localize $f^\#$ to obtain a local ring morphism
\[
f^\#_p: \OO_{\spec A, f(p)} \to \OO_{\spec B, p}
.\]
We also have a commutative diagram

\begin{tikzcd}
	A && B \\
	\\
	{A\localize{f(p)}} && {B\localize{p}}
	\arrow["{f^\#(\spec A) = \phi}", from=1-1, to=1-3]
	\arrow["{f^\#(\spec A)_p = \phi_p}"', from=3-1, to=3-3]
	\arrow["{\text{localize}}"{description}, from=1-1, to=3-1]
	\arrow["{\text{localize}}"{description}, from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJBIl0sWzIsMCwiQiJdLFswLDIsIkFcXGxvY2FsaXple2YocCl9Il0sWzIsMiwiQlxcbG9jYWxpemV7cH0iXSxbMCwxLCJmXlxcIyhcXHNwZWMgQSkgPSBcXHBoaSJdLFsyLDMsImZeXFwjKFxcc3BlYyBBKV9wID0gXFxwaGlfcCIsMl0sWzAsMiwiXFx0ZXh0e2xvY2FsaXplfSIsMV0sWzEsMywiXFx0ZXh0e2xvY2FsaXplfSIsMV1d)

Now we use locality in a key way to conclude $\phi\inv(p) = f(p)$ by commutativity: check that $(f^\#_p)\inv(\mfm_p) = \mfm_{f(p)} \mapsvia{\loc\inv} f(p)$, and $\loc\inv(\mfm_p) = p \mapsvia{\phi} \phi(p)$.
:::