# Friday, September 10

## Sections and Stalks of $\OO_{\Spec A}$ as Localizations

:::{.remark}
Last time: any affine scheme is quasicompact, so for each ring $A$ and an open cover $\mcu \covers D(f)$ then there is a finite subcover $\ts{D(h_i)}_{i\leq N} \covers D(f)$.
We were looking at proposition: for the ringed space $(\Spec A, \OO_A)$,

a. $\OO_\mfp \cong A_{\mfp}$ for all $\mfp \in \Spec A$,
b. $\OO(D(f))\cong A_{f}$ for all $f\in A$,
c. $\Globsec{\Spec A; \OO_A} \cong A$.

Note that $\OO_A$ is uniquely characterized by these properties.
:::

:::{.remark}
We can write $D(1) = \Spec A$ and write $\ts{ D(h_i) }_{i\leq N} \covers \spec A$ to obtain $1^n = \sum b_i h_i$.
This is analogous to a partition of unity, where each $b_i h_i$ vanishes on $D(h_i)^c = V(h_i)$
:::

:::{.proof title="of proposition 2.2b, continued"}
Let $\psi:A_{f} \injects \OO(D(f))$ where we just take restrictions of functions.
We know this is injective, and we want to show surjectivity.

**Step 1**:
Let $s\in \OO(D(f))$.
For each open $D(h_i)$, write $\ro{s}{D(h_i)} = a_i /h_i$ for some $a_i \in A$.

**Step 2**:
By quasicompactness, write $f^n = \sum_{1 \leq i\leq m} b_i h_i$.

**Step 3**:
Glue the $a_i/h_i$ into an element $a/f$ of $A_{f}$.

*Part 1*:
For any $1\leq i\neq j\leq m$, $D(h_i h_j) = D(h_i) \intersect D(h_j)$.
Note that $a_i/h_i = a_j/h_j$ in $\OO(D(h_i h_j))$, and these are elements of $A_{h_i h_j}$ since $a_i /h_i = a_ih_j/h_i h_j$.
Using injectivity of $\psi$ for $h_i h_j$, we get an equality $a_i/h_i = a_j/h_j$ in $A_{h_i h_j}$.
Then for $\ell$ large enough, $(h_i h_j)^\ell( a_i h_j - a_j h_i) = 0 \in A$.
Regrouping yields $h_j^{n+1}(h_i^n a_i) - h_i^{n+1}(h_j a_j) = 0$, so
\[
{a_i h_i^n \over h_i ^{n+1}} = {a_j h_j^r \over h_j^{n+1}} \da {\tilde a_i \over \tilde h_i} = {\tilde a_j \over \tilde h_j}
,\]
noting that $D(\tilde h_i) = D(\tilde h_i)$ since $\tilde h_i$ is a power of $h_i$.

Now use POU gluing to write $f^n = \sum_i \tilde b_i \tilde h_i$ and $a \da \sum \tilde a_i \tilde h_i\in A$ be a global function on $D(f)$.
Then (claim) $a_j/f^n = \tilde a_j/\tilde h_j$ on $D(\tilde h_j)$.
We can rewrite
\[
\tilde h_j a = \sum_i \tilde b_i \tilde a_i \tilde h_j = \sum_i \tilde b_i \tilde a_j \tilde h_i
.\]
But then $a/f^n = \ro{s}{D(\tilde h_i)}$, so $s= a/f^n$.
:::

:::{.example title="Projective space as a scheme covered by affine charts"}
Consider $\PP^1\slice k$ as a scheme -- we know the space, and the claim is that we can glue sheaves of affines to obtain a structure sheaf for it.
Cover $\PP^1$ by $U_0 = \PP^1\sm\ts{\infty} \cong \AA^1$ and $U_1 = \PP^1\sm\ts{0} \cong \AA^1$.
The gluing data is the following:

\begin{tikzcd}
	&& {\PP^1\slice k} \\
	\\
	{\AA^1} & {U_0} && {U_1\cong \AA^1} & {\AA^1} \\
	\\
	& {\AA^1\smz} & {U_0 \intersect U_1 \cong D(t) \subseteq \AA^1} & {\AA^1\smz}
	\arrow["{\phi_0}", hook, from=3-2, to=1-3]
	\arrow["{\phi_1}"', hook', from=3-4, to=1-3]
	\arrow[hook', from=5-3, to=3-2]
	\arrow[hook, from=5-3, to=3-4]
	\arrow[hook, two heads, from=5-3, to=5-4]
	\arrow[hook, two heads, from=5-3, to=5-2]
	\arrow[hook, from=5-2, to=3-1]
	\arrow[hook', two heads, from=3-1, to=3-2]
	\arrow[hook, from=5-4, to=3-5]
	\arrow[hook', two heads, from=3-5, to=3-4]
\end{tikzcd}

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

Here the transition maps are
\[
\phi_1 \circ \phi_0\inv: \phi_0( U_0 \intersect U_1) &\to \phi_1(U_0 \intersect U_1) \\
t &\mapsto t\inv
.\]
What is the map on sheaves?
We need a map $\ro{\OO}{U_0\smz} \iso \ro{\OO}{U_1 \smts{\infty}}$.
:::

:::{.definition title="Ringed Spaces"}
A **ringed space** $(X, \OO_X) \in \Top \times \Sh(X, \CRing)$ as a topological space with a sheaf of rings.
A morphism is a pair $(f, f^\#) \in C^0(X, Y) \times \in \Mor_\Sh(\OO_Y, f_* \OO_X)$.
:::

:::{.example title="?"}
The scheme $(\Spec A, \OO_{\Spec A})$ is a ringed space.
:::

:::{.example title="?"}
Consider $\RR$ in the Euclidean topology, then $(\RR, C^0(\wait, \RR))$ with the sheaf of continuous functions is a ringed space.
Then consider the morphism given by projection onto the first coordinate of $\RR^2$:
\[
(\pi, \pi^\#): (\RR^2, C^0(\wait, \RR)) &\to (\RR, C^{\infty }(\wait, \RR)) \\
(x, y) &\mapsto x
.\]
For $\pi^\#$, we can consider $\pi_* C^0(\wait, \RR)(U) \da C^0(\pi\inv(U)) = C^0(U\times \RR)$, so
\[
\pi^\#: C^\infty(U, \RR) &\to C^0(U\times \RR) \\
f &\mapsto f\circ \pi
,\]
which is a well-defined map of rings since smooth functions are continuous.
:::

:::{.warnings}
Not every scheme is built out of affine opens!
:::