# The structure sheaf $\OO$ (Wednesday, September 01)

## Ringed spaces are finer than topological spaces

:::{.example title="Polynomial rings"}
Let $k = \bar k$ be algebraically closed, then 
\[
\spec k[x] = \ts{ \gens{x-a} \st a\in k}\union \gens{0}
.\]
Similarly, 
\[
\spec k[x, y] = \ts{ \gens{x-a, y-b} \st a,b\in k} \union \ts{\gens{f} \st f\in k[x,y] \text{ irreducible}} \union \gens{0}
.\]
Note that both have non-closed, generic points $\eta = \gens{0}$.
:::

:::{.example title="Distinct ringed spaces which are homeomorphic"}
Consider $X \da \spec \ZZpadic$ and $Y\da \spec \CC\powerseries{t}$, then 
\[
X = \ts{\gens p, \gens 0},\qquad
Y = \ts{ \gens t, \gens 0 }
.\]
Both are two point spaces, with exactly one open/generic point $\gens 0$ and one closed point ($\gens p$ and $\gens t$ respectively).
These spaces are isomorphic as topological spaces (i.e. there is a homeomorphism between them), but later we'll see that they can be distinguished as ringed spaces.
:::

## Properties of $V$

:::{.remark}
Recall that for $A\in\CRing$, we defined $\spec A$ to have closed sets of the form 
\[
V(I) = \ts{p \in \spec(A) \st p\contains I} \qquad \forall I\normal A
.\]

:::

:::{.lemma title="$V$ sends finite products to unions"}
\[
V(IJ) = V(I) \union V(J)
,\]
:::

:::{.corollary title="Primeness of ideals in terms of ideals (and not just elements)"}
If a prime ideal $p$ contains $IJ$ then $p\contains I$ or $p\contains J$.
:::

:::{.proof title="of lemma"}
$\impliedby$: If $I \subseteq P$ or $J \subseteq P$, then $IJ \subseteq I$ and $IJ \subseteq J$, so $IJ \subset p$.

$\implies$:
Suppose $IJ \subset p$ but $J \not\subset p$, so pick $j\in J \sm p$.
Then for all $i\in I$, we have $ij\in IJ \subseteq p$, forcing $i\in p$.
:::

:::{.lemma title="$V$ sends arbitrary sums to intersections"}
An arbitrary intersection satisfies
\[
V\qty{ \sum_{i\in J} I_i} = 
\Intersect_{i\in J} V(I_i) 
.\]
:::

:::{.proof title="of lemma"}
$\implies$:
For $p\in \spec(A)$, we want to show that $p \contains \sum I_i$ iff $p \contains I_i$ for all $i$, so $I_i \subseteq \sum I_i \subset P$.

$\impliedby$:
Ideals are additive groups, regardless of whether or not they're prime!
:::

:::{.proof title="of proposition"}
\envlist

- $\emptyset$ is closed, since $\emptyset = V(A)$
- $X$ is closed, since $X = V(0)$ and $O$ is contained in every prime ideal.
- Closure under finite unions: by induction, it's enough to show that $V(I) \union V(J)$ is closed.
  This follows from the 1st lemma above.
 - Closure under arbitrary unions: this follows from the 2nd lemma.
:::

:::{.proposition title="$V(I) = V(\sqrt I)$"}
\[
V(I) = V(\sqrt I)
.\]
:::

:::{.proof title="?"}
The proof is simple: prime ideals are radical.
:::

:::{.example title="?"}
Note that 
\[
\spec \ZZ = \ts{\gens 0} \union \ts{ \gens{p} \st p\normal \ZZ \text{ is prime}}
.\]
In general, maximal ideals are always closed points, and $\gens 0$ is not a closed point.
This is homeomorphic to e.g. 
\[
\spec \QQbar[t] = \ts{\gens 0} \union \ts{\gens{t-a} \st a\in \QQbar}
,\]
since both are comprised of countably many closed points and a single open point.
:::

## Localization and the structure sheaf

:::{.definition title="Localization"}
Suppose $p \subseteq A$ is a prime ideal, then the **localization** of $A$ at $p$, is defined as
\[
A_p \da A\primelocalize{p} \da \ts{ {a\over f} \st a, f\in A,\, f\not\in p}\modiso \\ \\ \quad
{a\over f} \sim {b\over g}\iff \exists \, h\in A \text{ s.t. } h(ag-bf)=0
.\]
This makes the elements of $p^c$ invertible, and is a local ring with residue field $\kappa = \ff(A/p)$ and maximal ideal $pA_p$.
Ideals of $A_p$ biject with ideals of $A$ contained in $p$.
:::

:::{.remark}
Idea: $A_p$ should look like germs of functions at the point $p$.
Note that localizing at the ideal $p$ is like deleting $\cl_X(V(p))$, which is also useful.
We now want to construct a sheaf $\OO = \OO_{\spec A}$ which has stalks $A_p$. 
We'll construct something that's obviously a sheaf, at the cost of needing to work hard to prove things about it!
:::

:::{.definition title="Structure sheaf"}
For $U\in \spec(A)$ open, so $U = V(I)^c$, define the **structure sheaf of $X$** as the sheaf given 
\[
\OO(U) \da \ts{ s:U \to \Disjoint{p\in U} A_p \st s(p) \in A_p, \text{ and } s \text{ is locally a fraction}}
.\]
Here *locally a fraction* means that for all $p\in U$ there is an open $p\in V \subseteq U$ and elements $a, f\in A$ such that

1. $f\not\in Q$ for any $Q\in V$ and
2. $s(Q) = a/f$ for all $Q \in V$.

Restriction is defined for $V \subseteq U$ as honest function restriction on $\OO(U) \to \OO(V)$.
:::

:::{.remark}
Note that this is sheafifying the presheaf that sends $U = D_f$ for $f\in A$ to the ring $A_f$.
:::

:::{.example title="Structure sheaf of a field"}
Let $k \in \Field$ and $X\da \spec(k) = \ts{\gens 0}$.
Then $\OO_X$ is determined by 
\[
\globsec{X; \OO_X} = \ts{s: \spec k \to k \st \text{ conditions above} } = k
,\]
since the conditions are vacuous here.
:::

:::{.example title="Structure sheaf of formal power series rings"}
Let $X = \spec \CC\powerseries{t} = \ts{ \gens 0, \gens t}$. Then
\[
\OO_X(X) = \CC\powerseries{t}\qquad
\OO_X(\gens 0) = \CC\functionfield{t}
.\]
:::