# Sheaves, Stalks, Local Rings (Friday, August 20)

## Sheaves

:::{.definition title="Sheaf"}
Recall the definition of a presheaf, and the main 3 properties:

1. $\mcf( \cofinal ) = \final$ where $\final = 0 \in \Ab$,
2. $\rho_{UU} = \id_{\mcf(U)}$
3. For all $W \subseteq V \subseteq U$, a cocycle condition:
\[
\rho_{UW} = \rho_{VW} \circ \rho_{UV}
.\]
Write $s_i \in \mcf(U_i)$ to be a section.


    A presheaf is a **sheaf** if it additionally satisfies

4. When restrictions are compatible on overlaps, so 
\[
\ro{s_i}{U_i \intersect U_j} = \ro{s_j}{U_i \intersect U_j}
,\]
there exists a uniquely glued section $\mcf(\union U_i)$ such that $\ro{s}{U_i} = s_i$ for all $i$.

:::

:::{.example title="?"}
Take $C^0(\wait; \RR)$ the sheaf of continuous real-valued functions on a topological space.
For $f_i: U_i \to \RR$ agreeing on overlaps, there is a continuous function $f: \union U_i\to \RR$ restricting to $f_i$ on each $U_i$ by just defining $f(x) \da f_i(x)$ for $x\in U_i$ and assembling these into a piecewise function, which is well-defined by agreement of the $f_i$ on overlaps.
:::

:::{.example title="A presheaf which is not a sheaf"}
Let $X$ be a topological space and $A\in \CRing$, then take the constant sheaf 
\[
\ul{A}(U) \da
\begin{cases}
A & U\neq \emptyset 
\\
0 & \text{else}.
\end{cases}
.\]
This is not a sheaf -- let $X = \RR$ and $A = \ZZ/2$, let $U_1 = (0, 1)$ and $U_2 = (2, 3)$, and take $s_1 = 0$ on $U_1$ and $s_2 = 1$ on $U_2$.
Since $U_1 \intersect U_2 = \emptyset$, the sections trivially agree on overlaps, but there is no constant function on $U_1 \union U_2$ restricting to 1 on $U_2$ and 0 on $U_1$

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:::

:::{.definition title="Locally constant sheaves"}
The **(locally) constant sheaf** $\ul{A}$ on any $X\in \Top$ is defined as 
\[
\ul{A}(U) \da \ts{ f: U\to A \st f \text{ is locally constant} }
.\]

:::

:::{.remark}
As a general principle, this is a sheaf since the defining property can be verified locally.
:::

:::{.example title="?"}
Let $C^0_{\mathrm{bd}}$ be the presheaf of bounded continuous functions on $S^1$.
This is not a sheaf, but one needs to go to infinitely many sets: take the image of $[{1\over n}, {1\over n+1}]$ with (say) $f_n(x) = n$ for each $n$.
Then each $f_n$ is bounded (it's just constant), but the full collection is unbounded, so these can not glue to a bounded function.
:::

## Stalks and Local Rings

:::{.definition title="Stalks"}
Let $\mcf \in \Presh(X)$ and $p\in X$, then the **stalk** of $\mcf$ at $p$ is defined as 
\[
\mcf_p(U) \da \lim_{U\ni p} \da \ts{(s, U) \st U\ni p \text{ open}, \, s\in \mcf(U)}/\modiso
,\]
where $(s, U) \sim (t ,V)$ iff there exists a $W \ni p$ with $W \subset U \intersect V$ with $\ro{s}{W} = \ro{t}{W}$.
An equivalence class $[(s, U)] \in \mcf_p$ is referred to as a **germ**.
:::

:::{.example title="Stalks of sheaves of analytic functions"}
Let $C^\omega(\wait; \RR)$ be the sheaf of analytic functions, i.e. those locally expressible as convergent power series.
This is a sheaf because this condition can be checked locally.
What is the stalk $C_0^\omega$ at zero?
An example of a function in this germ is $[(f(x) = {1\over 1-x}, (-1, 1))$.
A first guess is $\RR\powerseries{t}$, but the claim is that this won't work.

Note that there is an injective map $C_0^\omega \injects \RR\powerseries{t}$ because $f, g$ have analytic power series expansions at zero, and if these expressions are equal then $\ro{f}{I} = \ro{g}{I}$ for some $I$ containing zero.
This map won't be surjective because there are power series with a non-positive radius of convergence, for example taking $f(t) \da \sum_{k=0}^\infty {kt}^k$ which only converges at $t=0$.
So the answer is that $C_0^\omega \leq \RR\powerseries{t}$ is the subring of power series with positive radius of convergence.
:::

:::{.definition title="Local ring of the structure sheaf, $\OO_p$"}
Let $X \in \Alg\Var$ and $\OO$ its sheaf of regular functions.
For $p\in X$, the stalk $\OO_p$ is the **local ring** of $X$ at $p$.
:::

:::{.example title="Local rings of affine space"}
For $X \da \AA^1\slice{k}$ for $k=\bar{k}$, the opens are cofinite sets and $\OO(U) = \ts{f/g \st f, g\in k[t]}$.
Consider the stalk $\OO_p$ for some fixed $p\in \AA^1\slice k$.
Applying the definition, we have
\[
\OO_p \da \ts{(f/g, U) \st p\in U,\, g\neq 0 \text { on } U} / \modiso
.\]
Given any $g\in k[t]$ with $g(p) \neq 0$, there is a Zariski open set $U = V(g)^c = D_g$, the distinguished open associated to $g$, where $g\neq 0$ on $U$ by definition.
Thus $p\in U$, and so any $f/g\in \ff(k[t]) = k\rff{t}$ with $p\neq 0$ defines an element $(f/g, D_g) \in \OO_p$.
Concretely:
\[
\ro{f/g}{W} = \ro{f/g}{W'} \implies f/g = f'/g' \in k(t)
,\]
and $fg' = f'g$ on the cofinite set $W$, making them equal as polynomials.
We can thus write
\[
\OO_p = \ts{f/g \in k(t) \st g(p) \neq 0} = k[t]\localize{\gens{t-p}}, \quad \gens{t-p}\in \mspec k[t]
,\]
recalling that $k[t]\localize{\mfp} \da \ts{f/g \st f,g\in k[t],\,\, g\not\in \mfp}$.

:::

:::{.remark}
Note that for $X\in \Aff\Var$, writing $X = V(f_i) = V(I)$ for a radical ideal $I$, we have the coordinate ring 
\[
k[X] \da \kx{n}/I = R
\implies 
\OO_p = R\localize{\mfm_p},\quad \mfm_p \da \ts{f\in R \st f(p) = 0}
.\]
We thus have the following:
:::

:::{.slogan}
The local ring at $p$ is the localization at the maximal ideal of all functions in the coordinate ring vanishing at $p$.
:::

:::{.warnings}
This doesn't quite hold for non-algebraically closed fields:
\[
f(x) \da x^p-x \in \FF_p[x] \implies f(x) = 0 \quad \forall x\in \FF_p \implies f \equiv 0 \in \FF_p[x]
.\]

> DZG: I missed something here, so I'm not sure **what** isn't supposed to hold!

:::

:::{.remark}
Next time: morphisms of sheaves/presheaves, and isomorphisms of sheaves can be checked on stalks.
:::