# Exactness for Sheaves (Wednesday, August 25)

## Some examples

:::{.remark}
Recall the definition of sheafification: let $\mcf\in \Presh(X; \Ab\Grp)$.
Construct a sheaf $\mcf^+\in \Sh(X, \Ab\Grp)$  and a morphism $\theta: \mcf \to \mcf^+$ of presheaves satisfying the appropriate universal property:

\begin{tikzcd}
	{\mcf^+} \\
	\\
	\mcf && \mcg \\
	\\
	{}
	\arrow["\psi", from=3-1, to=3-3]
	\arrow["\theta", from=3-1, to=1-1]
	\arrow["{\exists \tilde \psi}", dashed, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJcXG1jZiJdLFswLDRdLFswLDAsIlxcbWNmXisiXSxbMiwyLCJcXG1jZyJdLFswLDMsIlxccHNpIl0sWzAsMiwiXFx0aGV0YSJdLFsyLDMsIlxcZXhpc3RzIFxcdGlsZGUgXFxwc2kiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

So any presheaf morphism to a sheaf factors through the sheafification uniquely (via $\theta$).
Note that this is a instance of a general free/forgetful adjunction.

We can construct it as
\[
\mcf^+(U) \da \ts{s:U\to \disjoint_{p\in U} \mcf_p,\quad s(p) \in \mcf_p, \cdots}
.\]
where the addition condition is that for all $q\in U$ there exists a $V\nu q$ and $t\in \mcf(V)$ such that $t_p = s(p)$ for all $p\in V$.
Note that $\theta$ is defined by $\theta(U)(s) = \ts{s:p\to s_p}$, the function assigning points to germs with respect to the section $s$.
Idea: this is like replacing an analytic function on an interval with the function sending a point $p$ to its power series expansion at $p$.
:::

:::{.example title="?"}
Recall $\exp: C^0 \to (C^0)\units$ on $\CC\units$, then $\coker^\pre(\exp)(U) = \ts{1}$ on contractible $U$, using that one can choose a logarithm on such a set.
However $\coker^\pre(\exp)(\CC\units) \neq \ts{1}$ since $[z]\in (C^0)\units(\CC\units)/\exp(C^0(\CC\units))$.
:::

:::{.remark}
Letting $\phi: \mcf \to \mcg$ be a morphisms of sheaves, then we defined $\coker(\phi) \da (\coker^\pre(\phi))^+$ and $\im(\phi) \da (\im^\pre(\phi))^+$.
Then
\[
\coker^\pre(\exp) &\to \coker(\exp) \\
s\in \mcf(U) &\mapsto s(p) = s_p
.\]
The claim is that $[z]_p = 1$ for all $p\in \CC\units$, since we can replace $[([z], \CC\units)]$ with $([z]_U, U)$ for $U$ contractible.
:::

:::{.example title="?"}
A useful example to think about: $X = \ts{p, q}$ with

- $\mcf(p) = A$
- $\mcf(q) = B$
- $\mcf(X) = 0$

Then local sections don't glue to a global section, so this isn't a sheaf, but it is a presheaf.
The sheafification satisfies $\mcf^+(X) = A\cross B$.
:::

## Subsheaves

:::{.definition title="Subsheaves, injectivity, surjectivity"}
$\mcf'$ is a **subsheaf** of $\mcf$ if

- $\mcf'(U) \leq \mcf(U)$ for all $U$,
- $\Res'(U, V) = \ro{ \Res(U, V) }{\mcf'(U)}$.

$\phi: \mcf\to \mcg$ is **injective** iff $\ker \phi = 0$, **surjective** if $\im(\phi) = \mcg$ or $\coker \phi = 0$.
:::

:::{.exercise title="?"}
Check that $\ker \phi$ already satisfies the sheaf property.
:::

## Exact Sequences of sheaves

:::{.definition title="Exact sequences of sheaves"}
Let $\cdots \to \mcf^{i-1} \mapsvia{\phi^{i-1}} \mcf^i \mapsvia{\phi^i} \mcf^{i+1}\to \cdots$ be a sequence of morphisms in $\Sh(X)$, this is **exact** iff $\ker \phi^i = \im \phi^{i-1}$.
:::

:::{.lemma title="?"}
$\ker \phi$ is a sheaf.
:::

:::{.proof title="?"}
By definition, $\ker(\phi)(U) \da \ker \qty{ \phi(U): \mcf(U) \to \mcg(U) }$, satisfying part (a) in the definition of presheaves.
We can define restrictions $\ro{\Res(U, V)}{\ker(\phi)(U)} \subseteq \ker(\phi)(V)$.
Use the commutative diagram for the morphism $\phi: \mcf \to \mcg$.

Now checking gluing:
Let $s_i \in \ker(\phi)(U_i)$ such that $\Res(s_i, U_i \intersect U_j) = \Res(s_j, U_i \intersect U_j)$ for all $i, j$.
This holds by viewing $s_i \in \mcf(U_i)$, so $\exists ! s\in \mcf(\Union_i U_i)$ such that $\Res(s, U_i) = s_i$.
We want to show $s\in \ker(\phi)\qty{\Union U_i}$, so consider
\[
t\da \phi\qty{ \Union_i U_i}(s) \in \mcg\qty{\Union U_i}
,\]
which is zero.
Now 
\[
\Res(t, U_i) = \phi(U_i)(\Res(s, U_i)) = \phi(U_i)(s_i) = 0
\]
by assumption,
using the commutative diagram.
By unique gluing for $\mcg$, we have $t=0$, since $0$ is also a section restricting to $0$ everywhere.
:::

:::{.definition title="Quotients"}
For $\mcf' \leq \mcf$ a subsheaf, define the **quotient** $\mcf/\mcf' \da ((\mcf/\mcf')^\pre)^+$ where 
\[
(\mcf/\mcf')^\pre(U) \da \mcf(U)/ \mcf'(U)
.\]
:::