# Friday, January 28

:::{.remark}
Last time: 

- Morphisms of sheaves $\phi$, 
- $\ker \phi$ (already a sheaf), 
- $(\im \phi)^-, (\coker \phi)^-$ (need to sheafify),
  - All defined to commute with taking stalks: $(\ker \phi)_p = \ker(\phi_p)$, etc
- $(\im \phi)^-$ may fail the existence axioms for sheaves, using $\exp: \OO^\an \to (\OO^\an)\units$ for $X$ a complex analytic space,
- $(\coker \phi)^-$ may fail the uniqueness axioms for sheaves,
- $(\im \phi)^-$ satisfies existence $\iff (\coker \phi)^-$ satisfies uniqueness,
- For $\mcf\injects \mcg$ injective, the presheaf quotient $(\mcg/\mcf)^-$ may fail to be a sheaf.

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:::{.example title="of the last claim"}
For $X\in \Alg\Var\slice k$ for $k=\kbar$, let $\OO_X$ be its regular algebraic functions.
Take $X = \PP^1$ and $U \da \AA^1\sm\ts{\pt} \subseteq \AA^1 \subseteq \PP^1\sm\ts{a_1,\cdots, a_k}$.
Then $\OO_X(U) = k[x]\localize{f}$ for $f(x) \da \prod (x-a_k)$, $\OO_X(X) = k$, $K_X(U) = k(x)$, and $K_X\units(U) = k(x)\smz$ if $U \neq \emptyset$.
Define **Cartier divisors** as global sections of the sheaf $\Cart\Div \da K_X\units/\OO_X\units$.
Recall that Weil divisors are finite sums of codimension 1 subvarieties, and these notions coincide for nonsingular varieties.

For $p\in \AA^1 \subseteq \PP^1$, we have 
\[
(K_X\units/\OO_X\units)_p 
= {K_{X, p} \over \OO_{X, p}} 
= {k(x) \over \ts{f/g \st f(p)\neq 0 , g(p) \neq 0}}
\cong \ZZ
,\]
using that any element in the quotient can be written as $h(x) = (x-p)^n g(x)$ for some $g\in \OO_{X, p}\units$.
Here $\Cart\Div(X) = \sum n_p P$ are all finite sums with $n_p\in \ZZ$.
The claim is that sheaf existence fails for this quotient -- there are local sections that do not glue.
Here

- $K\units(\PP^1) = k(x)\units$
- $\OO\units(\PP^1) = k\units$
- $K\units(\PP^1)/\OO\units(\PP^1) = {k(x)\units \over k\units}$

For any $s$ in the quotient, we can associated $(s)_0 - (s)_\infty = \sum n_p P$, but not every Cartier divisor is of this form -- these are the *principal* divisors.
This form a group $\Pic(X) = \Cart\Div(X) / \Prin\Cart\Div(X)$, which may not be trivial.
This proof generalizes to locally Noetherian schemes, not necessarily reducible, with no embedded components.
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:::{.remark}
Note that $\Pic(X)$ is also the group of invertible sheaves on $X$, and for irreducible algebraic varieties these coincide.
Use the SES $0\to \OO\units \to K\units \to K\units/\OO\units \to 0$ to obtain
\[
1 \to H^0(\OO\units) \to H^0(K\units) \to \Prin\Cart\Div(X) \to H^1(\OO\units)\cong \text{invertible sheaves}/\sim \to 0,
,\]
where $H^1(K\units)$ vanishes since it's a constant sheaf on an irreducible scheme in the Zariski topology.
:::

:::{.proposition title="?"}
$(\im \phi)^-$ satisfies existence $\iff (\coker \phi)^-$ satisfies uniqueness.
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:::{.proof title="?"}
$\implies$:
Let $s\in \coker(F(U) \to G(U))$ and write $U = \union U_i$.
We want to show that $s_{U_i}$ implies $s\in \coker(F(U_i) \to G(U_i))$ for all $i$.
Note that $s = 0$ in $\coker(F(U) \to G(U))$ iff $s\in \im(F(U) \to G(U))$
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