# Monday, February 07

:::{.remark}
Examples of sheaves:

- $\OO_X^\cts$ for $X\in \Top$, where $\OO_X^\cts(\wait) = \Top(\wait, \RR)$
- $\OO_X^{\smooth}(\wait) = C^\infty(\wait, \RR)$.
- $\OO_X^{\hol}(\wait) = \Hol(\wait, \CC)$
- $\OO_X^{\an}(\wait) \subseteq \Top(\wait, \RR)$ the sheaf of analytic functions, those locally equal to power series.
- For $X\in \Alg\Var\slice k$, $\OO_X(\wait) = \Top((\wait)^\zar, k)$ the Zariski-regular $k\dash$valued functions.

In all cases, $\OO_X$ can be regarded as sheaves of *regular* sections to $X\times \AA^1\slice k \mapsvia{\pi}  X$.
Note that this doesn't necessarily coincide with sections of the espace etale, since e.g. the fibers are $\AA^1$ and not necessarily the stalks.
For $\OO\sumpower{d}$, one instead takes $X\times \AA^d\slice k\to X$.
:::

:::{.definition title="Locally free and invertible sheaves"}
A sheaf $\mcf \in \Sh(X)$ is **locally free** iff there exists an open cover $\mcu \covers X$ with $\ro{\mcf}{U_i} \cong \OO_{U_i}\sumpower{n}$.
The quantity $n$ is the **rank** of $\mcf$.
If $\rank \mcf = 1$, then $\mcf$ is **invertible**.

A **vector bundle** over $X$ is $V \mapsvia{\pi} X$ with $\pi\inv(U_i) \cong U_i \times \AA^r$.
For $r=1$, this is a **line bundle**.
:::

:::{.remark}
Maps between bundles are linear in the second coordinate.

Note that there is a correspondence between vector bundles and locally free sheaves.
Consider the rank 1 case, matching invertible sheaves and line bundles.
The necessary data:

- An open cover $\mcu\covers X$, where $\mcu = \ts{U_i}_{i\in I}$
- For all $i, j\in I$, transition functions $\phi_{ij}\in \OO\units(U_{ij}) = \Aut_{\mods{\OO_X}}(U_{ij})$.
- A cocycle condition: $\phi_{ii} = \id, \phi_{ij} = \phi_{ji}\inv$, and $\phi_{ij} \phi_{jk} \phi_{ki} =\id \in \OO\units(U_{ijk})$

Note that any morphism of sheaves $\OO_V \to \OO_V$ induces a morphism of $\OO_V\dash$modules on global sections
\[
\OO_V(V) &\iso \OO_V(V) \in \mods{\OO_V} \\
1 &\mapsto \phi
,\]
and this being an isomorphism manes $\phi$ is invertible.
Note that these are not isomorphic as rings.

Write $Z_1(\mcu; \OO\units) = \ts{\phi_{ij} \in \OO\units(U_{ij}) \st \cdots }$ for those $\phi_{ij}$ satisfying the conditions above, and $B_1(\mcu; \OO\units) = \ts{\phi_{ij} \in \OO\units(U_{ij}) \st \phi_{ij} \sim \phi_{ij}{\psi_j \over \psi_i} }$ for any ${\psi_i \over \psi_j} \in \GL_1(\OO) \cong \OO\units$.
More generally, we let $\phi_{ij} = \psi_j \phi_{ij} \psi_i\inv$ for $\psi_i, \psi_j \in \GL_n(\OO)$.
:::

:::{.remark}
Recall that for a given space $X$, the open covers of $X$ form a poset under refinement, where $\mcu \geq \mcv$ iff for every $U_i\in \mcu$ there is some $V_j \in \mcv$ with $U_i \contains V_j$.
This yields a system of maps $Z^1(\mcu; \OO\units) \to Z^1(\mcv; \OO\units)$ compatible with transition maps, so we define
\[
\Hc^1(X; \OO_X\units) \da \colim_{\mcu\covers X} \Hc^1(\mcu; \OO\units)
.\]
:::

:::{.exercise title="?"}
Compute $\Hc^1(\PP^1; \OO_{\PP^1}\units )$ using an open cover by two sets.
:::