# Wednesday, February 09

:::{.remark}
Plan:

- $\Hom_{\Sh(X)}(\wait, \wait)$
- $\Hom_{\mods{ \OO_X}}(\wait, \wait)$
- $(\wait) \tensor_{\OO_X}(\wait)$

:::

:::{.remark}
For $(X, \OO_X)\in \LRS$ and $\mcf, \mcg\in \Sh(X; \Ab\Grp)$, define $\Hom_{\OO_X}(\mcf, \mcg)$ to be natural transformations which are $\OO_X\dash$linear.
This forms an abelian group under pointwise operations, and more generally an $\OO_X\dash$module since one can act on morphisms by global sections.
There is a sheaf version, the local hom $\ul{\Hom}_{\OO_X}(\mcf, \mcg)(U) \da \Hom_{\OO_U}(\mcf_U, \mcg_U)$ where we write $\mcf_U \da \ro{\mcf}{U}$.
:::

:::{.proposition title="?"}
This forms a sheaf of $\OO_X\dash$modules.
:::

:::{.proof title="?"}
Let 
\[
f_i\in \Hom_{\OO_{U_i}}(\mcf_{U_i}, \mcg_{U_i}) \\
f_j\in \Hom_{\OO_{U_j}}(\mcf_{U_j}, \mcg_{U_j}) 
.\]
If $\ro{f_i}{U_{ij}} = \ro{f_j}{U_{ij}}$, then the claim is that there exists a unique $F\in \Hom_{\OO_{U_{ij}}}(\mcf_{U_{ij}}, \mcg_{U_{ij}} )$.
For $V \subseteq X$, decompose as $V = \Union_i U_i$.
:::

:::{.proposition title="?"}
If $\mcf \in \mods{\OO_X}^{\lf, \rank=r}$ and $\mcg\in \mods{\OO_X}^{\lf, \rank=s}$ then $\ul{\Hom}_{\OO_X}(\mcf, \mcg)\in \mods{\OO_X}^{\lf, \rank = rs}$.
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:::{.proof title="?"}
Choose trivializations $\mcf_{U_i} \iso \OO_{U_i}\sumpower{r}$ and $\mcg_{U_i} \iso \OO_{U_i}\sumpower{s}$.
The claim is that $\ul{\Hom}_{\OO_U}(\OO_U, \OO_U) = \OO_U$ for any $\OO_U$.
Given this, $\Hom_{\OO_X}(\OO_X\sumpower{r}, \OO_X\sumpower{s}) \cong \Mat_{r\times s}(\OO_X)$ split out as matrices.
To prove this, just check on global sections that $\ul{\Hom}_{\OO_X}(\OO_X, \OO_X) \cong \Hom_{\mods{R}}(R, R)\cong R$ for $R\da \globsec{R}$.
:::

:::{.remark}
Recall that $\Sh(X)^{\lf, \rank=1} \cong \Bun_{\GL}^{\rank = 1}$, i.e. we identify rank 1 locally free sheaves with line bundles.
We can write $\Hom_\OO(\mcf,\mcg) = \ts{ {\phi_{ij} \over \psi_{ij} } \st \phi_{ij} \in \OO_X\units(U_{ij}) \text{ satisfies the cocycle condition} }$.
What are the transition functions? 

We also define $\Hom_\OO(\mcf, \OO) \da \mcf\dual$, and there is a relation to $\Pic(X)$.
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:::{.remark}
Note also that $\ul{\Hom}_{\OO}(\OO, \mcf) \cong \mcf$, so global sections coincide with homs.
This will be useful later when defining $H^*$ in terms of derived functors.
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:::{.definition title="Tensor product"}
Define the tensor product of $\mcf,\mcg\in \mods{\OO_X}$ as the sheafification of
\[
(\mcf \tensor_{\OO_X} \mcg)^- \da U\mapsto  \mcf(U) \tensor_{\OO_U} \mcg(U)
.\]

Note that there is a formula for stalks:
\[
(\mcf\tensor_{\OO_X} \mcg)_x = \mcf_x \tensor_{\OO_x} \mcg_x
.\]
Moreover $\mcf\tensor_{\OO_X} \mcg \in \mods{\OO_X}^{\lf, \rank = rs}$.
This endows $\mods{\OO_X}$ with a symmetric monoidal structure with duals, so 

- $\mcf \tensor_{\OO_X }\mcf\dual \cong \OO_X$
- $\OO_X \tensor_{\OO_X} \mcf \cong \mcf$

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:::{.remark}
Recall that $f\in \Top(X, Y)$ for $X, Y\in \Aff\Sch$ induces $f\inv \in \Sh(X)(f\inv \OO_Y, \OO_X)$. 
For varieties, this is just given by pullback of regular functions.
More generally, for $X, Y\in \LRS$, define the **full pullback** $f^*$ as
\[
f^*\mcf = f\inv\mcf \tensor_{f\inv\OO_Y} \OO_X
.\]

:::

:::{.lemma title="?"}
For the full pullback,
\[
f^* \OO_Y \cong \OO_X
,\]
which is not true for $f\inv$.
This essentially follows from $R \tensor_R S \cong S$.
:::

:::{.remark}
Consider $f\in \kalg(S, R)$ for $k=\bar k$ where we only consider reduced algebra (no nonzero nilpotents).
This induces maps $\tilde f: \spec R\to \spec S$ and $\tilde f': \mspec R\to \mspec S$.
If $\mca \in \Sh(X; \algs{\OO_X})$, there are induced maps $\OO_X(U) \to \mca(U)$ and thus affine morphism $\pi: \spec \mca(U) \to U$ covering the affine open $U$.
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:::{.example title="?"}
\envlist

- $\mca = \OO_X[x_1,\cdots, x_n]$ yields a trivial vector bundle $\spec \mca = X\times \AA^n \to X$.
- For $\mcf \in \Sh(X, \mods{\OO_X}^{\lf, \rank = n})$, set 
\[
\mca = \Sym_\OO^*(\mcf) \da \OO_X \oplus \mcf \oplus \Sym^2(\mcf) \oplus \cdots
,\]
which yields a nontrivial vector bundle $\spec \mca \to X$.
- For $\mcf$ rank 1, $\mcf\tensorpower{\OO_X}{n} \iso \OO_X(-D)$, set 
\[
\mca \da T_{\OO}(\mca) \da \OO \oplus \mcf \oplus \mcf\tensorpower{\OO_X}{2} \oplus \cdots
,\]
then $\spec \mca \to X$ is a cyclic Galois cover for $G = \mu_n$.

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