# Friday, February 04

:::{.remark}
Last time: extension by zero, inverse image, pushforward on closed sets and adjunctions.
\[
f\in \Hom_{\Top}(X, Y) \leadsto \Hom_{\Sh(X)}(f\inv \mcg, \mcf) \cong \Hom_{\Sh(Y)}(\mcg, f_* \mcf)
.\]
:::

:::{.warnings}
Pushing forward open sets is not generally a good idea!
Take $X = \RR^\zar$, $Z = \ts{\pt}, U = X\sm Z$.
Then $(i_* \ul{\ZZ_U})_p = \ZZ\sumpower{2}$ if $p= \pt$, since any neighborhood of $p$ pulls back to two connected components.
:::

:::{.remark}
Consider $U \injectsvia{i} X$ with $U$ open and $Z \injectsvia{j} X$ with $Z$ closed, then for $\mcf \in \Sh(X), \mch\in \Sh(U), \mcg \in \Sh(Z)$,
\[
\Hom_{\Sh(Z)}( \ro{\mcf}{Z}, \mcg ) &\iso \Hom_{\Sh(X)}(\mcf, j_* \mcg) \\
\Hom_{\Sh(U)}(\mch, \ro{\mcf}{U} ) &\iso \Hom_{\Sh(X)}(i_! \mch, \mcf)
.\]
:::

:::{.remark}
We'll consider $(X, \OO_X) \in \LRS\slice{\CRing}$ with sheaves of reduced commutative rings -- note that noncommutative rings are also important, e.g. $\GL_n$ or $\liegl_n$.
:::

:::{.example title="?"}
Common examples of locally ringed spaces:

- $(X, \ul{R})$ any space with a constant sheaf.
- $(X, \mcf)$ for $\mcf \da \OO_X^\cts \da \Hom_\Top(\wait, \RR)$.
- $(X, \OO_X^\zar)$ for $X\in \Aff\Alg\Var\slice k$ and $\OO_X^\zar$ the sheaf of Zariski-regular functions.
  In this case, for $k=\kbar$, these are of the form $\mspec R \subseteq \AA^n\slice k$ for $R\da \kxn/\gens{f}$.
  Recall distinguished opens are $D(g) = \ts{g\neq 0}$ for $g\in \kxn$, and sections are $\OO_X(D(g)) = R\localize{g}$ are functions $\rho: X\to k$ of the form $\rho = h/g^k$ for some regular function $h$.
  It's a theorem that these assemble to a sheaf.
:::

:::{.remark}
Define algebraic varieties as locally ringed spaces $(X, \OO_X)$ that 

1. $X$ is covered by finitely affine algebraic varieties, so $X = \union U_i$ with $(U_i, \OO_{U_i})$ affine algebraic, and
2. $X$ is separated, i.e. $X \mapsvia{\Delta_X} X\fiberpower{X}{2}$ is closed.

Note that affine and even quasiprojective schemes are automatically separated.
We require the separated condition here to rule out things like $\AA^1$ with two origins, i.e. $X \da \AA^1\glue{\AA^1\smz}\AA^1$.
:::

:::{.example title="?"}
Affine schemes: for $R\in \CRing$, take $X\da \spec R$ with a basis $D(g)$ and define a presheaf by $\OO_X(D(g)) = R\localize{g}$. 
It's a theorem that this yields a sheaf.
:::

:::{.definition title="$\OO_X\dash$modules"}
For $(X, \OO_X) \in \LRS$, $\mcf$ is a **sheaf of $\OO_X\dash$modules** iff every section $F(U)$ is an $\OO_X(U)\dash$module and restriction is compatible with the module structures in the sense that $\ro{(rm)}{V} = \ro{r}{V} \ro{m}{V}$:

\begin{tikzcd}
	m\in & {\mcf(U)} && {\OO_X(U)} & {\ni r} \\
	\\
	& {\mcf(V)} && {\OO_X(V)}
	\arrow[from=1-2, to=3-2]
	\arrow[from=1-2, to=1-4]
	\arrow[from=3-2, to=3-4]
	\arrow[from=1-4, to=3-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMSwwLCJcXG1jZihVKSJdLFsxLDIsIlxcbWNmKFYpIl0sWzMsMCwiXFxPT19YKFUpIl0sWzMsMiwiXFxPT19YKFYpIl0sWzAsMCwibVxcaW4iXSxbNCwwLCJcXG5pIHIiXSxbMCwxXSxbMCwyXSxbMSwzXSxbMiwzXV0=)

:::

:::{.example title="?"}
Any constant sheaf $\ul{M}$ for $M\in \rmod$.
:::

:::{.definition title="Quasicoherent and coherent sheaves"}
An $\OO_X\dash$module is 

- **Quasicoherent** if locally $\mcf \cong \ul{M}$ (there exists an open cover $X = \union U_i$ with $\ro{\mcf}{U_i} \cong \ul{M_{U_i}}$),
- **Coherent** iff $\mcf$ is quasicoherent and $M \in \rmod^\fg$ and $X$ is locally Noetherian.

:::

:::{.example title="?"}
Of an $\OO_X\dash$module for a constant sheaf: $M = R/p$ for $\OO_X = \ul{R}$.
:::

:::{.example title="?"}
For complex analytic varieties, take $(X, \OO_X^\an)$ so $\OO_X(U)$ are locally meromorphic functions regular on $U$, i.e. whose denominator does not vanish on $U$.
This is the setting where Cartan, Serre, etc defined original notions of coherence, and e.g. Serre vanishing, and scheme theory is developed by analogy to this situation.
Here, $\mcf$ is a **coherent** sheaf iff $\mcf$ is a sheaf of $\OO_X^\an\dash$modules and admits a presentation
\[
\OO_X^\an \sumpower{m} \to \OO_X^\an\sumpower{n} \to \mcf\to 0
.\]
:::

:::{.remark}
Next time: locally free, invertible, tensor, and hom.
:::