# Monday, September 25


:::{.remark}
Recall the theory of sheaves, and in particular sheaves of $\OO_X\dash$modules.
Recall that given $f_{uv}: \mcf(U) \to \mcf(V)$, one asks for a commutative diagram of the following form: 

\[\begin{tikzcd}
	{\OO_X(U) \times \mcf(U)} && {\OO_X(V) \times \mcf(V)} \\
	\\
	{\mcf(U)} && {\mcf(V)}
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-3, to=3-3]
	\arrow["{(r, f_{uv})}", from=1-1, to=1-3]
	\arrow["{f_{uv}}", from=3-1, to=3-3]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXE9PX1goVSkgXFx0aW1lcyBcXG1jZihVKSJdLFsyLDAsIlxcT09fWChWKSBcXHRpbWVzIFxcbWNmKFYpIl0sWzAsMiwiXFxtY2YoVSkiXSxbMiwyLCJcXG1jZihWKSJdLFswLDJdLFsxLDNdLFswLDEsIihyLCBmX3t1dn0pIl0sWzIsMywiZl97dXZ9Il1d)

where $r: \OO_X(U) \to \OO_X(V)$ is the restriction.
For $X = \spec R$, there is a functor $\rmod\to\oxmod$ where $M\mapsto \mcf_M$ where $\mcf_M(U) \da M_f$ the localization of $M$ at $f$ where $U \da V(f)^c$.
We define $\QCoh(X) \leq \oxmod$ as those sheaves $\mcf$ such that on affine opens $U$ we have $\ro \mcf U = \mcf_M$, the sheaf associated to some module $M\in \rmod$ where $U = \spec R$, and $\Coh(X) \leq \QCoh(X)$ as those sheaves where every $M\in\rmod^{\fg}$.
:::


:::{.example title="?"}
Find a sheaf that is not quasicoherent.
:::


:::{.remark}
For $X = \spec \CC$, quasicoherent sheaves are $\CC\dash$vector spaces.
To produce a non-quasicoherent sheaf, take any $\CC\dash$algebra which is not finitely generated as a vector space, e.g. $\CC[x]$.
:::