modules to line bundles. We will mostly use the algebraic language as it is more elementary, but the reader is encouraged to keep the geometric viewpoint in mind.

Denote by $\Omega^1_{\mathscr{R}}$ the module of K[ä]{}hler differentials, which is a rank $1$ projective module, and comes with its canonical derivation $d: {\mathscr{R}}\rightarrow \Omega^1_{\mathscr{R}}$. A connection on an ${\mathscr{R}}$-module ${\mathscr{F}}$ is a map $\nabla_{\mathscr{F}}: {\mathscr{F}}\rightarrow \Omega^1_{\mathscr{R}}\otimes {\mathscr{F}}$ satisfying the Leibniz identity with respect to $d$. An equivalent viewpoint is as follows. For any ${\mathscr{F}}$ there is a canonical short exact sequence of modules $$\label{eq:pullback-sequence}
0 \rightarrow \Omega^1_{\mathscr{R}}\otimes {\mathscr{F}}\longrightarrow J^1({\mathscr{F}}) \longrightarrow {\mathscr{F}}\rightarrow 0,$$ where ${\mathscr{J}}^1({\mathscr{F}})$ is the one-jet module [@atiyah57]. Connections correspond to splittings of this sequence. ${\mathscr{F}}$ admits a connection if and only if its Atiyah class $\mathit{At}({\mathscr{F}}) \in \mathit{Ext}^1_{\mathscr{R}}({\mathscr{F}}, \Omega^1_{\mathscr{R}}\otimes {\mathscr{F}})$, which is the extension class of , vanishes. In particular, projective modules always admit connections (this can also be proved directly). In the other direction, one has [@bernstein-dmodules Lecture 2]:

\[th:locally-free\] Let ${\mathscr{F}}$ be a finitely generated ${\mathscr{R}}$-module which admits a connection. Then it is necessarily projective.

Let $R({\mathscr{S}})$ be the field of rational functions on our curve, which is the quotient field of ${\mathscr{R}}$. Tautologically, all projective modules of 