H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections

The fundamental sequence is exact. By [F1] the map $\mathcal{L}\to {\mathcal{L}}_{\eta }$ is injective and $\mathcal{P}(\mathcal{L})$ is its quotient, so $0\to \mathcal{L}\to {\mathcal{L}}_{\eta }\to \mathcal{P}(\mathcal{L})\to 0$ is exact. The stalk at the generic point is $0\to {\mathcal{L}}_{\eta }\stackrel{\mathrm{i}\mathrm{d}}{\to }{\mathcal{L}}_{\eta }\to 0\to 0$, and at each closed point $p$ it is $0\to {\mathcal{L}}_p\to {\mathcal{L}}_{\eta }\to {\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p\to 0$. These are exact; the curve has no other points, so the published stalkwise exactness criterion A sequence of abelian sheaves is exact exactly when it is exact on every stalk gives exactness of the sheaf sequence.

The middle sheaf is acyclic. Since $C$ is irreducible [F2], the constant sheaf ${\mathcal{L}}_{\eta }$ is flasque, so by [F2] its cohomology vanishes in positive degrees and $H^0(C,{\mathcal{L}}_{\eta })={\mathcal{L}}_{\eta }$ is the space of global meromorphic sections; the Axiom of Choice [F4] is inherited through the cited suppliers.

Take the long exact cohomology sequence. By [F3], step 1.1 and the vanishing of step 1.2 give the exact sequence $0\to H^0(C,\mathcal{L})\to {\mathcal{L}}_{\eta }\to H^0(C,\mathcal{P}(\mathcal{L}))\to H^1(C,\mathcal{L})\to H^1(C,{\mathcal{L}}_{\eta })=0$, so $H^1(C,\mathcal{L})$ is canonically isomorphic to the cokernel of the middle map ${\mathcal{L}}_{\eta }\to H^0(C,\mathcal{P}(\mathcal{L}))$.

Identify the two terms concretely. By [F1] the sheaf $\mathcal{P}(\mathcal{L})$ is the direct sum of the skyscraper sheaves with values ${\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$, so $H^0(C,\mathcal{P}(\mathcal{L}))={⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$ is the space of finite-support families of local principal parts, and the map of step 2.1 becomes the diagonal $s\mapsto (s+{\mathcal{L}}_p{)}_p$ whose image is the family of principal parts of the global meromorphic sections; the subgroup $H^0(C,\mathcal{L})$ maps to $0$ in $H^0(C,\mathcal{P}(\mathcal{L}))$ because it is the image of the subsheaf $\mathcal{L}$ inside ${\mathcal{L}}_{\eta }$.

Conclude. Combining steps 2.1 and 3.1, $H^1(C,\mathcal{L})$ is the cokernel of the diagonal map ${\mathcal{L}}_{\eta }\to {⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$, i.e. the space of finite-support families of local principal parts modulo the principal parts of global meromorphic sections of $\mathcal{L}$. The classes of global meromorphic sections are exactly the coboundaries of the long exact sequence. Each family whose entries are regular germs maps componentwise to zero in ${⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$; this requires no claim that an arbitrary family of germs comes from one global section. This proves the statement for every invertible sheaf $\mathcal{L}$ on $C$.