 Rational principal-parts resolution on an algebraic curve

On an irreducible algebraic curve, let $\mathcal{R}(\mathcal{F})$ be the constant rational-section sheaf and put $\mathcal{P}(\mathcal{F})(U)={⨁}_{P\in U}\mathrm{Rat}(\mathcal{F})/{\mathcal{F}}_P$. Every open subset is quasi-compact, so compatible finite-support families glue with finite support. Its stalk at $P$ is the displayed rational-principal-part quotient. A rational section is regular outside finitely many points, making its principal-part map well defined. The resulting sequence is stalkwise exact. Both right-hand sheaves are flasque, and consequently $H^1(X,\mathcal{F})$ is the cokernel of the global principal-part map and $H^q(X,\mathcal{F})=0$ for $q\ge 2$. The curve and Zariski-topology hypotheses distinguish finite support here from locally finite pole sets in complex analysis.