= Rational principal-parts resolution on an algebraic curve
{title2=$0\to\mathcal F\to\mathcal R(\mathcal F)\to\mathcal P(\mathcal F)\to0$}
On an irreducible algebraic curve, let $\mathcal R(\mathcal F)$ be the constant rational-section sheaf and put $\mathcal P(\mathcal F)(U)=\bigoplus_{P\in U}\operatorname{Rat}(\mathcal F)/\mathcal F_P$. 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 $q\ge2$. The curve and Zariski-topology hypotheses distinguish finite support here from locally finite pole sets in complex analysis.
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