= Sheaf of meromorphic principal parts
{title2=$\mathcal P=\mathcal M_X/\mathcal O_X$}
On a <Riemann surface>, the quotient sheaf $\mathcal P=\mathcal M_X/\mathcal O_X$ records the finite negative <Laurent series> tails of meromorphic germs. At $x$, a local coordinate $t$ identifies its stalk with $\bigoplus_{m\geq1}\mathbb C t^{-m}$. As a sheaf it is the <direct sum of sheaves> $\bigoplus_x(i_x)_*\mathcal P_x$. Global sections prescribe locally finite families of principal parts, possibly infinite on a noncompact surface. The <connecting homomorphism> to $H^1(X,\mathcal O_X)$ is exactly the obstruction to a global meromorphic function with those parts and no extra poles.
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