Take pairwise disjoint coordinate discs $U_m$ around $x_m$ and put $U_0=S\setminus\{x_1,\ldots,x_N\}$. A prescribed <principal part of a meromorphic function> $P_m$ is holomorphic on $U_0\cap U_m$. Define
$$
P_{0m}=P_m,\qquad P_{m0}=-P_m,
$$
with all other components zero. No three distinct cover members meet, so the <Čech cocycle condition> is automatic. Thus $(P_{ij})$ is a Čech one-cocycle for $\mathcal O$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 2 / b / i / Solution

{wiki=Mittag-Leffler's_theorem#Pole_prescription}

= Principal part
{synonym}

At a point $x$ of a <Riemann surface>, the principal part of a <meromorphic function> is the negative-degree part of its <Laurent series> in a local coordinate. It is equivalently a germ in the quotient $\mathcal K_x/\mathcal O_x$.

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 Incoming links: Principal part of a meromorphic function