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Let $X$ be a Riemann surface and let $\mathcal L$ be a line bundle on $X$. Here are two definitions of "meromorphic sections".

  1. Suppose $\mathcal L$ is trivializable on $\{ U_i \}$. Say $f_{ij}$ are the transition functions from $\mathcal O_X \mid_{U_i}$ to $\mathcal O_X \mid_{U_j}$. Then a meromorphic section of $\mathcal L$ is a choice of meromorphic functions $f_i$ on $U_i$ such that $f_i$ = $f_{ij} f_j$ are the same function on $U_{ij}$.
  2. A meromorphic function of $\mathcal L$ is a global section of the sheaf $\mathcal L \otimes_{\mathcal O_X} \mathcal M_X$, where $\mathcal M_X$ is the sheaf of meromorphic functions on $X$.

Are these definitions the same? How do we pass from one to the other? Thank you.

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This is one example of a general construction: computing cohomology of sheaves of abelian groups on topological spaces by Cech complex https://en.wikipedia.org/wiki/%C4%8Cech_cohomology

Here you just do it for $H^0$.

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