# Sheaf of meromorphic functions on non-compact Riemann surfaces

Why does the first cohomology group $H^1(X, K)$ of the sheaf of meromorphic functions on a non-compact Riemann surface $X$ vanish?

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It's a problem (26.6) from Forster's book (Lectures on Riemann surfaces). The first part of the exercise ($H^1(X,\mathcal{O}_D)=0$ for any divisor $D$) would follow from this fact and solvability of Mittag-Leffler problem on non-compact Riemann surfaces (plus some sheaf exact sequence), but I have no idea how to prove the statement above. – mathdonk Jan 6 '13 at 11:23

## 1 Answer

Since for $H^{1}$, derived-functor cohomology coincides with Čech cohomology, I'll use the latter and show that every $\xi \in {\check{H}^{1}}(X,K)$ is zero.

The cohomology class $\xi$ is represented in some locally finite open covering $\mathcal U =(U_{i})$ by some cocycle $(\xi_{ij})$ with $\xi_{ij} \in K(U_{ij})$, where $U_{ij} = U_{i} \cap U_{j}$. Letting $D$ be the divisor defined by the poles of all these $\xi_{ij}$’s, we see that actually $\xi_{ij} \in {\mathcal{O}_{X,D}}(U_{ij})$.

However, ${\check{H}^{1}}(X,\mathcal{O}_{X,D}) = 0$ because on a non-compact Riemann surface, the positive-dimensional cohomology groups of coherent sheaves are zero (open Riemann surfaces are Stein!).

Hence, we can write $(\xi_{ij})$ as the coboundary $\xi_{ij} = \xi_{j}|_{U_{ij}} - \xi_{i}|_{U_{ij}}$ of some $1$-cochain $(\xi_{i})$ with $\xi_{i} \in {\mathcal{O}_{X,D}}(U_{i})$. We can interpret a fortiori $(\xi_{i})$ as a $1$-cochain with $\xi_{i} \in K(U_{i})$ (because $\mathcal{O}_{X,D} \subseteq K$), thus proving that $\xi$ is the coboundary of a cochain of $K$ and hence that its class is zero in ${\check{H}^{1}}(X,K)$.

Conclusion: ${\check{H}^{1}}(X,K) = 0$.

A GAGA-type remark

If $X_{\text{alg}}$ is the unique algebraic variety underlying $X$ and $K_{\text{rat}}$ its sheaf of rational functions, we also have ${H^{1}}(X_{\text{alg}},K_{\text{rat}}) = 0$, but this time, the nullity is trivial because $K_{\text{rat}}$ is a flabby sheaf.

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