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Let $X$ be riemann surface (not supposed compact) and $\mathcal D$ be sheaf of divisors on $X$.Remind that this means for $U\subset X$ open then $\mathcal D(U)$ is group of divisors on $U$. How to prove that $H^1(X,\mathcal D)=0$ . This is exercise in §16 of Forster book Riemann Surfaces (but is not homework for me) and he says hint is to use discontinuous partitions of unit but I don't understand. Any other method for proving is for me also satisfying . I suppose $H^2(X,\mathcal D)$ is also zero, no?

Addedd: Problem is in analytic category, not algebraic ( where is trivial : $\mathcal D$ is flabby). So I don't understand Matt E's proof (but thank you very much for answer Matt) because I don't know if cohomology commutes with co-limits in non noetherian case.

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2 Answers 2

up vote 3 down vote accepted

The hint suggests modifying the proof that $H^1$ for $C^\infty$-sheaves vanishes. This also makes me guess that the problem is about Cech cohomology. The $C^\infty$ proof appears in many places, one reference is Proposition 4.1 in chapter IX of Miranda's book: Algebraic Curves and Riemann Surfaces.

There's probably a way of doing this more cleanly and in more generally but as a start: on $X$ fix a locally finite covering $\{U_i\}_{i \in I}$ where the index set is linearly ordered: $I \subset \mathbb{Z}$.

In this case, choose for each natural number $n$ a set of integers $a_1,..., a_n$ such that $\sum_{1}^n a_i = 1$; e.g. $a_1 = 1$ and all others $0$.

Then define functions $\phi_i \colon U_i \to \mathbb{Z}$ as follows:

1) For $p \in U_i$ let $U_{j_1}, ..., U_{j_n}$ be sets containing $p$ with $j_1 < ... < j_n$.

2) let $k$ be the index such that $j_k = i$

3) let $a_1,..., a_n$ be the previously determined integers summing to $1$

Define $\phi_i(p) = a_k$. These are discontinuous functions and by construction $\sum_i \phi_i \equiv 1$.

To show $H^1(X, \mathcal{D}) = H^1(\{U_i\}, \mathcal{D}) = 0$ we have to show that every $1$-cocycle $(f_{ij})$ is a co-boundary.

The proof now proceeds as in the $C^\infty$ case: on $U_i$ define $g_i = -\sum_j \phi_j f_{ij}$. Then

$g_i - g_j = -\sum_k \phi_kf_{ik} + \sum_k \phi_kf_{jk} = \sum_k \phi_k(f_{jk} - f_{ik})$

Now using that $f_{jk} - f_{ik} = f_{ji}$ we get

$g_i - g_j$ = $f_{ji}\sum_k \phi_k = f_{ji}$.

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The sheaf $\mathcal D$ is equal to $\bigoplus_{x \in X} i_{x*} \mathbb Z,$ where $\mathbb Z$ denotes the constant sheaf on the point $x$, and $i_x:\{x \} \to X$ is the embedding of the one point space $\{x\}$ into the Riemann surface $X$. (So $i_{x*}\mathbb Z$ is the skyscraper sheaf supported at $x$, whose stalk is $\mathbb Z$). The vanishing of $H^1$ now follows from the vanishing of higher cohomology for skyscraper sheaves.

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