Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p\_1,\ldots, p\_n\}) \ \mbox{with at worst a simple pole at}\ p\_i \} $$ where $\mathbb M(U)$ means the set of meromorphic functions on $U$ and $\mathbb O(U)$ means the set of holomorphic functions on $U$. What is the cohomology of $A$ on the Riemann sphere?



The cohomology of meromorphic functions
The cohomology of these sheaves are always 0. This follows for example from the case $\{p\_i\} = \emptyset$ by induction and using the sequences 
$A\_{p\_1,..,p\_{n-1}} \to A\_{p\_1,...,p\_n} \to \mathbb{C}\_{p\_n}$. 
In general, the isomorphism class of this sheaf depends only on the number of points, and a representative of the class for $n$ points is denoted $\mathcal{O}(n)$. The first cohomology of these sheaves is $0$ for $n > -2$ and of dimension 
$-n - 1$ for $n \le -2$ (for negative $n$ this is given by imposing zeros instead of allowing poles). Finally, there are no higher cohomologies since $\mathbb{P}^1$ is one dimensional. 


