Complex cohomology $\cong$ sheaf cohomology of constant sheaf on Riemann surface?

Question

I am currently reading in a rather down to earth book on Riemann surfaces. They define the first complex cohomology group $H^1(X, \mathbb{C})$ associated to a Riemann surface $X$ via $H^1(X, \mathbb{C}) = \mathrm{Hom}(H_1(X), \mathbb{C})$. Here $H_1(X)$ denotes the first singular homology group with integer coefficients.

From the properties they show (namely some exact sequence), I believe that this definition agrees with the first sheaf-cohomology of the constant sheaf $\underline{\mathbb{C}}$ on $X$. I would be interested in how to prove this.

Answer 1

Here's a method:

Show that if $C^i$ denotes the continuous cochain sheaf: $U\mapsto C^i(U)$ ($C^i(U)$ are continuous $i$-cochains) then

$$0\to\underline{\mathbb{C}}\to C^0\to C^1\to\cdots$$

is a resolution, with the $C^i$ acyclic.