Suppose $X$ is a compact Riemann surface of genus $g$, $\mathcal{O}$ represents the sheaf of holomorphic functions and $\Omega$ the sheaf of holomorphic 1-forms
I want to show the dimension of $H^1(X, \mathbb{C})$ equals $2g$ by using the short exact sequence $0 \to \mathbb{C} \to \mathcal{O} \xrightarrow{d} \Omega \to 0$.
My approach so far is to use the long sequence in cech cohomology of the given short exact sequence $$0 \to \mathbb{C} \xrightarrow{i} \check{H}^0(X,\mathcal{O}) \xrightarrow{d} \check{H}^0(X,\Omega) \xrightarrow{\Delta} \check{H}^1(X, \mathbb{C}) \xrightarrow{i^1} \check{H}^1(X, \mathcal{O}) \xrightarrow{d^1} H^1(X, \Omega)$$ where $\Delta$ is the connecting homomorphism and the maps $i^1$ and $d^1$ are the induced maps from the respective sheaf maps.
By the Serre-Duality Theorem, I know that dim$\check{H}^0(X,\Omega) = g$ and dim$\check{H}^1(X, \Omega) = 1$. After repeated applications of the Rank-Nullity theorem, I can show that dim$($Ker$(i^1)) = g$. But I am having trouble showing that dim$($Im$(i^1)) = g$ as well. It seems like I'm missing something really really straightforward.
The books I am using are Rick Miranda's Algebraic Curves and Riemann Surfaces and Otto Forster's Lectures on Riemann Surfaces.
Would greatly appreciate any help or hints! Thank you so much!