I have been reading this paper about de-Rham cohomology of hyperelliptic curves, and I have been trying to recompute some of what has been done in section 3.
In particular, I am trying to see why the first de-Rham cohomology group is as described.
I have read about (and am relatively happy) with (algebraic) de-Rham cohomology and Čech cohmology seperately, but I seem to be struggling to put them together.
It appears that in the paper the author has taken the cohomology of the total complex of the following square \begin{equation} \begin{array}[ccc] ~ {\mathcal O}_X(U_1) \times {\mathcal O}_X(U_1) & \rightarrow & \Omega_X(U_1)\times \Omega_X(U_2) \\ \downarrow & ~ & \downarrow \\ {\mathcal O}_X(U_1\cap U_2) & \rightarrow & \Omega_X(U_1\cap U_2) \end{array} \end{equation} where $X$ is the hyperelliptic curve, $U_1$ and $U_2$ form the cover of $X$ being used (obtained by removing distinct points $P_1$ and $P_2$ respectively), and zeroes outside of the diagram.
It seems to me that this is just computing the Čech cohomology of the complex \begin{equation} \begin{array}[cc] w0 \rightarrow {\mathcal O}_X \rightarrow \Omega_X \rightarrow 0 && {(1)} \end{array} \end{equation} and I don't see where algebraic de-Rham cohomology is coming in to this.
It appears to me that one would need to take the Čech cohomology of the bicomplex associated to (1) obtained by taking the Godement resolution of $\mathcal O_X$ and $\Omega_X$, but this would give some sort of 3-dimensional complex, and I do not know how to take the cohomology of such a thing.
My guess is that I have some gross misunderstanding, and if anyone could help me with that it would be much appreciated.