Let $X=\mathbb P^1\times\mathbb P^1$, and let $\pi_1$ and $\pi_2$ be the projection maps. For each $a,b\in\mathbb Z$, we have a sheaf of $\mathcal O_X$-modules $\mathscr F_{a,b} = \pi_1^*\mathcal O(a)\otimes \pi_2^*\mathcal O(b)$, and I want to compute its cohomology using the open affine with four open sets obtained from the usual cover $\{U_0,U_1\}$ of $\mathbb P^1$, or otherwise.
Unless I am missing something, on a product $U=U_i\times U_j$, $\mathscr F_{a,b}(U)$ consists of bihomogeneous quotients $f(x)g(y)/x_i^ry_j^s$ such that $\deg f = r+a$ and $\deg g = s+b$. On sets of the form $U_{ij}\times U_k$ and the remaining others there is an analogous description.
Using the above, I was trying to compute $H^*(X,\mathscr F_{a,b})$, but quickly run into cumbersome computations. I did get that $d^2$ is zero since all triple and cuadruple intersections are the same, so $d^2=0$ since there are four triple intersections and everything cancels, hence $H^3$ is just $O(a)(U_{01})\otimes O(b)(U_{01})$ unless I am missing something.
I can also describe the kernels of $d^0$ and $d^1$, so probably after some long computations arrive at an answer. Does anyone have a hint on how to move on? Perhaps a more clever approach? Perhaps the Segre embedding could help out here?
Note. This is the last exercise in Chapter 8 of these notes.
The following two references deal with this: Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves, $\S4$, by Kempf, and A Kunneth formula for coherent algebraic sheaves, by Sampson and Washnitzer.