Ytukyg

Let $X=mathbb P^1timesmathbb P^1$, and let $pi_1$ and $pi_2$ be the projection maps. For each $a,binmathbb Z$, 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 $U=U_itimes U_j$, $mathscr F_{a,b}(U)$ 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.