Calculating cohomology of $\mathscr{O}(d,e)$ on $\mathbb{P}^n\times\mathbb{P}^m$. Fix a Noetherian ring $A$, so that we have the twisting sheafs $\mathscr{O}_{\mathbb{P}^n}(d)$ and $\mathscr{O}_{\mathbb{P}^m}(e)$ on $\mathbb{P}_A^n$ and $\mathbb{P}_A^m$, respectively. If we fix the projection maps $\pi_1:\mathbb{P}^n\times\mathbb{P}^m\to\mathbb{P}^n$ and $\pi_2:\mathbb{P}^n\times\mathbb{P}^m\to\mathbb{P}^m$ , then we define the double twisting sheaf $$\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^m}(d,e) = \pi_1^*\mathscr{O}_{\mathbb{P}^n}(d)\otimes_{\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^m}}\pi_2^*\mathscr{O}_{\mathbb{P}^m}(e)$$ then is there any simple statement about the cohomology $H^i(\mathbb{P}^n\times\mathbb{P}^m,\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^m}(d,e))$? I'm not so interested in the proof (if it's long), I'm more interested in having a reference for this.
Additionally, is is true that $\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^m}(0,0)\cong\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^m}$?
 A: There is a Künneth Formula for sheaf cohomology, see this MathOverflow question for example which asks for a reference. So we have
$$H^i(\mathbb{P}^n\times\mathbb{P}^m, \pi_1^*\mathscr{O}_{\mathbb{P}^n}(d)\otimes_{\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^m}}\pi_2^*\mathscr{O}_{\mathbb{P}^m}) \cong \bigoplus_{p+q=i}H^p(\mathbb{P}^n, \mathscr{O}_{\mathbb{P}^n}(d))\otimes H^q(\mathbb{P}^m, \mathscr{O}_{\mathbb{P}^m}(e)).$$
The cohomology of line bundles on complex projective space is fairly simple: 


*

*if $k \geq 0$, $\dim H^0(\mathbb{P}^n, \mathscr{O}_{\mathbb{P}^n}(k)) = \binom{n+k}{k}$,

*if $k \leq -n - 1$, $\dim H^n(\mathbb{P}^n, \mathscr{O}_{\mathbb{P}^n}(k)) = \binom{-k-1}{-k-1-n}$,

*in all other cases, $H^q(\mathbb{P}^n, \mathscr{O}_{\mathbb{P}^n}(k)) = 0$.


See page $8$ of Okonek, Schneider, & Spindler's Vector Bundles on Complex Projective Spaces. I don't know if there are analogous results for $\mathbb{P}_A^n$.
These facts combine to give you the desired cohomology groups, at least for $\mathbb{P}^n_{\mathbb{C}}$, for any $d$ and $e$.
