a question on higher direct images in a product I want to compute $H^i(X,\pi_1^* O(a)\otimes \pi_2^*O(b))$ where $X=\mathbb{P}^1\times \mathbb{P}^1$ and $\pi_i$ is the projection onto the $i$-th $\mathbb{P}^1$ factor. I realize this is a special case of the Kunneth formula, but I'm trying to get some experience with higher direct images. 
So I want to approach it by Leray's spectral sequence (coboundary maps are zero). So I need to compute $H^i(X,R^1{\pi_{1}}_*(\pi_2^*O(b))\otimes O(a))$ (using the projection formula), but it's not getting anywhere. Can someone show (or give a reference for) the following computations?
$$
\bullet H^1(X,R^1{\pi_{1}}_*(\pi_2^*O(b))\otimes O(a))=H^1(\mathbb{P}^1,O(b))\otimes H^1(\mathbb{P}^1,O(a))
\bullet H^0(X,R^1{\pi_{1}}_*(\pi_2^*O(b))\otimes O(a))=H^1(\mathbb{P}^1,O(b))\otimes H^0(\mathbb{P}^1,O(a))
$$
 A: We want to show that 
$$ R^1{\pi_1}_*(\pi_2^* \mathcal{O}(b))=H^1(\mathbb{P}^1,\mathcal{O}(b))\otimes \mathcal{O}_{\mathbb{P}^1} $$
Consider the fibered square
$$
\require{AMScd}
\begin{CD}
\mathbb{P}^1 \times \mathbb{P}^1 @>{\pi_2}>> \mathbb{P^1}\\
@V{\pi_1}VV @VV{g}V \\
\mathbb{P}^1 @>{f}>> \text{Spec } k
\end{CD}
$$
then, as in exercise 2.B.(a) of http://math.stanford.edu/~vakil/0708-216/216class38.pdf we have a morphism of sheaves on $\mathbb{P}^1$
$$ f^*(R^ig_* \mathcal{O}(b)) = H^1(\mathbb{P}^1,\mathcal{O}(b))\otimes \mathcal{O}_{\mathbb{P^1}} \overset{\phi}{\to} R^1{\pi_1}_*(\pi_2^* \mathcal{O}(b))   $$
and we want to show that this is an isomorphism. It is enough to check this for any affine open subset $\text{Spec} A\subseteq \mathbb{P}^1$. The restriction to $\text{Spec} A$ of the morphism $\phi$ is the morphism induced by the cartesian square
$$
\require{AMScd}
\begin{CD}
\text{Spec }A \times \mathbb{P}^1 @>{\pi_2}>> \mathbb{P^1}\\
@V{\pi_1}VV @VV{g}V \\
\text{Spec }A @>{f}>> \text{Spec } k
\end{CD}
$$
and now, since the map $f\colon \text{Spec } A \to \text{Spec } k$ is affine and flat, we can use exercise 2.B of http://math.stanford.edu/~vakil/0708-216/216class38.pdf that tells us that $\phi_{|\text{Spec } A}$ is an isomorphism.
Since this holds for every open subset $\text{Spec} A\subseteq \mathbb{P}^1$ the claim follows. 
