Push forward and pullback in products I am reading this Questions about Serre duality, and there is one part in the answer that I'd like to know how it works. But after many tries I didn't get anywhere. So here is the problem. 
Let $X$ and $B$ be algebraic varieties over an algebraically closed field, $\pi_1$ and $\pi_2$ be the projections from $X\times B$ onto $X$ and $B$, respectively.  Then it was claimed that $R^q\pi_{2,*} \pi_1^* \Omega_X^p \cong H^q(X, \Omega^p_X)\otimes \mathcal{O}_B$. 
I am guessing it works for any (quasi)coherent sheaf on $X$. 
Basically, I have two tools available, either Proposition III8.1 of Hartsshorne or going through the definition of the derived functors.  
Thank you.  

Using flat base-change (Prop. III.9.3 of Hartshorne), one sees that $$R^1\pi_{2 *} \pi_1^*\Omega_X^p = \pi_1^* H^q(X,\Omega^p_X) = H^q(X,\Omega^p_X)\otimes \mathcal O_B.$$
