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.