Let $S$ be a smooth projective surface and $p:X\to S\times S$ be a blowup along the diagonal with the exceptional divisor $E$. How to compute $Rp_*\mathcal{O}_X(-2E)$?
Is it true that $p_*\mathcal{O}_X(-2E)=I^2$, where $I$ is the ideal sheaf of the diagonal?
You can think of sections of $\mathcal O_X(-2E)$ on an open set as rational functions with zero of order two along $E$. By definition, a section of its push forward on an open set $U\subseteq S\times S$ is a function in the neighborhood $p^{-1}U$ of $E$ which is zero of order at least $2$ on $E$. It then has to be regular on $U$, and vanish to second order on the blowup locus. So, yes, $p_*\mathcal O_X(-2E)=I^2$.
I think that $R^{>0}p_*\mathcal O_X(-nE)=0$ for all $n\geq 0$, but my brain is foggy right now, and I don't have a quick argument or reference for it.