Let $f\colon A \rightarrow S$ be an abelian scheme of dimension $d$. I would like a reference or an argument for the fact that $R^1f_* \mathcal{O}_A$ is locally free of dimension $d$ and that its formation commutes with arbitrary base change. Using usual limit arguments one may reduce to the case when $S$ is the spectrum of a Noetherian ring. I would be especially happy if the answer could mention whether similar local freeness and cohomological flatness claims hold true for $R^if_* \mathcal{O}_A$ for arbitrary $i \ge 0$.

I think that the key input should be a theorem of Grauert reducing the claim to the fibers, but I couldn't isolate a precise reference that would apply in the desired generality. I know that there is a slew of related results in EGA III, section 7 but I have not studied them in detail, so due to heavy notation used there it is difficult for me to gauge their applicability. There are also improvements to this part of EGA in SGA 6, Expose III, but those are mostly formulated in terms of total derived functors, so I couldn't see how to extract information about a single $R^i f_*$.

Théorie de Dieudonné cristalline, II(LNM 930), Prop. 2.5.2 compute the De Rham cohomology of an abelian scheme - the proof uses a bit of spectral sequences. $\endgroup$ – ACL Dec 17 '14 at 23:17