Coherent cohomology of an abelian scheme and base change 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 $R^i f_*$.
 A: It suffices to prove the vector bundle property for all cohomological degrees when the base is an artin local ring $R$ (by considerations with direct limits to pass to the noetherian case and then using the standard base change formalism).  We can make a flat local extension $R \rightarrow R'$ to an artin local ring with algebraically closed residue field, so we may assume the residue field $k$ of $R$ is algebraically closed.  If $k$ has characteristic 0 then we can make $R$ into a $k$-algebra and pass to the case $k = \mathbf{C}$, so the self-contained slick argument of Deligne from his Hodge degeneration paper can be applied to conclude the vector bundle property (as $\Omega^1$ is globally free for an abelian scheme over a local ring).  
If $k$ has characteristic $p > 0$ then by picking an ample line bundle on the special fiber and raising it to a sufficiently high $p$-power (depending on the nilpotence order of the maximal ideal of $R$) we can lift it to the abelian $R$-scheme so as to make a polarization over $R$. (This trick doesn't work for residue characteristic 0.)  It is harmless to pass to $(A \times A^{\vee})^4$ (as a direct summand of a finite free $R$-module is finite free), so by Zarhin's trick we can assume $A$ is principally polarized over $R$.  But as explained in Theorem 2.4.1 of Oort's 1970 article "Finite group schemes, local moduli for abelian varieties, and lifting problems", the formal deformation ring of an abelian variety equipped with a polarization of degree prime to the characteristic is formally smooth. (This is a theorem of Grothendieck.) Hence, by formal GAGA we get a lift over a formal power series ring over $W(k)$.  Now the base is reduced, so we can use Grauert's theorem to conclude.
