For an abelian scheme, $R^pf_* \Omega^q$ is locally free and its formation is compatible with any base change Let $k$ be a field, $\bar{R} \to R$ a local homomorphism of artinian local rings with the residue fields $k$, $I$ its kernel, $A/R$ an abelian scheme, and $\mathscr{T}$ its tangent sheaf.
Let $A_0 = A \times_R k$.
Assume that $\mathfrak{m}_\bar{R} I = 0$.
Then $H^2(A, \mathscr{T}_{A/R} \otimes_R I) \cong H^2(A_0, \mathscr{T}_{A_0/k}) \otimes_k I$?
This is a part of the proof of (2.2.4.1) of Kai-Wen Lan's "Arithmetic Compactifications of PEL-Type Shimura Varieties".
To show it, I need the following proposition:

Let $S$ be a scheme, $f : A \to S$ an abelian scheme of relative dimension $g$.
  Then the sheaf $R^pf_* \Omega^q$ is locally free.
  And this formation commutes with any base change.

Are there "elementary" proof of this?
I know this is (2.5.2) of Berthelot, Breen, Messing's Théorie de Dieudonné Cristalline.
But its proof is too hard for me, since it heavily relies on the theory which I don't know.
And I know that this post shows it elementary.
But it uses the formally smoothness and the pro-representability of the deformation of "abelian schemes + polarization", which is what I want to show using this highlighted statement.
So it is a circular reasoning for me.
 A: Let $A/R$ be an abelian scheme of relative dimension $g$ over an Artinian local ring $(R, \mathfrak m, \kappa)$. I am going to give you a proof that works if the characteristic of $\kappa$ is not $2$.
Denote $f : A \to \text{Spec}(R)$ the structure morphism. By the usual trick (see for example here) we have $\Omega_{A/R} \cong \mathcal{O}_A^{\oplus g}$. Thus $\Omega_{A/R}^q$ is isomorphic to the free $\mathcal{O}_A$-module of rank ${g \choose q}$. Hence it suffices to prove that $H^i(A, \mathcal{O}_A)$ is a free $R$-module of rank ${g \choose i}$. Namely, we already know that formation of $K = Rf_*\mathcal{O}_A$ in the derived category $D(R)$ commutes with base change (by very general cohomology and base change results, see for example the exposition in Mumford's book on Abelian varieties) and freeness of its cohomology will imply it is the direct sum of its cohomology sheaves.
Denote $[2] : K \to K$ the pullback by multiplication by $2$ on $A$. By cohomology and base change (see above) we know that $K \otimes_R^\mathbf{L} \kappa$ is isomorphic to $\wedge^*(\kappa^{\oplus g})$. It follows that $K$ can be represented in $D(R)$ by a complex of the shape
$$
K^\bullet :
R \to R^{\oplus g} \to \ldots \to R^{\oplus g} \to R
$$
See for example here. Moreover, the map $[2] : K \to K$ in $D(R)$ can be represented by a map of complexes $t^\bullet : K^\bullet \to K^\bullet$ by usual homological algebra. Calculating on the special fibre we see that $t^i \bmod \mathfrak m$ is multiplication by $2^i$ on $\wedge^i(\kappa)$. A bit of elementary algebra then shows that the differentials of $K^\bullet$ have to be zero (look at what happens to the ``leading terms'').
PS: In char 2 you may be able to use the trick with the shearing map, but I didn't try.
