Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes

Question

Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism $$ \kappa : T_{S/k} \to R^1p_*T_{A/S} $$ where $T_{S/k}$ (resp. $T_{A/S}$) denotes the dual module of $\Omega^1_{S/k}$ (resp. $\Omega^1_{A/S}$).

Let $\text{Lie}_SA$ be the $\mathcal{O}_S$-dual of $p_*\Omega^1_{A/S}$. If I didn't misunderstand it, in Faltings-Chai, page 80, one identifies $R^1p_*T_{A/S}$ with $$ \text{Lie}_SA \otimes_{\mathcal{O}_S} R^1p_*\mathcal{O}_A $$ and I recall that $R^1p_*\mathcal{O}_A$ is naturally isomorphic to $\text{Lie}_SA^t$, where $A^t\to S$ denotes the dual abelian scheme.

The authors seem to give no justification for the isomorphism $R^1p_*T_{A/S} \cong \text{Lie}_SA \otimes R^1p_*\mathcal{O}_A$. How to prove it?

Answer 1

Posting my comment as an answer:

Since $T_{A/S}$ is trivial locally on $S$, we have $T_{A/S} = p^* p_* T_{A/S} = p^* {\rm Lie}_S A$. By the projection formula, we get $$ R^1 p_* T_{A/S} = R^1 p_* p^* {\rm Lie}_S A = (R^1 p_* \mathcal{O}_A)\otimes {\rm Lie}_S A. $$