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?



Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes
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
$$ R^1 p\_\* T\_{A/S} = R^1 p\_\* p^\* {\rm Lie}\_S A = (R^1 p\_\* \mathcal{O}\_A)\otimes {\rm Lie}\_S A. $$


