Fibres of abelian schemes and unit-preserving morphisms

The fibre $A_s=A{\times }_S\mathrm{Spec}\kappa (s)$ is smooth, proper and geometrically connected of dimension $g_A$ over $\kappa (s)$ by [F1], hence an abelian variety of dimension $g_A$; this is (a).

For commutativity, let $c:A{\times }_{S}A\to A$ be the commutator morphism $c(a,b)=ab{a}^{-1}{b}^{-1}$, using the group law; it sends the unit sections to the unit. For each geometric point $\bar{t}$ of the base, the fibre of $A{\times }_{S}A\to S$ over $\bar{t}$ is $A_{\bar{t}}{\times }_{\bar{t}}{A}_{\bar{t}}$, and by the field-level commutativity [F2] the commutator is constant, equal to the identity, on each geometric fibre of the second projection; by [F3] applied to the base change $A_A\to A$ (second projection), $c$ factors through the base, and evaluating at the first unit section gives $c=1$, hence $ab=ba$ as morphisms. This proves the first claim of (b), including nilpotents.

For the inverse: $m({\mathrm{id}}_A,-1_A)$ and $e\circ f$ agree on the closed subscheme $A$ by the group axioms, so the inverse is $-1_A$ as defined; this is the second claim of (b).

For (c), let $u:A\to B$ satisfy $u\circ e_A=e_B$ and consider the defect morphism $d(a,b)=u(a+b)-u(a)-u(b)$ on $A{\times }_{S}A$, using the group law of $B$. On each geometric fibre of the first projection, the field-level pointed-morphism theorem [F2] makes $d$ constant, equal to $0$; by [F3] it factors through the base and evaluation at $a=e_A$ gives $d\equiv 0$, so $u$ is additive; compatibility with the unit is assumed, so $u$ is a homomorphism of $S$-group schemes. For (d), two group structures on the same $S$-scheme with the same unit section have an identity morphism which preserves the unit, hence is a homomorphism by (c), and being an isomorphism of underlying schemes it identifies the two structures.