Cube-derived square over DVR

Write the cube identity for $\mathcal{L}$ in the standard form $\mathcal{L}(x+y+z)\otimes \mathcal{L}(z)\cong \mathcal{L}(x+z)\otimes \mathcal{L}(y+z)\otimes \mathcal{L}(x+y)\otimes \mathcal{L}(x{)}^{-1}\otimes \mathcal{L}(y{)}^{-1},$ read as an isomorphism of pullbacks on $A{\times }_{K}A{\times }_{K}A$ modulo the constant identity-fibre factors. This is a literal line-bundle pullback identity, and specialising $z=0$ and cancelling the constant factors gives the theorem of the square on the first two factors, proving (a) over $K$ without any Picard or duality input.

Now let $\mathcal{L}$ be an invertible sheaf on $H$ and consider the square defect line bundle on $H^0{\times }_R{H}^0{\times }_{R}H$: the restriction of the square identity to the generic fibre is supplied by step 1.1 for the abelian generic fibre, and the difference of the two sides extends to a line bundle on the smooth total space. Extend the generic base line bundle on $H^0{\times }_R{H}^0$ by regular divisor closure using [F3]: the closure of a generic Cartier divisor is Cartier because the regular local rings of the smooth total spaces are UFDs, and Hartogs extends the defining equations in codimension one. The residual square obstruction is then a line bundle with a vertical divisor.

Because $H^0$ has geometrically irreducible fibres by [F2], every vertical prime divisor on $H^0{\times }_R{H}^0{\times }_{R}H$ is the inverse image of a special-fibre component of $H$, hence is pulled back from the last factor. Restricting the square obstruction to $(\text{unit},\text{unit},{\mathrm{id}}_H)$ makes the square identity trivial, so the residual line bundle pulled back from $H$ is pulled back from $R$; it can therefore be absorbed into the base line bundle on $H^0{\times }_R{H}^0$. Hence the square identity holds for $\mathcal{L}$ on $H$ with the $H^0$-translation action, proving (b).

The argument uses only the published cube theorem, divisor closure in regular total spaces and the component structure of [F2]; no Picard scheme, dual abelian variety, Chevalley decomposition or Raynaud theorem is used. The same statement applies after the base changes used later, since the hypotheses are stable under flat base change of DVRs.