Birational group law

Let $\xi$ be a generic point of the special fibre of the first copy of $X$, and put $R^{\mathrm{\prime }}={\mathcal{O}}_{X,\xi }$ and $K^{\mathrm{\prime }}=\mathrm{Frac}{R}^{\mathrm{\prime }}$. The canonical map $\mathrm{Spec}{K}^{\mathrm{\prime }}\to X_K=A$ is an $A(K^{\mathrm{\prime }})$-point $a$ coming from the first, parameter factor. Apply [F1] to $t_a$ and $t_{-a}$ on the second copy $X_{R^{\mathrm{\prime }}}$. Their rational-domain open immersions are inverse on fibre-dense open subsets. By finite-presentation spreading (Finite-stage descent of finitely presented schemes and their morphisms) their domains, maps and inverse identities spread over a neighbourhood of $\xi$ in the parameter $X$. Thus $(x,y)\mapsto (x,m(x,y))$ and its inverse are defined near all special-fibre generic points of $X{\times }_{R}X$ projecting to $\xi$: after localization these are generic points of the special fibre of $X_{R^{\mathrm{\prime }}}$, and the local domains are $R^{\mathrm{\prime }}$-dense. Every special-fibre component of the product projects dominantly onto a special component of the first factor, since both factors are smooth and their fibre components are geometrically regular. Repeating for its finitely many $\xi$, and adjoining the generic translation and its inverse on the open generic fibre, gives fibre-dense domains for a rational map $\mathrm{\Phi }$ and its inverse. On overlaps the maps agree on the generic fibre and hence agree by separatedness and flatness. Restricting the two domains to where the compositions are defined gives inverse open immersions; these restrictions remain fibre-dense by the local inverse construction. Hence $\mathrm{\Phi }$ is $R$-birational, and its second projection defines an $R$-rational extension of $m_K$.

Apply the same argument with the second factor as parameter: its canonical $K^{\mathrm{\prime }}$-point, not a point of the variable first factor, supplies the translation. This constructs the other universal map $\mathrm{\Psi }(x,y)=(m(x,y),y)$ and its rational inverse. Both are $R$-birational on fibre-dense open domains.

The two constructions agree generically on the common dense open where both are defined, because both restrict to the generic multiplication of $A$; by separatedness of $X$ and schematic density of the domain ([F2]) they define a single $R$-rational map $m$. Associativity holds wherever the composed expressions are defined: both sides restrict to the associative law of $A$ on a schematically dense open, so by [F2] they agree; consequently $m$ is an $R$-birational group law with birational universal translations, as claimed.