Separated translate gluing

Let $\mathrm{\Gamma }\subseteq X{\times }_{R}X$ be the graph closure of the section translation $t_a$; by [F1] its two projections are open immersions onto $R$-dense open subschemes. Gluing $X$ to a copy $X(a)$ along $\mathrm{\Gamma }$ is therefore gluing along an open subscheme, giving a smooth finite-type $R$-scheme $X^{\mathrm{\prime }}$ containing $X$ as an $R$-dense open subscheme; the closedness of the graph in the separated product makes $X^{\mathrm{\prime }}$ separated.

Write $j:X\stackrel{\sim }{\to }X(a)$ for the canonical copy map, which extends the left translation by $a$ on its original domain. Let $U$ be the strict-law domain in $X^2$, with translation images $V$ and $W$. On $U_1=(j\times \mathrm{id})(U)$ define $m^{\mathrm{\prime }}(j(x),y)=j(m(x,y))$. On $U_2\subset X\times X(a)$, consisting of $(x,j(y))$ with $(x,a)\in U$ and $(m(x,a),y)\in U$, define $m^{\mathrm{\prime }}(x,j(y))=m(m(x,a),y)$. Together with $m$ on $U$ these morphisms agree on overlaps by associativity and schematic density [F2], and give $m^{\mathrm{\prime }}$ on $U^{\mathrm{\prime }}=U\cup U_1\cup U_2$. For any fixed $y$, strictness makes the conditions on $x$ in $U_2$ dense: right translation by $a$ is birational, and the domain of right translation by $y$ is dense. Thus $U_2$ is dense along its second projection. The domains $U$ and $U_1$ are dense along the first projection; since $X$ is fibre-dense in $X^{\mathrm{\prime }}$, these facts make $U^{\mathrm{\prime }}$ dense along both projections of $(X^{\mathrm{\prime }}{)}^2$. No definition on the whole fourth chart $X(a{)}^2$ is needed for this density assertion.

The universal translations on each of $U,U_1,U_2$ are open immersions: on $U_1$ conjugate the original translations by the copy isomorphism $j$; on $U_2$ compose the original translations with the open-immersion right translation by $a$ and with $j$. Hence the glued translations are quasi-finite. They are birational and separated, and their source and target are regular componentwise; [F3] makes them open immersions. The left-translation image contains $V$ and $(j\times j)(V)$, so it is dense along both projections. The right-translation image contains $W$ and $(j\times \mathrm{id})(W)$, giving first-projection density and second-projection density over $X$. Over a second coordinate $j(y)$, its image from $U_2$ is the set of products $(xa)y$ on the dense domain just described; the composite of the birational right translations by $a$ and $y$ has dense image in $X$, hence in $X^{\mathrm{\prime }}$. This gives second-projection density over $X(a)$ as well. Thus $m^{\mathrm{\prime }}$ is strict. Associativity follows from its agreement with $m$ on schematically dense domains.