Strict law graph calculus

For an $R$-scheme $T$, let ${\mathrm{B}\mathrm{i}\mathrm{r}}_{X\mathrm{/}R}(T)$ be the group of $T$-birational self-maps of $X_T=X{\times }_{R}T$. Strictness [F1] makes each section $a\in X(T)$ act by a $T$-birational left translation ${\tau }_a$, naturally in $T$. This defines $X\to {\mathrm{B}\mathrm{i}\mathrm{r}}_{X\mathrm{/}R}$. It is a monomorphism: if ${\tau }_a={\tau }_b$, then on the common dense domain the maps $({\tau }_a,\mathrm{id})$ and $({\tau }_b,\mathrm{id})$ from $T{\times }_{R}X$ to $X_T{\times }_T{X}_T$ agree. They factor as the universal right translation $(x,y)\mapsto (m(x,y),y)$ after $(a,\mathrm{id})$ and $(b,\mathrm{id})$, respectively. The right translation is an open immersion by [F1], so cancellation gives $(a,\mathrm{id})=(b,\mathrm{id})$ on that dense open; [F2] makes the open schematically dense, hence the equality holds on $T{\times }_{R}X$. Since $T{\times }_{R}X\to T$ is faithfully flat, [F4] gives $a=b$. Associativity also gives ${\tau }_a{\tau }_b={\tau }_c$ whenever $m(a,b)=c$ is defined.

Let $\mathrm{\Gamma }\subseteq X^3$ be the schematic closure of the graph of $m{\mathrm{\mid }}_U$, with coordinates $(x,y,z)$. On the open locus in $X^4$ where $(y,w),(x,m(y,w)),(z,w)\in U$, the maps $m(x,m(y,w))$ and $m(z,w)$ are defined. This locus is dense over the first three coordinates by the two-projection density of $U$ and the open-immersion property of the strict translations in [F1]. On the intersection with the graph over $U$, associativity makes the two morphisms agree on the dense open where the associative identity is represented; as the target $X$ is separated, [F2] gives equality on their common domain. The graph over $U$, after product with the flat scheme $X$, is schematically dense in $\mathrm{\Gamma }{\times }_{R}X$; hence [F2] extends this equality to the whole common domain in $\mathrm{\Gamma }{\times }_{R}X$. Pulling back along any $T$-valued triple $(a,b,c):T\to \mathrm{\Gamma }$ shows ${\tau }_a{\tau }_b={\tau }_c$ as $T$-birational maps. In particular, if $m(a,b)=c$ is defined, then $t_a{t}_b=t_c$, proving (c); conversely, for fixed any two coordinates of a triple in $\mathrm{\Gamma }(T)$, the third is unique by the monomorphism of step 1.1 and invertibility in ${\mathrm{B}\mathrm{i}\mathrm{r}}_{X\mathrm{/}R}(T)$.

Each projection $q_{ij}:\mathrm{\Gamma }\to X^2$ is a monomorphism: for every $T$, a triple $(a,b,c)\in \mathrm{\Gamma }(T)$ satisfies the translation relation of step 2.1, so any two of its coordinates determine the third. It is finite type, hence quasi-finite by [F3]. On the graph over $U$, $q_{12}$ is the identity onto $U$, while $q_{13}$ and $q_{23}$ are the universal left and right translations; these are open immersions with dense images by [F1]. Thus each $q_{ij}$ is birational on every component it meets. The target $X^2$ is normal because it is smooth over the regular DVR $R$. Applying [F3] componentwise shows each $q_{ij}$ is an open immersion. Its image contains respectively $U$, the left-translation image, and the right-translation image, all dense over both projections, so the images are dense over both projections. This proves (a).

The open immersion $q_{12}$ identifies $\mathrm{\Gamma }$ with an open $W\subseteq X^2$, and the third coordinate defines $m$ on $W$. This is the full domain: any local morphism extending $m$ has graph in the closed $\mathrm{\Gamma }$, since it agrees with the graph over $U$ on a schematically dense open. For a section $a:T\to X$ and a $T$-point $b$ of $X_T$, if $m$ is defined at $(a,b)$, pullback along $a\times \mathrm{id}$ shows $t_a$ is defined at $b$. Conversely, if $t_a$ is defined at $b$, choose an open neighbourhood $D$ on which it is a morphism and consider $D\to X^3$, $y\mapsto (a(p_T(y)),y,t_a(y))$, where $p_T:X_T\to T$. On the T-dense open where the strict law defines $t_a$, this graph factors through $\mathrm{\Gamma }$; universal schematic density [F2] therefore makes it factor through $\mathrm{\Gamma }$ on $D$. Hence $(a,b)\in W$, so the law is defined there. This proves (b) for every test scheme. For an $R$-section $a$, the graph closure ${\mathrm{\Gamma }}_a\subseteq X^2$ of $t_a$ maps into $\mathrm{\Gamma }$. Its two projections are finite-type monomorphisms by the monomorphism of $q_{12}$ and $q_{13}$, and are birational because $t_a$ is an $R$-birational map. The target $X$ is normal; [F3] makes both projections open immersions with dense images, as used for translate gluing.