Weil's extension theorem for rational maps into smooth separated group schemes

Work locally on $S$ and $Z$, with $S$ affine and $Z$ of finite type, so the finite-type descent lemma [F1] applies. The total spaces $Z$ and $Z{\times }_{S}Z$ are regular by [F3]. Form $v(z_1,z_2)=u(z_1)u(z_2{)}^{-1}$ on $U{\times }_{S}U$ and let $V$ be its maximal domain. On $V\cap {\mathrm{\Delta }}_Z$, $v$ is the unit: it is the unit on the dense open $U\subset {\mathrm{\Delta }}_Z$, so separatedness gives equality wherever both morphisms are defined.

Suppose $x\in {\mathrm{\Delta }}_Z\setminus V$, with image $s\in S$. Choose an affine open $H\subset G$ containing $e(s)$ and shrink around $s$ so $e$ lands in $H$. Choose an integral regular affine neighbourhood $W$ of $x$ in $Z{\times }_{S}Z$. The open $V\cap W\cap v^{-1}(H)$ is nonempty: every neighbourhood of $x$ meets ${\mathrm{\Delta }}_Z\cap U$, where $v=e$. It is therefore dense in $W$ and represents an ordinary rational map $v^{\mathrm{\prime }}:W⇢H$. Let $V^{\mathrm{\prime }}$ be its maximal domain. Then $V^{\mathrm{\prime }}\subset V\cap W$, and $V^{\mathrm{\prime }}\cap {\mathrm{\Delta }}_Z=V\cap W\cap {\mathrm{\Delta }}_Z$, since at a diagonal point where $v$ is defined its value lies in $H$. By [F2], $F^{\mathrm{\prime }}=W\setminus V^{\mathrm{\prime }}$ is pure codimension one. Its intersection with the diagonal is contained in ${\mathrm{\Delta }}_Z\setminus U$, which has codimension at least two in ${\mathrm{\Delta }}_Z$.

At $x$ the reduced support of $F^{\mathrm{\prime }}$ is cut out by a product $f$ of prime elements in the regular local UFD ${\mathcal{O}}_{W,x}$. Its restriction to the regular local ring of the diagonal is nonzero, because $U$ is dense in the diagonal, and is a nonunit, because $x\in F^{\mathrm{\prime }}$. The principal ideal theorem [F3] then gives a codimension-one component of $F^{\mathrm{\prime }}\cap {\mathrm{\Delta }}_Z$ locally at $x$, contradicting step 2.1. Hence $V$ contains the diagonal.

Put $Z^{\mathrm{\prime }}=V\cap (Z{\times }_{S}U)$. Its first projection $f:Z^{\mathrm{\prime }}\to Z$ is flat, as a restriction of a smooth projection. For every geometric point $z$ of $Z$, the open $V_z$ in the corresponding fibre of the second factor contains the diagonal point and is nonempty; it meets the fibrewise dense open $U$, so $f$ is surjective. Thus $f$ is faithfully flat, and $(z_1,z_2)\mapsto v(z_1,z_2)u(z_2)$ on $Z^{\mathrm{\prime }}$ represents $u\circ f$ everywhere. Both $Z^{\mathrm{\prime }}$ and $Z$ are smooth of finite type over the locally Noetherian base in this local calculation, so [F1] descends it to a morphism $Z\to G$. These local extensions glue uniquely, since they agree on the schematically dense domain of $u$ and $G$ is separated.