The Stacks project

Lemma 39.7.13. Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$. Assume that $G$ is Jacobson and that all closed points are $k$-rational. Let $T = \mathop{\mathrm{Spec}}(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any morphism $f : T \to G$ there exists an affine open $U \subset G$ containing $f(T)$.

Proof. Let $G^0 \subset G$ be the closed subgroup scheme found in Proposition 39.7.11. The first two paragraphs serve to reduce to the case $G = G^0$.

Observe that $T$ is a directed inverse limit of finite topological spaces (Limits, Lemma 32.4.6), hence profinite as a topological space (Topology, Definition 5.22.1). Let $W \subset G$ be a quasi-compact open containing the image of $T \to G$. After replacing $W$ by the image of $G^0 \times W \to G \times G \to G$ we may assume that $W$ is invariant under the action of left translation by $G^0$, see Lemma 39.7.2. Consider the composition

\[ \psi = \pi \circ f : T \xrightarrow {f} W \xrightarrow {\pi } \pi _0(W) \]

The space $\pi _0(W)$ is profinite (Topology, Lemma 5.23.9 and Properties, Lemma 28.2.4). Let $F_\xi \subset T$ be the fibre of $T \to \pi _0(W)$ over $\xi \in \pi _0(W)$. Assume that for all $\xi $ we can find an affine open $U_\xi \subset W$ with $F \subset U$. Since $\psi : T \to \pi _0(W)$ is universally closed as a map of topological spaces (Topology, Lemma 5.17.7), we can find a quasi-compact open $V_\xi \subset \pi _0(W)$ such that $\psi ^{-1}(V_\xi ) \subset f^{-1}(U_\xi )$ (easy topological argument omitted). After replacing $U_\xi $ by $U_\xi \cap \pi ^{-1}(V_\xi )$, which is open and closed in $U_\xi $ hence affine, we see that $U_\xi \subset \pi ^{-1}(V_\xi )$ and $U_\xi \cap T = \psi ^{-1}(V_\xi )$. By Topology, Lemma 5.22.4 we can find a finite disjoint union decomposition $\pi _0(W) = \bigcup _{i = 1, \ldots , n} V_ i$ by quasi-compact opens such that $V_ i \subset V_{\xi _ i}$ for some $i$. Then we see that

\[ f(T) \subset \bigcup \nolimits _{i = 1, \ldots , n} U_{\xi _ i} \cap \pi ^{-1}(V_ i) \]

the right hand side of which is a finite disjoint union of affines, therefore affine.

Let $Z$ be a connected component of $G$ which meets $f(T)$. Then $Z$ has a $k$-rational point $z$ (because all residue fields of the scheme $T$ are isomorphic to $k$). Hence $Z = G^0 z$. By our choice of $W$, we see that $Z \subset W$. The argument in the preceding paragraph reduces us to the problem of finding an affine open neighbourhood of $f(T) \cap Z$ in $W$. After translation by a rational point we may assume that $Z = G^0$ (details omitted). Observe that the scheme theoretic inverse image $T' = f^{-1}(G^0) \subset T$ is a closed subscheme, which has the same type. After replacing $T$ by $T'$ we may assume that $f(T) \subset G^0$. Choose an affine open neighbourhood $U \subset G$ of $e \in G$, so that in particular $U \cap G^0$ is nonempty. We will show there exists a $g \in G^0(k)$ such that $f(T) \subset g^{-1}U$. This will finish the proof as $g^{-1}U \subset W$ by the left $G^0$-invariance of $W$.

The arguments in the preceding two paragraphs allow us to pass to $G^0$ and reduce the problem to the following: Assume $G$ is irreducible and $U \subset G$ an affine open neighbourhood of $e$. Show that $f(T) \subset g^{-1}U$ for some $g \in G(k)$. Consider the morphism

\[ U \times _ k T \longrightarrow G \times _ k T,\quad (t, u) \longrightarrow (uf(t)^{-1}, t) \]

which is an open immersion (because the extension of this morphism to $G \times _ k T \to G \times _ k T$ is an isomorphism). By our assumption on $T$ we see that we have $|U \times _ k T| = |U| \times |T|$ and similarly for $G \times _ k T$, see Lemma 39.7.12. Hence the image of the displayed open immersion is a finite union of boxes $\bigcup _{i = 1, \ldots , n} U_ i \times V_ i$ with $V_ i \subset T$ and $U_ i \subset G$ quasi-compact open. This means that the possible opens $Uf(t)^{-1}$, $t \in T$ are finite in number, say $Uf(t_1)^{-1}, \ldots , Uf(t_ r)^{-1}$. Since $G$ is irreducible the intersection

\[ Uf(t_1)^{-1} \cap \ldots \cap Uf(t_ r)^{-1} \]

is nonempty and since $G$ is Jacobson with closed points $k$-rational, we can choose a $k$-valued point $g \in G(k)$ of this intersection. Then we see that $g \in Uf(t)^{-1}$ for all $t \in T$ which means that $f(t) \in g^{-1}U$ as desired. $\square$


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