Lemma 44.5.1 (0B9Q)—The Stacks project

Lemma 44.5.1. Let $k$ be a field. Let $G : (\mathit{Sch}/k)^{opp} \to \textit{Groups}$ be a functor. With terminology as in Schemes, Definition 26.15.3, assume that

$G$ satisfies the sheaf property for the Zariski topology,
there exists a subfunctor $F \subset G$ such that
1. $F$ is representable,
2. $F \subset G$ is representable by open immersion,
3. for every field extension $K$ of $k$ and $g \in G(K)$ there exists a $g' \in G(k)$ such that $g'g \in F(K)$.

Then $G$ is representable by a group scheme over $k$.

Proof. This follows from Schemes, Lemma 26.15.4. Namely, take $I = G(k)$ and for $i = g' \in I$ take $F_ i \subset G$ the subfunctor which associates to $T$ over $k$ the set of elements $g \in G(T)$ with $g'g \in F(T)$. Then $F_ i \cong F$ by multiplication by $g'$. The map $F_ i \to G$ is isomorphic to the map $F \to G$ by multiplication by $g'$, hence is representable by open immersions. Finally, the collection $(F_ i)_{i \in I}$ covers $G$ by assumption (2)(c). Thus the lemma mentioned above applies and the proof is complete. $\square$

Comments (2)

Comment #4852 by Kazuki Masugi on December 22, 2019 at 18:05

I cannot understand why the collection $(F_i)_{i\in I}$ covers $G$ .

Comment #4853 by Weixiao Lu on December 23, 2019 at 07:48

@#4852, for every $g \in G(T)$ , $\{U_{gg^\prime} \vert g^\prime \in G(k)\}$ is an open covering of $T$ such that $g|_{U_{gg^\prime}} \in F_{g^\prime}(U_{gg^\prime})$ , where $U_{gg^\prime}$ is defined as in Definition 01JI. ( $\{U_{gg^\prime} \vert g^\prime \in G(k)\}$ covers $T$ because for any point $x \in T$ there exists $g^\prime \in G(k)$ such that $j^\ast(g)g^\prime \in F(\mathop{\mathrm{Spec}}(\kappa(x)))$ by assumption (2)(c), where $j: \mathop{\mathrm{Spec}}(\kappa(x)) \to T$ . Then $x \in U_{gg^\prime}$ by Definition 01JI (3).)

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