The Stacks project

Lemma 83.5.5. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\mathop{\mathrm{Spec}}(k) \to B$ be a geometric point of $B$. Let $j : R \to U \times _ B U$ be a pre-equivalence relation over $B$. In this case the weak orbit of $\overline{u} \in U(k)$ is simply

\[ \{ \overline{u}' \in U(k) \text{ such that } \exists \overline{r} \in R(k), \ s(\overline{r}) = \overline{u}, \ t(\overline{r}) = \overline{u}' \} \]

and the orbit of $\overline{u} \in U(k)$ is

\[ \{ \overline{u}' \in U(k) : \exists \text{ field extension }K/k, \ \exists \ r \in R(K), \ s(r) = \overline{u}, \ t(r) = \overline{u}'\} \]

Proof. This is true because by definition of a pre-equivalence relation the image $j(R(k)) \subset U(k) \times U(k)$ is an equivalence relation. $\square$

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