The Stacks project

Lemma 40.10.3. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \mathop{\mathrm{Spec}}(k)$ with $k$ a field. For any points $r, r' \in R$ there exists a field extension $k'/k$ and points $r_1, r_2 \in R \times _{s, \mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k')$ and a diagram

\[ \xymatrix{ R & R \times _{s, \mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k') \ar[l]_-{\text{pr}_0} \ar[r]^\varphi & R \times _{s, \mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k') \ar[r]^-{\text{pr}_0} & R } \]

such that $\varphi $ is an isomorphism of schemes over $\mathop{\mathrm{Spec}}(k')$, we have $\varphi (r_1) = r_2$, $\text{pr}_0(r_1) = r$, and $\text{pr}_0(r_2) = r'$.

Proof. This is a special case of Lemma 40.7.1 parts (1) and (2). $\square$

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