Lemma 40.11.9. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field and assume $R$ is quasi-compact (equivalently $s, t$ are quasi-compact). Let $Z \subset U \times _ S U$ be the scheme theoretic image (see Morphisms, Definition 29.6.2) of $j = (t, s) : R \to U \times _ S U$. Then

\[ (U, Z, \text{pr}_0|_ Z, \text{pr}_1|_ Z, \text{pr}_{02}|_{Z \times _{\text{pr}_1, U, \text{pr}_0} Z}) \]

is a groupoid scheme over $S$.

**Proof.**
As $(U, U \times _ S U, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$ is a groupoid scheme over $S$ this is a special case of Lemma 40.11.6. But we can also prove it directly as follows.

The main difficulty is to show that $\text{pr}_{02}|_{Z \times _{\text{pr}_1, U, \text{pr}_0} Z}$ maps into $Z$. Write $U = \mathop{\mathrm{Spec}}(k)$. Denote $R_ s$ (resp. $Z_ s$, resp. $U^2_ s$) the scheme $R$ (resp. $Z$, resp. $U \times _ S U$) viewed as a scheme over $k$ via $s$ (resp. $\text{pr}_1|_ Z$, resp. $\text{pr}_1$). Similarly, denote ${}_ tR$ (resp. ${}_ tZ$, resp. ${}_ tU^2$) the scheme $R$ (resp. $Z$, resp. $U \times _ S U$) viewed as a scheme over $k$ via $t$ (resp. $\text{pr}_0|_ Z$, resp. $\text{pr}_0$). The morphism $j$ induces morphisms of schemes $j_ s : R_ s \to U^2_ s$ and ${}_ tj : {}_ tR \to {}_ tU^2$ over $k$. Consider the commutative diagram

\[ \xymatrix{ R_ s \times _ k {}_ tR \ar[r]^ c \ar[d]_{j_ s \times {}_ tj} & R \ar[d]^ j \\ U^2_ s \times _ k {}_ tU^2 \ar[r] & U \times _ S U } \]

By Varieties, Lemma 33.24.3 we see that the scheme theoretic image of $j_ s \times {}_ tj$ is $Z_ s \times _ k {}_ tZ$. By the commutativity of the diagram we conclude that $Z_ s \times _ k {}_ tZ$ maps into $Z$ by the bottom horizontal arrow. As in the proof of Lemma 40.11.8 it is also true that $\sigma (Z) \subset Z$ and that $\Delta _{U/S}$ factors through $Z$. Hence we conclude as in the proof of that lemma.
$\square$

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