Lemma 40.11.8. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a perfect field. Let $Z \subset U \times _ S U$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image 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 40.11.4. But we can also prove it directly as follows.

We first explain why the statement makes sense. Since $U$ is the spectrum of a perfect field $k$, the scheme $Z$ is geometrically reduced over $k$ (via either projection), see Varieties, Lemma 33.6.3. Hence the scheme $Z \times _{\text{pr}_1, U, \text{pr}_0} Z \subset Z$ is reduced, see Varieties, Lemma 33.6.7. Hence by Lemma 40.11.7 we see that $\text{pr}_{02}$ induces a morphism $Z \times _{\text{pr}_1, U, \text{pr}_0} Z \to Z$. Finally, it is clear that $\Delta _{U/S}$ factors through $Z$ and that the map $\sigma : U \times _ S U \to U \times _ S U$, $(x, y) \mapsto (y, x)$ preserves $Z$. Since $(U, U \times _ S U, \text{pr}_0, \text{pr}_1, \text{pr}_{02}, \Delta _{U/S}, \sigma )$ satisfies the axioms of a groupoid, it follows that after restricting to $Z$ they satisfy the axioms.
$\square$

## Comments (0)