Lemma 40.11.4. Notation and assumptions as in Situation 40.11.1. Assume that $k$ is perfect. Let $Z \subset R_2$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image of $a : R_1 \to R_2$. Then

\[ (U, Z, s_2|_ Z, t_2|_ Z, c_2|_ Z) \]

is a groupoid scheme over $S$.

**Proof.**
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 $Z \times _{s_2, U, t_2} Z \subset Z$ is reduced, see Varieties, Lemma 33.6.7. Hence by Lemma 40.11.3 we see that $c$ induces a morphism $Z \times _{s_2, U, t_2} Z \to Z$. Finally, it is clear that $e_2$ factors through $Z$ and that the map $i_2 : R_2 \to R_2$ preserves $Z$. Since the morphisms of the septuple $(U, R_2, s_2, t_2, c_2, e_2, i_2)$ satisfies the axioms of a groupoid, it follows that after restricting to $Z$ they satisfy the axioms.
$\square$

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