The Stacks project

Lemma 40.11.6. Notation and assumptions as in Situation 40.11.1. Assume that $a : R_1 \to R_2$ is a quasi-compact morphism. Let $Z \subset R_2$ be the scheme theoretic image (see Morphisms, Definition 29.6.2) 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. The main difficulty is to show that $c_2|_{Z \times _{s_2, U, t_2} Z}$ maps into $Z$. Consider the commutative diagram

\[ \xymatrix{ R_1 \times _{s_1, U, t_1} R_1 \ar[r] \ar[d]^{a \times a} & R_1 \ar[d] \\ R_2 \times _{s_2, U, t_2} R_2 \ar[r] & R_2 } \]

By Varieties, Lemma 33.24.3 we see that the scheme theoretic image of $a \times a$ is $Z \times _{s_2, U, t_2} Z$. By the commutativity of the diagram we conclude that $Z \times _{s_2, U, t_2} Z$ maps into $Z$ by the bottom horizontal arrow. As in the proof of Lemma 40.11.4 it is also true that $i_2(Z) \subset Z$ and that $e_2$ factors through $Z$. Hence we conclude as in the proof of that lemma. $\square$

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