Lemma 40.11.6 (04QA)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 2: Schemes
Chapter 40: More on Groupoid Schemes
Section 40.11: Morphisms of groupoids on fields
Lemma 40.11.6 (cite)

numberstags

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. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment