Lemma 78.19.4 (044M)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.19: Quotient sheaves
Lemma 78.19.4 (cite)

previous definition
next lemma

numberstags

history
statistics

Lemma 78.19.4. In the situation of Definition 78.19.1. Assume there is an algebraic space $M$ over $S$ , and a morphism $U \to M$ such that

the morphism $U \to M$ equalizes $s, t$ ,
the map $U \to M$ is a surjection of sheaves, and
the induced map $(t, s) : R \to U \times _ M U$ is a surjection of sheaves.

In this case $M$ represents the quotient sheaf $U/R$ .

Proof. Condition (1) says that $U \to M$ factors through $U/R$ . Condition (2) says that $U/R \to M$ is surjective as a map of sheaves. Condition (3) says that $U/R \to M$ is injective as a map of sheaves. Hence the lemma follows. $\square$

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