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. $\square$
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)