## Tag `02VG`

Chapter 38: Groupoid Schemes > Section 38.20: Quotient sheaves

Definition 38.20.1. Let $\tau$, $S$, and the pre-relation $j : R \to U \times_S U$ be as above. In this setting the

quotient sheaf $U/R$associated to $j$ is the sheafification of the presheaf (38.20.0.1) in the $\tau$-topology. If $j : R \to U \times_S U$ comes from the action of a group scheme $G/S$ on $U$ as in Lemma 38.16.1 then we sometimes denote the quotient sheaf $U/G$.

The code snippet corresponding to this tag is a part of the file `groupoids.tex` and is located in lines 3385–3394 (see updates for more information).

```
\begin{definition}
\label{definition-quotient-sheaf}
Let $\tau$, $S$, and the pre-relation $j : R \to U \times_S U$ be as above.
In this setting the {\it quotient sheaf $U/R$} associated
to $j$ is the sheafification of the presheaf
(\ref{equation-quotient-presheaf}) in the $\tau$-topology.
If $j : R \to U \times_S U$ comes from the action of a group scheme
$G/S$ on $U$ as in Lemma \ref{lemma-groupoid-from-action} then we
sometimes denote the quotient sheaf $U/G$.
\end{definition}
```

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