Definition 39.20.1. 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 (39.20.0.1) 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 39.16.1 then we sometimes denote the quotient sheaf $U/G$.

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