The Stacks project

Lemma 77.22.3. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition 77.20.1 (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\mathit{Isom}(x, y)$ on $(\mathit{Sch}/T)_{fppf}$ has the following property: There exists a fppf covering $\{ T_ i \to T\} _{i \in I}$ such that $\mathit{Isom}(x, y)|_{(\mathit{Sch}/T_ i)_{fppf}}$ is representable by an algebraic space.

Proof. Follows immediately from Lemma 77.22.1 and the fact that both $x$ and $y$ locally in the fppf topology come from objects of $\mathcal{S}_ U$ by construction of the quotient stack. $\square$


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