The Stacks project

Lemma 94.18.1. Suppose given big sites $\mathit{Sch}_{fppf}$ and $\mathit{Sch}'_{fppf}$. Assume that $\mathit{Sch}_{fppf}$ is contained in $\mathit{Sch}'_{fppf}$, see Topologies, Section 34.12. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Let $f : (\mathit{Sch}'/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ the morphism of sites corresponding to the inclusion functor $u : (\mathit{Sch}/S)_{fppf} \to (\mathit{Sch}'/S)_{fppf}$. Let $\mathcal{X}$ be a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

1. if $\mathcal{X}$ is representable by some $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, then $f^{-1}\mathcal{X}$ is representable too, in fact it is representable by the same scheme $X$, now viewed as an object of $(\mathit{Sch}'/S)_{fppf}$,

2. if $\mathcal{X}$ is representable by $F \in \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$ which is an algebraic space, then $f^{-1}\mathcal{X}$ is representable by the algebraic space $f^{-1}F$,

3. if $\mathcal{X}$ is an algebraic stack, then $f^{-1}\mathcal{X}$ is an algebraic stack, and

4. if $\mathcal{X}$ is a Deligne-Mumford stack, then $f^{-1}\mathcal{X}$ is a Deligne-Mumford stack too.