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.

Proof. Let us prove (3). By Lemma 94.16.2 we may write $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces $(U, R, s, t, c)$. By Groupoids in Spaces, Lemma 78.28.1 we see that $f^{-1}[U/R] = [f^{-1}U/f^{-1}R]$. Of course $(f^{-1}U, f^{-1}R, f^{-1}s, f^{-1}t, f^{-1}c)$ is a smooth groupoid in algebraic spaces too. Hence (3) is proved.

Now the other cases (1), (2), (4) each mean that $\mathcal{X}$ has a presentation $[U/R]$ of a particular kind, and hence translate into the same kind of presentation for $f^{-1}\mathcal{X} = [f^{-1}U/f^{-1}R]$. Whence the lemma is proved. $\square$

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