94.18 Change of big site
In this section we briefly discuss what happens when we change big sites. The upshot is that we can always enlarge the big site at will, hence we may assume any set of schemes we want to consider is contained in the big fppf site over which we consider our algebraic space. We encourage the reader to skip this section.
Pullbacks of stacks is defined in Stacks, Section 8.12.
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 $(\mathit{Sch}/S)_{fppf}$.
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 $(\mathit{Sch}'/S)_{fppf}$,
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 $f^{-1}F$,
if $\mathcal{X}$ is an algebraic stack, then $f^{-1}\mathcal{X}$ is an algebraic stack, and
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.
$\square$
It is not true (in general) that the restriction of an algebraic space over the bigger site is an algebraic space over the smaller site (simply by reasons of cardinality). Hence we can only ever use a simple lemma of this kind to enlarge the base category and never to shrink it.
Lemma 94.18.2. Suppose $\mathit{Sch}_{fppf}$ is contained in $\mathit{Sch}'_{fppf}$. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Denote $\textit{Algebraic-Stacks}/S$ the $2$-category of algebraic stacks over $S$ defined using $\mathit{Sch}_{fppf}$. Similarly, denote $\textit{Algebraic-Stacks}'/S$ the $2$-category of algebraic stacks over $S$ defined using $\mathit{Sch}'_{fppf}$. The rule $\mathcal{X} \mapsto f^{-1}\mathcal{X}$ of Lemma 94.18.1 defines a functor of $2$-categories
\[ \textit{Algebraic-Stacks}/S \longrightarrow \textit{Algebraic-Stacks}'/S \]
which defines equivalences of morphism categories
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Algebraic-Stacks}/S}(\mathcal{X}, \mathcal{Y}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Algebraic-Stacks}'/S}(f^{-1}\mathcal{X}, f^{-1}\mathcal{Y}) \]
for every objects $\mathcal{X}, \mathcal{Y}$ of $\textit{Algebraic-Stacks}/S$. An object $\mathcal{X}'$ of $\textit{Algebraic-Stacks}'/S$ is equivalence to $f^{-1}\mathcal{X}$ for some $\mathcal{X}$ in $\textit{Algebraic-Stacks}/S$ if and only if it has a presentation $\mathcal{X} = [U'/R']$ with $U', R'$ isomorphic to $f^{-1}U$, $f^{-1}R$ for some $U, R \in \textit{Spaces}/S$.
Proof.
The statement on morphism categories is a consequence of the more general Stacks, Lemma 8.12.12. The characterization of the “essential image” follows from the description of $f^{-1}$ in the proof of Lemma 94.18.1.
$\square$
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