The Stacks project

Lemma 93.12.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be categories over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are equivalent as categories over $(\mathit{Sch}/S)_{fppf}$. Then $\mathcal{X}$ is an algebraic stack if and only if $\mathcal{Y}$ is an algebraic stack. Similarly, $\mathcal{X}$ is a Deligne-Mumford stack if and only if $\mathcal{Y}$ is a Deligne-Mumford stack.

Proof. Assume $\mathcal{X}$ is an algebraic stack (resp. a Deligne-Mumford stack). By Stacks, Lemma 8.5.4 this implies that $\mathcal{Y}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$. Choose an equivalence $f : \mathcal{X} \to \mathcal{Y}$ over $\mathit{Sch}_{fppf}$. This gives a $2$-commutative diagram

\[ \xymatrix{ \mathcal{X} \ar[r]_ f \ar[d]_{\Delta _\mathcal {X}} & \mathcal{Y} \ar[d]^{\Delta _\mathcal {Y}} \\ \mathcal{X} \times \mathcal{X} \ar[r]^{f \times f} & \mathcal{Y} \times \mathcal{Y} } \]

whose horizontal arrows are equivalences. This implies that $\Delta _\mathcal {Y}$ is representable by algebraic spaces according to Lemma 93.9.3. Finally, let $U$ be a scheme over $S$, and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a $1$-morphism which is surjective and smooth (resp. ├ętale). Considering the diagram

\[ \xymatrix{ (\mathit{Sch}/U)_{fppf} \ar[r]_{\text{id}} \ar[d]_ x & (\mathit{Sch}/U)_{fppf} \ar[d]^{f \circ x} \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \]

and applying Lemma 93.10.2 we conclude that $f \circ x$ is surjective and smooth (resp. ├ętale) as desired. $\square$

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