The Stacks project

97.19 Algebraic stacks in the étale topology

Let $S$ be a scheme. Instead of working with stacks in groupoids over the big fppf site $(\mathit{Sch}/S)_{fppf}$ we could work with stacks in groupoids over the big étale site $(\mathit{Sch}/S)_{\acute{e}tale}$. All of the material in Algebraic Stacks, Sections 94.4, 94.5, 94.6, 94.7, 94.8, 94.9, 94.10, and 94.11 makes sense for categories fibred in groupoids over $(\mathit{Sch}/S)_{\acute{e}tale}$. Thus we get a second notion of an algebraic stack by working in the étale topology. This notion is (a priori) weaker than the notion introduced in Algebraic Stacks, Definition 94.12.1 since a stack in the fppf topology is certainly a stack in the étale topology. However, the notions are equivalent as is shown by the following lemma.

Lemma 97.19.1. Denote the common underlying category of $\mathit{Sch}_{fppf}$ and $\mathit{Sch}_{\acute{e}tale}$ by $\mathit{Sch}_\alpha $ (see Sheaves on Stacks, Section 96.4 and Topologies, Remark 34.11.1). Let $S$ be an object of $\mathit{Sch}_\alpha $. Let

\[ p : \mathcal{X} \to \mathit{Sch}_\alpha /S \]

be a category fibred in groupoids with the following properties:

  1. $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{\acute{e}tale}$,

  2. the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces1, and

  3. there exists $U \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha /S)$ and a $1$-morphism $(\mathit{Sch}/U)_{\acute{e}tale}\to \mathcal{X}$ which is surjective and smooth.

Then $\mathcal{X}$ is an algebraic stack in the sense of Algebraic Stacks, Definition 94.12.1.

Proof. Note that properties (2) and (3) of the lemma and the corresponding properties (2) and (3) of Algebraic Stacks, Definition 94.12.1 are independent of the topology. This is true because these properties involve only the notion of a $2$-fibre product of categories fibred in groupoids, $1$- and $2$-morphisms of categories fibred in groupoids, the notion of a $1$-morphism of categories fibred in groupoids representable by algebraic spaces, and what it means for such a $1$-morphism to be surjective and smooth. Thus all we have to prove is that an étale stack in groupoids $\mathcal{X}$ with properties (2) and (3) is also an fppf stack in groupoids.

Using (2) let $R$ be an algebraic space representing

\[ (\mathit{Sch}_\alpha /U) \times _\mathcal {X} (\mathit{Sch}_\alpha /U) \]

By (3) the projections $s, t : R \to U$ are smooth. Exactly as in the proof of Algebraic Stacks, Lemma 94.16.1 there exists a groupoid in spaces $(U, R, s, t, c)$ and a canonical fully faithful $1$-morphism $[U/R]_{\acute{e}tale}\to \mathcal{X}$ where $[U/R]_{\acute{e}tale}$ is the étale stackification of presheaf in groupoids

\[ T \longmapsto (U(T), R(T), s(T), t(T), c(T)) \]

Claim: If $V \to T$ is a surjective smooth morphism from an algebraic space $V$ to a scheme $T$, then there exists an étale covering $\{ T_ i \to T\} $ refining the covering $\{ V \to T\} $. This follows from More on Morphisms, Lemma 37.38.7 or the more general Sheaves on Stacks, Lemma 96.19.10. Using the claim and arguing exactly as in Algebraic Stacks, Lemma 94.16.2 it follows that $[U/R]_{\acute{e}tale}\to \mathcal{X}$ is an equivalence.

Next, let $[U/R]$ denote the quotient stack in the fppf topology which is an algebraic stack by Algebraic Stacks, Theorem 94.17.3. Thus we have $1$-morphisms

\[ U \to [U/R]_{\acute{e}tale}\to [U/R]. \]

Both $U \to [U/R]_{\acute{e}tale}\cong \mathcal{X}$ and $U \to [U/R]$ are surjective and smooth (the first by assumption and the second by the theorem) and in both cases the fibre product $U \times _\mathcal {X} U$ and $U \times _{[U/R]} U$ is representable by $R$. Hence the $1$-morphism $[U/R]_{\acute{e}tale}\to [U/R]$ is fully faithful (since morphisms in the quotient stacks are given by morphisms into $R$, see Groupoids in Spaces, Section 78.24).

Finally, for any scheme $T$ and morphism $t : T \to [U/R]$ the fibre product $V = T \times _{U/R} U$ is an algebraic space surjective and smooth over $T$. By the claim above there exists an étale covering $\{ T_ i \to T\} _{i \in I}$ and morphisms $T_ i \to V$ over $T$. This proves that the object $t$ of $[U/R]$ over $T$ comes étale locally from $U$. We conclude that $[U/R]_{\acute{e}tale}\to [U/R]$ is an equivalence of stacks in groupoids over $(\mathit{Sch}/S)_{\acute{e}tale}$ by Stacks, Lemma 8.4.8. This concludes the proof. $\square$

[1] Here we can either mean sheaves in the étale topology whose diagonal is representable and which have an étale surjective covering by a scheme or algebraic spaces as defined in Algebraic Spaces, Definition 65.6.1. Namely, by Bootstrap, Lemma 80.12.1 there is no difference.

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