The Stacks project

109.54 A non-algebraic classifying stack

Let $S = \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ and let $\mu _ p$ denote the group scheme of $p$th roots of unity over $S$. In Groupoids in Spaces, Section 77.20 we have introduced the quotient stack $[S/\mu _ p]$ and in Examples of Stacks, Section 94.15 we have shown $[S/\mu _ p]$ is the classifying stack for fppf $\mu _ p$-torsors: Given a scheme $T$ over $S$ the category $\mathop{\mathrm{Mor}}\nolimits _ S(T, [S/\mu _ p])$ is canonically equivalent to the category of fppf $\mu _ p$-torsors over $T$. Finally, in Criteria for Representability, Theorem 96.17.2 we have seen that $[S/\mu _ p]$ is an algebraic stack.

Now we can ask the question: “How about the category fibred in groupoids $\mathcal{S}$ classifying étale $\mu _ p$-torsors?” (In other words $\mathcal{S}$ is a category over $\mathit{Sch}/S$ whose fibre category over a scheme $T$ is the category of étale $\mu _ p$-torsors over $T$.)

The first objection is that this isn't a stack for the fppf topology, because descent for objects isn't going to hold. For example the $\mu _ p$-torsor $\mathop{\mathrm{Spec}}(\mathbf{F}_ p(t)[x]/(x^ p - t))$ over $T = \mathop{\mathrm{Spec}}(\mathbf{F}_ p(T))$ is fppf locally trivial, but not étale locally trivial.

A fix for this first problem is to work with the étale topology and in this case descent for objects does work. Indeed it is true that $\mathcal{S}$ is a stack in groupoids over $(\mathit{Sch}/S)_{\acute{e}tale}$. Moreover, it is also the case that the diagonal $\Delta : \mathcal{S} \to \mathcal{S} \times \mathcal{S}$ is representable (by schemes). This is true because given two $\mu _ p$-torsors (whether they be étale locally trivial or not) the sheaf of isomorphisms between them is representable by a scheme.

Thus we can finally ask if there exists a scheme $U$ and a smooth and surjective $1$-morphism $U \to \mathcal{S}$. We will show in two ways that this is impossible: by a direct argument (which we advise the reader to skip) and by an argument using a general result.

Direct argument (sketch): Note that the $1$-morphism $\mathcal{S} \to \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ satisfies the infinitesimal lifting criterion for formal smoothness. This is true because given a first order infinitesimal thickening of schemes $T \to T'$ the kernel of $\mu _ p(T') \to \mu _ p(T)$ is isomorphic to the sections of the ideal sheaf of $T$ in $T'$, and hence $H^1_{\acute{e}tale}(T, \mu _ p) = H^1_{\acute{e}tale}(T', \mu _ p)$. Moreover, $\mathcal{S}$ is a limit preserving stack. Hence if $U \to \mathcal{S}$ is smooth, then $U \to \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ is limit preserving and satisfies the infinitesimal lifting criterion for formal smoothness. This implies that $U$ is smooth over $\mathbf{F}_ p$. In particular $U$ is reduced, hence $H^1_{\acute{e}tale}(U, \mu _ p) = 0$. Thus $U \to \mathcal{S}$ factors as $U \to \mathop{\mathrm{Spec}}(\mathbf{F}_ p) \to \mathcal{S}$ and the first arrow is smooth. By descent of smoothness, we see that $U \to \mathcal{S}$ being smooth would imply $\mathop{\mathrm{Spec}}(\mathbf{F}_ p) \to \mathcal{S}$ is smooth. However, this is not the case as $\mathop{\mathrm{Spec}}(\mathbf{F}_ p) \times _\mathcal {S} \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ is $\mu _ p$ which is not smooth over $\mathop{\mathrm{Spec}}(\mathbf{F}_ p)$.

Structural argument: In Criteria for Representability, Section 96.19 we have seen that we can think of algebraic stacks as those stacks in groupoids for the étale topology with diagonal representable by algebraic spaces having a smooth covering. Hence if a smooth surjective $U \to \mathcal{S}$ exists then $\mathcal{S}$ is an algebraic stack, and in particular satisfies descent in the fppf topology. But we've seen above that $\mathcal{S}$ does not satisfies descent in the fppf topology.

Loosely speaking the arguments above show that the classifying stack in the étale topology for étale locally trivial torsors for a group scheme $G$ over a base $B$ is algebraic if and only if $G$ is smooth over $B$. One of the advantages of working with the fppf topology is that it suffices to assume that $G \to B$ is flat and locally of finite presentation. In fact the quotient stack (for the fppf topology) $[B/G]$ is algebraic if and only if $G \to B$ is flat and locally of finite presentation, see Criteria for Representability, Lemma 96.18.3.

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