The Stacks project

Lemma 100.3.7. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. If $\mathcal{X} \to S$ is locally of finite presentation, then $\mathcal{X}$ is limit preserving in the sense of Artin's Axioms, Definition 96.11.1 (equivalently: the morphism $\mathcal{X} \to S$ is limit preserving).

Proof. Choose a surjective smooth morphism $U \to \mathcal{X}$ for some scheme $U$. Then $U \to S$ is locally of finite presentation, see Morphisms of Stacks, Section 99.27. We can write $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma 92.16.2. Since $U$ is locally of finite presentation over $S$ it follows that the algebraic space $R$ is locally of finite presentation over $S$. Recall that $[U/R]$ is the stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ obtained by stackyfying the category fibred in groupoids whose fibre category over $T$ is the groupoid $(U(T), R(T), s, t, c)$. Since $U$ and $R$ are limit preserving as functors (Limits of Spaces, Proposition 68.3.10) this category fibred in groupoids is limit preserving. Thus it suffices to show that fppf stackyfication preserves the property of being limit preserving. This is true (hint: use Topologies, Lemma 34.13.2). However, we give a direct proof below using that in this case we know what the stackyfication amounts to.

Let $T = \mathop{\mathrm{lim}}\nolimits T_\lambda $ be a directed limit of affine schemes over $S$. We have to show that the functor

\[ \mathop{\mathrm{colim}}\nolimits [U/R]_{T_\lambda } \longrightarrow [U/R]_ T \]

is an equivalence of categories. Let us show this functor is essentially surjective. Let $x \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_ T)$. In Groupoids in Spaces, Lemma 76.23.1 the reader finds a description of the category $[U/R]_ T$. In particular $x$ corresponds to an fppf covering $\{ T_ i \to T\} _{i \in I}$ and a $[U/R]$-descent datum $(u_ i, r_{ij})$ relative to this covering. After refining this covering we may assume it is a standard fppf covering of the affine scheme $T$. By Topologies, Lemma 34.13.2 we may choose a $\lambda $ and a standard fppf covering $\{ T_{\lambda , i} \to T_\lambda \} _{i \in I}$ whose base change to $T$ is equal to $\{ T_ i \to T\} _{i \in I}$. For each $i$, after increasing $\lambda $, we can find a $u_{\lambda , i} : T_{\lambda , i} \to U$ whose composition with $T_ i \to T_{\lambda , i}$ is the given morphism $u_ i$ (this is where we use that $U$ is limit preserving). Similarly, for each $i, j$, after increasing $\lambda $, we can find a $r_{\lambda , ij} : T_{\lambda , i} \times _{T_\lambda } T_{\lambda , j} \to R$ whose composition with $T_{ij} \to T_{\lambda , ij}$ is the given morphism $r_{ij}$ (this is where we use that $R$ is limit preserving). After increasing $\lambda $ we can further assume that

\[ s \circ r_{\lambda , ij} = u_{\lambda , i} \circ \text{pr}_0 \quad \text{and}\quad t \circ r_{\lambda , ij} = u_{\lambda , j} \circ \text{pr}_1, \]


\[ c \circ (r_{\lambda , jk} \circ \text{pr}_{12}, r_{\lambda , ij} \circ \text{pr}_{01}) = r_{\lambda , ik} \circ \text{pr}_{02}. \]

In other words, we may assume that $(u_{\lambda , i}, r_{\lambda , ij})$ is a $[U/R]$-descent datum relative to the covering $\{ T_{\lambda , i} \to T_\lambda \} _{i \in I}$. Then we obtain a corresponding object of $[U/R]$ over $T_\lambda $ whose pullback to $T$ is isomorphic to $x$ as desired. The proof of fully faithfulness works in exactly the same way using the description of morphisms in the fibre categories of $[U/T]$ given in Groupoids in Spaces, Lemma 76.23.1. $\square$

Comments (2)

Comment #2251 by Niels Borne on

There is a typo at the start of the third sentence of the proof.

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