The Stacks project

108.50 Limit preserving on objects, not limit preserving

Let $S$ be a nonempty scheme. Let $\mathcal{G}$ be an injective abelian sheaf on $(\mathit{Sch}/S)_{fppf}$. We obtain a stack in groupoids

\[ \mathcal{G}\textit{-Torsors} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

over $S$, see Examples of Stacks, Lemma 93.14.2. This stack is limit preserving on objects over $(\mathit{Sch}/S)_{fppf}$ (see Criteria for Representability, Section 95.5) because every $\mathcal{G}$-torsor is trivial. On the other hand, $\mathcal{G}\textit{-Torsors}$ is in general not limit preserving (see Artin's Axioms, Definition 96.11.1) as $\mathcal{G}$ need not be limit preserving as a sheaf. For example, take any nonzero injective sheaf $\mathcal{I}$ and set $\mathcal{G} = \prod _{n \in \mathbf{Z}} \mathcal{I}$ to get an example.

Lemma 108.50.1. Let $S$ be a nonempty scheme. There exists a stack in groupoids $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ such that $p$ is limit preserving on objects, but $\mathcal{X}$ is not limit preserving.

Proof. See discussion above. $\square$


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