Lemma 110.54.1. 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.
110.54 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 95.14.2. This stack is limit preserving on objects over (\mathit{Sch}/S)_{fppf} (see Criteria for Representability, Section 97.5) 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 98.11.1) 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.
Proof. See discussion above. \square
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