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.

## 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

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.

**Proof.**
See discussion above.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)