The Stacks project

110.53 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.

Lemma 110.53.1. 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. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07Z0. Beware of the difference between the letter 'O' and the digit '0'.