The Stacks project

Lemma 88.15.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. The following are equivalent:

  1. for $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ the functor $f : \mathcal{X}_ U \to \mathcal{Y}_ U$ is faithful,

  2. the functor $f$ is faithful, and

  3. $f$ is representable by algebraic spaces.

Proof. Parts (1) and (2) are equivalent by general properties of $1$-morphisms of categories fibred in groupoids, see Categories, Lemma 4.34.8. We see that (3) implies (2) by Lemma 88.9.2. Finally, assume (2). Let $U$ be a scheme. Let $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$. We have to prove that

\[ \mathcal{W} = (\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X} \]

is representable by an algebraic space over $U$. Since $(\mathit{Sch}/U)_{fppf}$ is an algebraic stack we see from Lemma 88.14.3 that $\mathcal{W}$ is an algebraic stack. On the other hand the explicit description of objects of $\mathcal{W}$ as triples $(V, x, \alpha : y(V) \to f(x))$ and the fact that $f$ is faithful, shows that the fibre categories of $\mathcal{W}$ are setoids. Hence Proposition 88.13.3 guarantees that $\mathcal{W}$ is representable by an algebraic space. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 88.15: Algebraic stacks, overhauled

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Your email address will not be published.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 04Y5. Beware of the difference between the letter 'O' and the digit '0'.