The Stacks project

Lemma 94.18.1. The category $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ endowed with the functor $p$ above defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Proof. As usual, the hardest part is to show descent for objects. To see this let $\{ U_ i \to U\} $ be a covering of $(\mathit{Sch}/S)_{fppf}$. Let $\xi _ i = (U_ i, Z_ i, y_ i, x_ i, \alpha _ i)$ be an object of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ lying over $U_ i$, and let $\varphi _{ij} : \text{pr}_0^*\xi _ i \to \text{pr}_1^*\xi _ j$ be a descent datum. First, observe that $\varphi _{ij}$ induces a descent datum $(Z_ i/U_ i, \varphi _{ij})$ which is effective by Descent, Lemma 35.37.1 This produces a scheme $Z/U$ which is finite locally free of degree $d$ by Descent, Lemma 35.23.30. From now on we identify $Z_ i$ with $Z \times _ U U_ i$. Next, the objects $y_ i$ in the fibre categories $\mathcal{Y}_{U_ i}$ descend to an object $y$ in $\mathcal{Y}_ U$ because $\mathcal{Y}$ is a stack in groupoids. Similarly the objects $x_ i$ in the fibre categories $\mathcal{X}_{Z_ i}$ descend to an object $x$ in $\mathcal{X}_ Z$ because $\mathcal{X}$ is a stack in groupoids. Finally, the given isomorphisms

\[ \alpha _ i : (y|_ Z)_{Z_ i} = y_ i|_{Z_ i} \longrightarrow F(x_ i) = F(x|_{Z_ i}) \]

glue to a morphism $\alpha : y|_ Z \to F(x)$ as the $\mathcal{Y}$ is a stack and hence $\mathit{Isom}_\mathcal {Y}(y|_ Z, F(x))$ is a sheaf. Details omitted. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 94.18: Finite Hilbert stacks

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Your email address will not be published.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 05WB. Beware of the difference between the letter 'O' and the digit '0'.