Lemma 95.16.1. Up to a replacement as in Stacks, Remark 8.4.9 the functor

defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Lemma 95.16.1. Up to a replacement as in Stacks, Remark 8.4.9 the functor

\[ \mathcal{P}\! \mathit{ic}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

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 $\xi _ i = (U_ i, b_ i, \mathcal{L}_ i)$ be an object of $\mathcal{P}\! \mathit{ic}_{X/B}$ lying over $U_ i$, and let $\varphi _{ij} : \text{pr}_0^*\xi _ i \to \text{pr}_1^*\xi _ j$ be a descent datum. This implies in particular that the morphisms $b_ i$ are the restrictions of a morphism $b : U \to B$. Write $X_ U = U \times _{b, B} X$ and $X_ i = U_ i \times _{b_ i, B} X = U_ i \times _ U U \times _{b, B} X = U_ i \times _ U X_ U$. Observe that $\mathcal{L}_ i$ is an invertible $\mathcal{O}_{X_ i}$-module. Note that $\{ X_ i \to X_ U\} $ forms an fppf covering as well. Moreover, the descent datum $\varphi _{ij}$ translates into a descent datum on the invertible sheaves $\mathcal{L}_ i$ relative to the fppf covering $\{ X_ i \to X_ U\} $. Hence by Descent on Spaces, Proposition 74.4.1 we obtain a unique invertible sheaf $\mathcal{L}$ on $X_ U$ which recovers $\mathcal{L}_ i$ and the descent data over $X_ i$. The triple $(U, b, \mathcal{L})$ is therefore the object of $\mathcal{P}\! \mathit{ic}_{X/B}$ over $U$ we were looking for. Details omitted.
$\square$

Your email address will not be published.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)

There are also: