93.16 The Picard stack

In this section we introduce the Picard stack in complete generality. In the chapter on Quot and Hilb we will show that it is an algebraic stack under suitable hypotheses, see Quot, Section 97.10.

Let $S$ be a scheme. Let $\pi : X \to B$ be a morphism of algebraic spaces over $S$. We define a category $\mathcal{P}\! \mathit{ic}_{X/B}$ as follows:

1. An object is a triple $(U, b, \mathcal{L})$, where

1. $U$ is an object of $(\mathit{Sch}/S)_{fppf}$,

2. $b : U \to B$ is a morphism over $S$, and

3. $\mathcal{L}$ is in invertible sheaf on the base change $X_ U = U \times _{b, B} X$.

2. A morphism $(f, g) : (U, b, \mathcal{L}) \to (U', b', \mathcal{L}')$ is given by a morphism of schemes $f : U \to U'$ over $B$ and an isomorphism $g : f^*\mathcal{L}' \to \mathcal{L}$.

The composition of $(f, g) : (U, b, \mathcal{L}) \to (U', b', \mathcal{L}')$ with $(f', g') : (U', b', \mathcal{L}') \to (U'', b'', \mathcal{L}'')$ is given by $(f \circ f', g \circ f^*(g'))$. Thus we get a category $\mathcal{P}\! \mathit{ic}_{X/B}$ and

$p : \mathcal{P}\! \mathit{ic}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}, \quad (U, b, \mathcal{L}) \longmapsto U$

is a functor. Note that the fibre category of $\mathcal{P}\! \mathit{ic}_{X/B}$ over $U$ is the disjoint union over $b \in \mathop{\mathrm{Mor}}\nolimits _ S(U, B)$ of the categories of invertible sheaves on $X_ U = U \times _{b, B} X$. Hence the fibre categories are groupoids.

Lemma 93.16.1. 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 72.4.1 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. $\square$

Comment #1428 by yogesh more on

minor typo: line 1499 $\xi_i = (U_i, b_i, \mathcal{L}_i)$ is an object lying over $U_i$, not $U$.

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