The Stacks project

95.16 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 99.10.

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

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.