The Stacks project

108.8 Properties of the Picard stack

Let $f : X \to B$ be a morphism of algebraic spaces which is flat, proper, and of finite presentation. Then the stack $\mathcal{P}\! \mathit{ic}_{X/B}$ parametrizing invertible sheaves on $X/B$ is algebraic, see Quot, Proposition 99.10.2.

Lemma 108.8.1. The diagonal of $\mathcal{P}\! \mathit{ic}_{X/B}$ over $B$ is affine and of finite presentation.

Proof. In Quot, Lemma 99.10.1 we have seen that $\mathcal{P}\! \mathit{ic}_{X/B}$ is an open substack of $\mathcal{C}\! \mathit{oh}_{X/B}$. Hence this follows from Lemma 108.4.1. $\square$

Lemma 108.8.2. The morphism $\mathcal{P}\! \mathit{ic}_{X/B} \to B$ is quasi-separated and locally of finite presentation.

Proof. In Quot, Lemma 99.10.1 we have seen that $\mathcal{P}\! \mathit{ic}_{X/B}$ is an open substack of $\mathcal{C}\! \mathit{oh}_{X/B}$. Hence this follows from Lemma 108.4.2. $\square$

Lemma 108.8.3. Assume $X \to B$ is smooth in addition to being proper. Then $\mathcal{P}\! \mathit{ic}_{X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition 101.39.10).

Proof. Taking base change, this immediately reduces to the following problem: given a valuation ring $R$ with fraction field $K$ and an algebraic space $X$ proper and smooth over $R$ and an invertible $\mathcal{O}_{X_ K}$-module $\mathcal{L}_ K$, show there exists an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ whose generic fibre is $\mathcal{L}_ K$. Observe that $X_ K$ is Noetherian, separated, and regular (use Morphisms of Spaces, Lemma 67.28.6 and Spaces over Fields, Lemma 72.16.1). Thus we can write $\mathcal{L}_ K$ as the difference in the Picard group of $\mathcal{O}_{X_ K}(D_ K)$ and $\mathcal{O}_{X_ K}(D'_ K)$ for two effective Cartier divisors $D_ K, D'_ K$ in $X_ K$, see Divisors on Spaces, Lemma 71.8.4. Finally, we know that $D_ K$ and $D'_ K$ are restrictions of effective Cartier divisors $D, D' \subset X$, see Divisors on Spaces, Lemma 71.8.5. $\square$

Lemma 108.8.4. Assume $f_{T, *}\mathcal{O}_{X_ T} \cong \mathcal{O}_ T$ for all schemes $T$ over $B$. Then the inertia stack of $\mathcal{P}\! \mathit{ic}_{X/B}$ is equal to $\mathbf{G}_ m \times \mathcal{P}\! \mathit{ic}_{X/B}$.

Proof. This is explained in Examples of Stacks, Example 95.17.2. $\square$

Lemma 108.8.5. Assume $f : X \to B$ has relative dimension $\leq 1$ in addition to the other assumptions in this section. Then $\mathcal{P}\! \mathit{ic}_{X/B} \to B$ is smooth.

Proof. We already know that $\mathcal{P}\! \mathit{ic}_{X/B} \to B$ is locally of finite presentation, see Lemma 108.8.2. Thus it suffices to show that $\mathcal{P}\! \mathit{ic}_{X/B} \to B$ is formally smooth, see More on Morphisms of Stacks, Lemma 106.8.7. Taking base change, this immediately reduces to the following problem: given a first order thickening $T \subset T'$ of affine schemes, given $X' \to T'$ proper, flat, of finite presentation and of relative dimension $\leq 1$, and for $X = T \times _{T'} X'$ given an invertible $\mathcal{O}_ X$-module $\mathcal{L}$, prove that there exists an invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ whose restriction to $X$ is $\mathcal{L}$. Since $T \subset T'$ is a first order thickening, the same is true for $X \subset X'$, see More on Morphisms of Spaces, Lemma 76.9.8. By More on Morphisms of Spaces, Lemma 76.11.1 we see that it suffices to show $H^2(X, \mathcal{I}) = 0$ where $\mathcal{I}$ is the quasi-coherent ideal cutting out $X$ in $X'$. Denote $f : X \to T$ the structure morphism. By Cohomology of Spaces, Lemma 69.22.9 we see that $R^ pf_*\mathcal{I} = 0$ for $p > 1$. Hence we get the desired vanishing by Cohomology of Spaces, Lemma 69.3.2 (here we finally use that $T$ is affine). $\square$


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