The Stacks project

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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