Lemma 99.11.7. In Situation 99.11.1 let $\sigma : B \to X$ be a section. If $\mathcal{O}_ T \to f_{T, *}\mathcal{O}_{X_ T}$ is an isomorphism for all $T$ over $B$, then $\mathcal{P}\! \mathit{ic}_{X/B, \sigma } \to (\mathit{Sch}/S)_{fppf}$ is fibred in setoids with set of isomorphism classes over $T$ given by
Proof. If $\xi = (T, h, \mathcal{L}, \alpha )$ is an object of $\mathcal{P}\! \mathit{ic}_{X/B, \sigma }$ over $T$, then an automorphism $\varphi $ of $\xi $ is given by multiplication with an invertible global section $u$ of the structure sheaf of $X_ T$ such that moreover $\sigma _ T^*u = 1$. Then $u = 1$ by our assumption that $\mathcal{O}_ T \to f_{T, *}\mathcal{O}_{X_ T}$ is an isomorphism. Hence $\mathcal{P}\! \mathit{ic}_{X/B, \sigma }$ is fibred in setoids over $(\mathit{Sch}/S)_{fppf}$. Given $T$ and $h : T \to B$ the set of isomorphism classes of pairs $(\mathcal{L}, \alpha )$ is the same as the set of isomorphism classes of $\mathcal{L}$ with $\sigma _ T^*\mathcal{L} \cong \mathcal{O}_ T$ (isomorphism not specified). This is clear because any two choices of $\alpha $ differ by a global unit on $T$ and this is the same thing as a global unit on $X_ T$. $\square$
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