Lemma 44.4.3. Let f : X \to S be as in Definition 44.4.1. Assume f has a section \sigma and that \mathcal{O}_ T \to f_{T, *}\mathcal{O}_{X_ T} is an isomorphism for all T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}). Then
0 \to \mathop{\mathrm{Pic}}\nolimits (T) \to \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathrm{Pic}_{X/S}(T) \to 0
is a split exact sequence with splitting given by \sigma _ T^* : \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathop{\mathrm{Pic}}\nolimits (T).
Proof.
Denote K(T) = \mathop{\mathrm{Ker}}(\sigma _ T^* : \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathop{\mathrm{Pic}}\nolimits (T)). Since \sigma is a section of f we see that \mathop{\mathrm{Pic}}\nolimits (X_ T) is the direct sum of \mathop{\mathrm{Pic}}\nolimits (T) and K(T). Thus by Lemma 44.4.2 we see that K(T) \subset \mathrm{Pic}_{X/S}(T) for all T. Moreover, it is clear from the construction that \mathrm{Pic}_{X/S} is the sheafification of the presheaf K. To finish the proof it suffices to show that K satisfies the sheaf condition for fppf coverings which we do in the next paragraph.
Let \{ T_ i \to T\} be an fppf covering. Let \mathcal{L}_ i be elements of K(T_ i) which map to the same elements of K(T_ i \times _ T T_ j) for all i and j. Choose an isomorphism \alpha _ i : \mathcal{O}_{T_ i} \to \sigma _{T_ i}^*\mathcal{L}_ i for all i. Choose an isomorphism
\varphi _{ij} : \mathcal{L}_ i|_{X_{T_ i \times _ T T_ j}} \longrightarrow \mathcal{L}_ j|_{X_{T_ i \times _ T T_ j}}
If the map
\alpha _ j|_{T_ i \times _ T T_ j} \circ \sigma _{T_ i \times _ T T_ j}^*\varphi _{ij} \circ \alpha _ i|_{T_ i \times _ T T_ j} : \mathcal{O}_{T_ i \times _ T T_ j} \to \mathcal{O}_{T_ i \times _ T T_ j}
is not equal to multiplication by 1 but some u_{ij}, then we can scale \varphi _{ij} by u_{ij}^{-1} to correct this. Having done this, consider the self map
\varphi _{ki}|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \circ \varphi _{jk}|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \circ \varphi _{ij}|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \quad \text{on}\quad \mathcal{L}_ i|_{X_{T_ i \times _ T T_ j \times _ T T_ k}}
which is given by multiplication by some regular function f_{ijk} on the scheme X_{T_ i \times _ T T_ j \times _ T T_ k}. By our choice of \varphi _{ij} we see that the pullback of this map by \sigma is equal to multiplication by 1. By our assumption on functions on X, we see that f_{ijk} = 1. Thus we obtain a descent datum for the fppf covering \{ X_{T_ i} \to X\} . By Descent, Proposition 35.5.2 there is an invertible \mathcal{O}_{X_ T}-module \mathcal{L} and an isomorphism \alpha : \mathcal{O}_ T \to \sigma _ T^*\mathcal{L} whose pullback to X_{T_ i} recovers (\mathcal{L}_ i, \alpha _ i) (small detail omitted). Thus \mathcal{L} defines an object of K(T) as desired.
\square
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