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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