Lemma 44.4.2. Let $f : X \to S$ be as in Definition 44.4.1. If $\mathcal{O}_ T \to f_{T, *}\mathcal{O}_{X_ T}$ is an isomorphism for all $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, then

$0 \to \mathop{\mathrm{Pic}}\nolimits (T) \to \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathrm{Pic}_{X/S}(T)$

is an exact sequence for all $T$.

Proof. We may replace $S$ by $T$ and $X$ by $X_ T$ and assume that $S = T$ to simplify the notation. Let $\mathcal{N}$ be an invertible $\mathcal{O}_ S$-module. If $f^*\mathcal{N} \cong \mathcal{O}_ X$, then we see that $f_*f^*\mathcal{N} \cong f_*\mathcal{O}_ X \cong \mathcal{O}_ S$ by assumption. Since $\mathcal{N}$ is locally trivial, we see that the canonical map $\mathcal{N} \to f_*f^*\mathcal{N}$ is locally an isomorphism (because $\mathcal{O}_ S \to f_*f^*\mathcal{O}_ S$ is an isomorphism by assumption). Hence we conclude that $\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_ S$ is an isomorphism and we see that $\mathcal{N}$ is trivial. This proves the first arrow is injective.

Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module which is in the kernel of $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathrm{Pic}_{X/S}(S)$. Then there exists an fppf covering $\{ S_ i \to S\}$ such that $\mathcal{L}$ pulls back to the trivial invertible sheaf on $X_{S_ i}$. Choose a trivializing section $s_ i$. Then $\text{pr}_0^*s_ i$ and $\text{pr}_1^*s_ j$ are both trivialising sections of $\mathcal{L}$ over $X_{S_ i \times _ S S_ j}$ and hence differ by a multiplicative unit

$f_{ij} \in \Gamma (X_{S_ i \times _ S S_ j}, \mathcal{O}_{X_{S_ i \times _ S S_ j}}^*) = \Gamma (S_ i \times _ S S_ j, \mathcal{O}_{S_ i \times _ S S_ j}^*)$

(equality by our assumption on pushforward of structure sheaves). Of course these elements satisfy the cocycle condition on $S_ i \times _ S S_ j \times _ S S_ k$, hence they define a descent datum on invertible sheaves for the fppf covering $\{ S_ i \to S\}$. By Descent, Proposition 35.5.2 there is an invertible $\mathcal{O}_ S$-module $\mathcal{N}$ with trivializations over $S_ i$ whose associated descent datum is $\{ f_{ij}\}$. Then $f^*\mathcal{N} \cong \mathcal{L}$ as the functor from descent data to modules is fully faithful (see proposition cited above). $\square$

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