The Stacks project

Proof. We may replace $B$ by $T$ and $X$ by $X_ T$ and assume that $B = T$ to simplify the notation. Let $\mathcal{N}$ be an invertible $\mathcal{O}_ B$-module. If $f^*\mathcal{N} \cong \mathcal{O}_ X$, then we see that $f_*f^*\mathcal{N} \cong f_*\mathcal{O}_ X \cong \mathcal{O}_ B$ 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 $\mathcal{O}_ B \to f_*f^*\mathcal{O}_ B$ is an isomorphism by assumption). Hence we conclude that $\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_ B$ 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 $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathrm{Pic}_{X/B}(B)$. Then there exists an fppf covering $\{ B_ i \to B\} $ such that $\mathcal{L}$ pulls back to the trivial invertible sheaf on $X_{B_ i}$. 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 $X_{B_ i \times _ B B_ j}$ and hence differ by a multiplicative unit

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