Line bundles are detected off codimension two

For any line bundle $\mathcal{M}$ on $X$, the natural restriction morphism $\mathcal{M}\to j_{\ast }(\mathcal{M}{\mathrm{\mid }}_U)$ is an isomorphism. Indeed, on every open set $V$ where $\mathcal{M}\cong {\mathcal{O}}_V$, it identifies with the restriction of the isomorphism in [F1] to $V$. Such opens cover $X$, and a morphism of sheaves that is an isomorphism locally is an isomorphism: its local inverses agree on overlaps and glue. In particular, sections of $\mathcal{M}{\mathrm{\mid }}_U$ extend uniquely to sections of $\mathcal{M}$ on $X$. The dual ${\mathcal{L}}^{\vee }$ is also a line bundle, since the dual of a local copy of ${\mathcal{O}}_X$ is a local copy of ${\mathcal{O}}_X$.

Suppose $\varphi :{\mathcal{O}}_U\stackrel{\sim }{\to }\mathcal{L}{\mathrm{\mid }}_U$ is a trivialization. Set $s_U=\varphi (1)$ and let $t_U$ be the section of ${\mathcal{L}}^{\vee }{\mathrm{\mid }}_U$ corresponding to ${\varphi }^{-1}$; thus $t_U(s_U)=1$. By step 1.1, these sections extend uniquely to $s\in \mathrm{\Gamma }(X,\mathcal{L})$ and $t\in \mathrm{\Gamma }(X,{\mathcal{L}}^{\vee })$. The evaluation $t(s)$ restricts to $1$ on $U$, so [F1] implies $t(s)=1$ on $X$. Locally write $s=ae$ and $t=b{e}^{\vee }$ in a basis $e$ and its dual. Then $ba=1$, so the maps ${\mathcal{O}}_X\to \mathcal{L}$, $1\mapsto s$, and $\mathcal{L}\to {\mathcal{O}}_X$ defined by $t$ are inverse isomorphisms locally, hence globally.

Restriction respects tensor products and the identity by their local constructions, so it is a homomorphism of the Picard groups in [F2]. If a class restricts to the identity, its representative admits a trivialization on $U$; step 2.1 makes that representative trivial on $X$. Thus the kernel is trivial, and the restriction homomorphism is injective.