The globally normalized first-layer Picard map is determinant

The sheaf $S$ is connected: every rank-zero perfect class is locally zero on affine free-complex charts, where its object class is its alternating free rank. Its fundamental-group sheaf is ${\mathbb{G}}_m$ through determinant by the local-ring calculation in [F2]; positive homotopy groups of the rank fiber agree with those of K by [F3]. Thus its pointed first Postnikov truncation is $B{\mathbb{G}}_m$, with its identification fixed by the actual determinant construction. These are local statements about sheaves of K-spaces and their isomorphisms, not statements that the global Picard group vanishes.

The composite $q_1\kappa :B{\mathbb{G}}_m\to B{\mathbb{G}}_m$ is a pointed stack map. By [F2] it is classified by a sheaf-group endomorphism of ${\mathbb{G}}_m$. This group is represented on the qcqs $k$-site by $\mathrm{Spec}k[t,t^{-1}]$; Yoneda and the Laurent-unit calculation over a field make the endomorphism $t\mapsto t^n$. The normalization of [F1] on $\mathcal{O}(-1)$ over ${\mathbb{P}}_k^1$ gives $\mathcal{O}(-n)=\mathcal{O}(-1)$ in $\mathrm{Pic}({\mathbb{P}}_k^1)=\mathbb{Z}$, hence $n=1$. The same universal test applies on this larger site, so $q_1\kappa$ is identity as an actual pointed stack map.

On fundamental-group sheaves, $\kappa$ is the unit section, and the determinant of that section is identity by [F2]. Hence ${\kappa }_{\ast }$ is the inverse of the determinant isomorphism on local K₁. Step1.2 forces $(q_1{)}_{\ast }$ to be determinant. Maps from the connected sheaf $S$ into the1-type $B{\mathbb{G}}_m$ factor through ${\tau }_{\le 1}S$; the full faithfulness in [F2] now identifies $q_1$ with the actual determinant/Picard map. Evaluate this equality of stack maps at any $X\mathrm{/}k$ to retain its full line-bundle descent data. No inference from agreement of Picard groups on affine opens occurs.