Shifted Nisnevich units and the line-bundle groupoid

The chosen site is full in schemes over its base and has a terminal object and Zariski refinements. Its covers are jointly surjective and etale by [F1], and are therefore fpqc by [F2]; the four structure-sheaf hypotheses of [F2] therefore hold, including on slices. A line bundle $L$ gives the $G$-torsor of frames $\mathrm{Isom}(\mathcal{O},L)$. Conversely use the torsor convention in [F2] and choose local sections using [A1]; their transition units $g_{ij}$ glue the free rank-one modules with frames $e_j=g_{ij}{e}_i$ by effective descent in [F2]. Finite local freeness descends, and its rank is one because it is one on the covering members. A torsor morphism, in these frames, is multiplication by units compatible on overlaps; full faithfulness of descent gives exactly one line-bundle isomorphism, and the reverse construction takes a line isomorphism to its equivariant frame map. Thus frames give an equivalence of groupoids, compatible with pullback and tensor products. The same effective and fully faithful descent makes these groupoids a Nisnevich stack. It is a pointed connected one-type locally, since lines trivialize on Zariski covers and all its arrows are isomorphisms. Multiplication by a global unit is every automorphism of ${\mathcal{O}}_X$; for another line bundle the same formula holds in frames and glues uniquely, so its automorphism sheaf is also $G$. Its component set is the Picard group of [F2].

Choose a single nonnegative injective resolution $G\to I^{\bullet }$ in abelian sheaves on the chosen bounded site. Apply [F4] to this site and this sheaf. It constructs the pointed sheaf of spaces $D_I(X)={\mathrm{\Omega }}^{\mathrm{\infty }}H(I^{\bullet }(X)[1])$, with coherent restrictions from that one sheaf complex, representing $D(X)={\mathrm{\Omega }}^{\mathrm{\infty }}H(R{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{i}\mathrm{s}}(X,G)[1])$ on the same site. Thus $D$ is a Nisnevich sheaf of spaces, and its loop sheaf is canonically $G$ through the identity normalization of [F4]. By [F1], ${\pi }_{n}D(X)=H_{\mathrm{N}\mathrm{i}\mathrm{s}}^{1-n}(X,G)$ for $n\ge 0$. For $n\ge 2$ this is zero. A nonnegative resolution gives $H_{\mathrm{N}\mathrm{i}\mathrm{s}}^0(X,G)=G(X)$ because evaluation preserves kernels; hence the loop sheaf is canonically $G$. To prove local connectedness, represent a class in ${\pi }_{0}D(X)=H_{\mathrm{N}\mathrm{i}\mathrm{s}}^1(X,G)$ by $z\in I^1(X)$ with $d^{1}z=0$. The injective resolution is exact as a complex of abelian sheaves on this same site. By [F5], some Nisnevich cover $\{U_i\to X\}$ admits $t_i\in I^0(U_i)$ with $d^0{t}_i=z{\mathrm{\mid }}_{U_i}$. Thus every component becomes the base component locally, so the sheaf of components of $D$ is terminal. This proves connectedness without a comparison of slice cohomology with derived evaluation. Apply [F3] to $D$ and the stack in step 1.1: the identity of their common loop sheaf $G$ determines a pointed equivalence, naturally on the site. Evaluating this equivalence gives the asserted equivalences on every $X$, not merely isomorphisms of component sets. The tensor/addition structures agree as well: their multiplication maps have common loop homomorphism $G\times G\to G$, and [F3] identifies these maps fully faithfully. Together with step 1.1 this proves all component and automorphism assertions.