Units and Picard groups in Nisnevich degree zero and one

The site has the full-subcategory, Zariski-refinement and terminal-object properties required by [F2]. A covering sieve contains a Nisnevich family, and that family is jointly surjective étale by [F1], hence fpqc by [F2]. The sheaf condition can therefore be checked on those generating families. The structure presheaf is a sheaf, and its units form the abelian sheaf ${\mathbb{G}}_m$ by [F3]. Evaluation at the terminal object $X$ and [F3] give the stated degree-zero equality.

Let $L$ be a line bundle. On each étale $U\to X$, take frames of its pullback $L{\mathrm{\mid }}_U$. The structure presheaf on the slice site over $U$ is a sheaf by the same argument as in step 1.1. Frames form a ${\mathbb{G}}_m$-torsor by [F4]: Zariski trivializations of $L{\mathrm{\mid }}_U$ are Nisnevich covers by [F1]. Thus every line bundle determines a torsor.

Conversely, let $P$ be a ${\mathbb{G}}_m$-torsor. Choose a Nisnevich covering family $U_i\to X$ with sections $s_i$. Its transition units $g_{ij}$ satisfy the cocycle equation by [F4]. On each $U_i$ take the free rank-one module with basis $e_i$; use the isomorphisms determined by $e_j=g_{ij}{e}_i$ on double fibre products. The cocycle equation is exactly the descent compatibility on triple fibre products.

The family is fpqc by [F1] and [F2]. Effective descent gives a quasi-coherent module $L$ with these pullbacks and transition maps; finite local freeness descends by [F2]. At every point of $X$, choose a point of some $U_i$ over it. The fibre of $L$ becomes one-dimensional after extension to that point's residue field, so the rank of $L$ there is one. Hence $L$ is a line bundle.

The descended frames have the same transitions as the $s_i$, so [F4] gives an equivariant isomorphism from the frame torsor of $L$ to $P$. Conversely, starting with a line bundle and using its local frames, full faithfulness of descent in [F2] identifies the descended module with the original bundle. These constructions are independent of choices: pass to a common refinement, use the change-of-section formula in [F4], and apply full faithfulness of descent. Thus line bundles up to isomorphism and torsors up to equivariant isomorphism are in bijection.

By [F4], the torsor bijection gives a canonical bijection with $H^1$. It is compatible with the group laws: using an injective embedding ${\mathbb{G}}_m\hookrightarrow I$, the construction described in the cited torsor-classification input represents a class by a global section of $I\mathrm{/}{\mathbb{G}}_m$, with its torsor the sheaf of local lifts. Adding two representatives adds their local lifts; their transition units multiply. By [F4], tensoring line bundles also multiplies their frame transitions. Therefore the bijection sends tensor products to sums in $H^1$.

The unit line bundle has the trivial frame torsor by [F4]. Together with [F5] and steps 1.1 and 6.1, this proves both identifications and the stated inverse.