Motivic weight one

PDF p. 4: Property (2) and equation (2.3)

Assume the Axiom of Choice and the site-size bounds of Low-degree consequences of a weight-one comparison. Let $k$ be a field and $X$ a quasi-compact quasi-separated $k$-scheme. There is a natural map $R{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{i}\mathrm{s}}(X,{\mathbb{G}}_m)[-1]⟶\mathbb{Z}(1{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)$ that is an equivalence in degrees at most three: it induces an isomorphism on cohomology in every degree $i\le 3$. Here ${\mathbb{G}}_m$ is the sheaf of invertible regular functions on the small Nisnevich site, and $\mathbb{Z}(1{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)$ is the complex of Motivic complexes and cohomology. Consequently, for $i\le 2$, $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i,1}(X,\mathbb{Z})\cong H_{\mathrm{N}\mathrm{i}\mathrm{s}}^{i-1}(X,{\mathbb{G}}_m)\cong \{\begin{array}{ll}0& i\le 0,\\ \mathrm{\Gamma }(X,{\mathcal{O}}_X{)}^{\times }& i=1,\\ \mathrm{Pic}(X)& i=2.\end{array}$ The Picard group is the group of isomorphism classes of line bundles, meaning locally free rank-one ${\mathcal{O}}_X$-modules, under tensor product.

The article's property (2) and equation (2.3) were read. The comparison is used through the explicitly identified external input; its underlying proof is not reconstructed here.