The root map through derived sections, truncation and localization

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Assume the Axiom of Choice and compatible universe bounds for the small étale sites, big sites, slices and products used below. Fix a prime $p$, a field $k$ in which $p$ is invertible, and an element $\zeta \in k$ of multiplicative order $p$. Put $S=\mathrm{Spec}\mathbb{Z}[1\mathrm{/}p]$ and $U=\mathrm{Spec}k$. Let $\mathcal{C}$ be the size-bounded category of qcqs $S$-schemes with BEM's big cdh topology, generated by finite Nisnevich covers and finitely presented abstract blowup covers. Equip $\mathcal{C}\mathrm{/}U$, the category of qcqs $k$-schemes, with the induced topology. Let $j:\mathcal{C}\mathrm{/}U\to \mathcal{C}$ forget the structure map to $U$, and let $j^{\ast }$ be restriction of presheaves. Write $L_{\mathcal{C}}$ and $L_{\mathcal{C}\mathrm{/}U}$ for the respective ordinary reflective sheafifications of $D(\mathbb{Z})$-valued presheaves, using derived Čech descent for covering families on both sites.

For $V\in \mathcal{C}$, compute $R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}$ on the size-bounded small étale site of $V$, with constant sheaf $\mathbb{Z}\mathrm{/}p$ and Tate coefficient ${\mu }_p^{\otimes 3}$, where the tensor product is over $\mathbb{Z}\mathrm{/}p$. Write $T(V)={\tau }^{\le 3}R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}(V,{\mu }_p^{\otimes 3})$ and $A(V)={\tau }^{\le 3}R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}(V,\mathbb{Z}\mathrm{/}p)$. Their actual coefficient map on the slice is an equivalence $j^{\ast }A\stackrel{t_{\zeta }^{(3)}}{\to }{j}^{\ast }T$. It induces a specified equivalence $j^{\ast }{L}_{\mathcal{C}}T\stackrel{\sim }{\to }{L}_{\mathcal{C}\mathrm{/}U}{j}^{\ast }A$, namely the slice/ambient comparison followed by the localized inverse of $t_{\zeta }^{(3)}$. Here ${\tau }^{\le 3}$ is objectwise cohomological truncation before sheafification.

The actual big cdh slice over a field

Derived global sections on a scheme site