Singular Beilinson–Lichtenbaum input

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Assume the Axiom of Choice and compatible universe bounds for the small étale sites, big cdh sites, slices and products in the supplied constructions. Let $k$ be a characteristic-zero field, let $p$ be a prime, and choose a primitive $p$th root $\zeta \in k$ (in particular this can be done for the article’s algebraically closed $k$). On all qcqs $k$-schemes there is the specified natural equivalence $\mathbb{Z}(3{)}_{\mathrm{c}\mathrm{d}\mathrm{h}}\mathrm{/}p\simeq L_{\mathrm{c}\mathrm{d}\mathrm{h},\mathrm{k}}{\tau }^{\le 3}R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}(-,\mathbb{Z}\mathrm{/}p)\mathrm{.}$ Here the left side is the restriction of the actual EM cdh-local weight-three complex on qcqs $\mathbb{Q}$-schemes, with its objectwise derived quotient; $L_{\mathrm{c}\mathrm{d}\mathrm{h},\mathrm{k}}$ is BEM big-site complex-valued cdh localization on qcqs $k$-schemes, with the induced topology and ordinary reflective sheafification specified by the slice-site construction. Equivalently the right side is the restriction of the ambient localized Tate-twisted complex followed by the specified root trivialization. No comparison with a separately fixed separated finite-type or hypercomplete site is asserted.

The actual big cdh slice over a field