Exact cdh factorization inputs for the source étale map

PDF p. 5: Prime-to-characteristic comparison

For $F=\mathbb{Q}$ or ${\mathbb{F}}_p$, $j\ge 0$, $n\ge 1$ invertible in $F$, use the derived cdh sheafification $L_{\mathrm{c}\mathrm{d}\mathrm{h}}$ on the source’s entire qcqs site of §3.3. It is the ordinary stable-valued sheafification of source §2.1, left adjoint to inclusion of derived cdh sheaves satisfying covering descent; no identification with unrestricted hypercompletion on the entire qcqs site is asserted. The source supplies: (i) the canonical $M_j\to \mathbb{Z}(j{)}_{\mathrm{c}\mathrm{d}\mathrm{h}}$, an equivalence modulo $n$ in characteristic zero and an equivalence after inverting $p$ in characteristic $p$; (ii) the natural equivalence ${\theta }_j:\mathbb{Z}(j{)}_{\mathrm{c}\mathrm{d}\mathrm{h}}\mathrm{/}n\simeq L_{\mathrm{c}\mathrm{d}\mathrm{h}}{\tau }^{\le j}{B}_j$; (iii) cdh descent of $B_j=R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}(-,{\mu }_n^{\otimes j})$. The last datum permits the induced map ${ϵ}_j$ adjoint to the ordinary truncation map.

Source version: arXiv:2309.08463v1, September 2023. Theorem 4.33(1) and its proof are on printed p. 31 (zero-based PDF page 30). Its proof sentence says “fiber” where the cofiber bound is needed. That sentence is not imported as a fiber bound; the map, descent and comparison data here are unaffected.

Import precisely (i)–(iii) and the source derived-localization convention under the recorded hypotheses. The role of this item is to supply substantial established cdh foundations after the five authored local applications; it does not claim objectwise cofiber bounds merely from local equivalences. No new bound or mixed-characteristic extension is imported.