The actual composite and its objectwise cofiber

PDF p. 5: Prime-to-characteristic comparison

Let $F=\mathbb{Q}$ or ${\mathbb{F}}_p$, let $j$ be a nonnegative integer and let $n\ge 1$ be an integer invertible in $F$. Work with $D(\mathbb{Z})$-valued presheaves on all qcqs $F$-schemes. Write $M_j=\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t},F}$ for the source motivic functor, $B_j=R{\mathrm{\Gamma }}_{\mathrm{e}\mathrm{t}}(-,{\mu }_n^{\otimes j})$ for the derived small-étale coefficient functor, and $\mathbb{Z}(j{)}_{\mathrm{c}\mathrm{d}\mathrm{h},F}$ for the source cdh-local classical-input complex. The slash denotes the objectwise derived quotient $A\mathrm{/}n=\mathrm{cofib}(n:A\to A)=A{\otimes }^{\mathbf{L}}\mathbb{Z}\mathrm{/}n$. Use the source qcqs cdh site generated by Nisnevich covers and abstract blowup families whose closed immersion and proper morphism are finitely presented. Here $L_{\mathrm{c}\mathrm{d}\mathrm{h}}$ is the source ordinary stable-valued sheafification, left adjoint to inclusion of derived cdh sheaves satisfying covering descent. Let ${\theta }_j:\mathbb{Z}(j{)}_{\mathrm{c}\mathrm{d}\mathrm{h},F}\mathrm{/}n\stackrel{\sim }{\to }{L}_{\mathrm{c}\mathrm{d}\mathrm{h}}{\tau }^{\le j}{B}_j$ be the source natural comparison. The map of interest is the specified composite $f_j:M_j\mathrm{/}n\stackrel{{\alpha }_j}{\to }\mathbb{Z}(j{)}_{\mathrm{c}\mathrm{d}\mathrm{h},F}\mathrm{/}n\stackrel{{\theta }_j}{\to }{L}_{\mathrm{c}\mathrm{d}\mathrm{h}}{\tau }^{\le j}{B}_j\stackrel{{ϵ}_j}{\to }{B}_j\mathrm{.}$ For each qcqs $F$-scheme $X$, its actual cofiber $C_j(X)=\mathrm{cofib}(f_j(X))$ has $H^a(C_j(X))=0$ for $a\le j$. This is an objectwise statement about this composite, not only vanishing of its cdh sheafification.