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$\mathit{H}\mathrm{9}\mathrm{0}\mathrm{(}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{)}\mathrm{Z}\mathrm{/}\mathrm{\ell }\mathrm{\to }\mathrm{Z}\mathrm{/}\mathrm{\ell }\mathrm{\left[}\mathrm{1}\mathrm{\right]}\mathit{i}\mathrm{\in }\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{.}\mathrm{.}\mathrm{,}\mathrm{}\mathit{n}\mathrm{-}\mathrm{3}\mathrm{\}}$ k-varieties, $\mathit{m}\mathrm{\in }\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{}\mathrm{.}\mathrm{.}\mathrm{,}\mathrm{}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{\}}\mathrm{:}\mathrm{Z}\mathrm{/}{\mathrm{\ell }}^{\mathrm{2}}$-coefficients

Spec $\mathrm{\left(}\mathit{L}\mathrm{\right)}\mathrm{\to }\mathit{S}\mathit{p}\mathit{e}\mathit{c}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{)}\mathrm{0}\mathrm{\to }\mathrm{Z}\mathrm{/}\mathrm{\ell }\mathrm{\to }\mathrm{Z}\mathrm{/}{\mathit{l}}^{\mathrm{2}}\mathrm{\to }\mathrm{Z}\mathrm{/}\mathrm{\ell }\mathrm{\to }\mathrm{0}\mathrm{)}\mathrm{.}{\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}}\mathrm{\left(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\right(}\mathit{X}\mathrm{)}\mathrm{;}\mathrm{Z}\mathrm{/}\mathrm{\ell }\mathrm{(}\mathrm{*}\mathrm{\left)}\mathrm{\right)}$

$\mathit{k}\mathit{k}\mathit{p}\mathrm{\ell }\mathit{L}\mathit{m}\mathit{m}\mathrm{\square }\mathit{X}\mathit{X}\mathit{k}\mathrm{)}\mathrm{\ell }\mathrm{,}$ $\mathrm{\ell }\mathrm{,}$ ( $\mathit{X}\mathrm{\ell }$. $\mathit{\pi }$ : $\mathrm{\ell }\mathrm{.}\mathit{m}\mathrm{\ell }\mathrm{.}\mathit{m}\mathit{n}\mathrm{=}\mathrm{0}\mathit{n}\mathrm{\ge }\mathrm{0}\mathit{n}\mathrm{\ge }\mathrm{1}{\mathit{Q}}_{\mathit{n}}$. ${\mathit{Q}}_{\underset{\mathrm{‾}}{\mathit{a}}}{\mathrm{\Theta }}_{\mathit{k}}^{\mathit{n}}$$\mathit{X}\mathrm{|}\mathit{L}\mathit{\beta }\mathrm{=}{\mathit{Q}}_{\mathrm{0}}\mathrm{\circ }{\mathit{Q}}_{\mathrm{1}}$. mod $\mathrm{\ell }\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{}\mathit{p}\mathit{n}\mathrm{>}\mathrm{0}$. ${\mathit{Q}}_{\mathit{i}\mathrm{+}\mathrm{1}}{\mathit{Q}}_{\mathit{i}\mathrm{+}\mathrm{1}}{\mathit{Q}}_{\mathit{i}\mathrm{+}\mathrm{1}}{\mathrm{\Theta }}_{\mathit{k}}^{\mathit{n}}$. $\mathit{D}\mathit{M}\mathrm{\left(}\mathit{k}\mathrm{\right)}{\mathit{Q}}_{\mathit{i}\mathrm{+}\mathrm{1}}$. $\mathit{S}\mathit{m}\mathrm{\left(}\mathit{k}\mathrm{\right)}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$$\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\left(}\mathit{X}\mathrm{\right)}{\mathrm{2}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{-}\mathrm{1}\mathit{D}\mathit{M}\mathrm{\left(}\mathit{k}\mathrm{\right)}$ : $\mathit{S}\mathit{H}\mathrm{\left(}\mathit{k}\mathrm{\right)}$. ${\mathrm{A}}^{\mathrm{1}}$-weak $\stackrel{\mathrm{ˇ}}{\mathit{C}}\mathrm{\left(}{\mathit{Q}}_{\underset{\mathrm{‾}}{\mathit{a}}}\mathrm{\right)}{\mathit{p}}^{\mathrm{2}}\mathit{\iota }\mathrm{.}\mathit{e}$. ${\mathit{Q}}_{\mathit{n}\mathrm{-}\mathrm{2}}\mathrm{\circ }\mathrm{.}\mathrm{.}\mathit{D}{\mathit{M}}^{\mathit{e}\mathit{f}\mathit{f}}\mathrm{\left(}\mathit{k}\mathrm{\right)}{\mathit{Q}}_{\mathit{i}}\mathrm{\circ }\mathrm{.}\mathrm{.}\mathrm{o}{\mathit{Q}}_{\mathrm{1}}$$\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\left(}{\mathit{Q}}_{\underset{\mathrm{‾}}{\mathit{a}}}\mathrm{\right)}$. $\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\left(}{\mathit{Q}}_{\underset{\mathrm{‾}}{\mathit{a}}}\mathrm{\right)}$. ${\mathrm{v}}_{\mathit{m}}$-points

${\mathit{H}}_{\mathit{B}}^{\mathit{n}\mathrm{+}\mathrm{1}}\mathrm{\left(}\stackrel{\mathrm{ˇ}}{\mathit{C}}\mathrm{\right(}{\mathit{Q}}_{\underset{\mathrm{‾}}{\mathit{a}}}\mathrm{)}\mathrm{,}\mathrm{}{\mathrm{Z}}_{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}\mathrm{(}\mathit{n}\mathrm{\left)}\mathrm{\right)}$

$\mathit{H}\mathit{o}{\mathit{m}}_{\mathit{D}\mathit{M}\mathrm{\left(}\mathit{k}\mathrm{\right)}}\mathrm{(}\mathrm{Z}\mathrm{/}\mathit{l}\mathrm{,}\mathrm{}\mathrm{Z}\mathrm{/}\mathit{p}\mathrm{(}{\mathrm{\ell }}^{\mathit{n}}\mathrm{-}\mathrm{1}\mathrm{\left)}\mathrm{\right)}\mathrm{[}\mathrm{2}\mathit{p}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{]}\mathrm{)}\mathrm{\ell }$ : ${\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}}\mathrm{\left(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\right(}\mathit{X}\mathrm{)}\mathrm{;}\mathrm{Z}\mathrm{/}{\mathrm{\ell }}^{\mathrm{2}}\mathrm{(}\mathrm{*}\mathrm{\left)}\mathrm{\right)}\mathrm{\to }{\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}\mathrm{+}\mathrm{1}}\mathrm{\left(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\right(}\mathit{X}\mathrm{)}\mathrm{;}\mathrm{Z}\mathrm{/}{\mathrm{\ell }}^{\mathrm{2}}\mathrm{(}\mathrm{*}\mathrm{\left)}\mathrm{\right)}$

${\mathit{n}}_{\mathrm{*}}$ : ${\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}}\mathrm{\left(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\right(}\mathit{X}\mathrm{\left|}\mathit{L}\mathrm{\right)}\mathrm{;}\mathrm{Z}\mathrm{/}{\mathrm{\ell }}^{\mathrm{2}}\mathrm{(}\mathrm{*}\mathrm{)}\mathrm{)}\mathrm{\to }{\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}}\mathrm{(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{;}\mathrm{Z}\mathrm{/}{\mathrm{\ell }}^{\mathrm{2}}\mathrm{(}\mathrm{*}\mathrm{)}\mathrm{)}$

${\mathit{n}}^{\mathrm{*}}$ : ${\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}}\mathrm{\left(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\right(}\mathit{X}\mathrm{)}\mathrm{;}\mathrm{Z}\mathrm{/}{\mathit{l}}^{\mathrm{2}}\mathrm{(}\mathrm{*}\mathrm{\left)}\mathrm{\right)}\mathrm{\to }{\tilde{\mathit{H}}}_{\mathit{B}}^{\mathrm{*}}\mathrm{\left(}\underset{\mathrm{‾}}{\mathit{C}}\mathrm{\right(}\mathit{X}\mathrm{\left|}\mathit{L}\mathrm{\right)}\mathrm{;}\mathrm{Z}\mathrm{/}{\mathit{p}}^{\mathrm{2}}\mathrm{(}\mathrm{*}\mathrm{)}\mathrm{)}$

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