]> No Title

a a a a a a a a a a a above above a: all all also analogous analogue analogue and

and and and and and any any any appears are argument argument) As as as as as asts associated assume as sumption as sumption

be because because bide- bigraded Bockstein Bockstein. But But by by by by by by byby

case case category category category category category category chain changes. classes.

closed coho- coho- cohomology cohomology cohomology cohomology coincide complete

complexes composition composition composition composition conclusion considered

construction. correct corresponding corresponding corresponding covered

cleduce defines degree degree degree denoted dimension dimension divisible do do

equivalences, equivalent exact extension extension extension extension

fact for for for form $\mathrm{F}$ from from

general get give grees group group group group groups groups

hand, has has has has has Here holds. homology homology homology. homotopy homotopgif image implies In In in in in in in in in in in in Indeed, inductive injective injective.injectivity injectivity integral introduce is is is is is is is is is is is is it it it its

kernel

latter Let Let Let Let Let like localization localizes long

Margolis Margolis means means middle mo- mology mology MOREL morphism morphismmorphism motivic motivic motivic motivic motivic much much multiplication

multiplication multiplication

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

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

$\mathit{k}\mathit{k}\mathit{p}\mathrm{\ell }\mathit{L}\mathit{m}\mathit{m}\mathrm{\square }\mathit{X}\mathit{X}\mathit{k}\mathrm{\right)}\mathrm{\ell }\mathrm{,}$ $\mathrm{\ell }\mathrm{,}$ ( $\mathit{X}\mathrm{\ell }$. $\mathit{\pi }$ : $\mathrm{\ell }\mathrm{.}\mathit{m}\mathrm{\ell }\mathrm{.}\mathit{m}\mathit{n}\mathrm{=}\mathrm{0}\mathit{n}\mathrm{\ge }\mathrm{0}\mathit{n}\mathrm{\ge }\mathrm{1}{\mathit{Q}}_{\mathit{n}}$. ${\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

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

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

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

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

$\mathrm{\left[}\mathrm{\right]}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{c}\mathrm{h}$ Nis- Nisnevich not notion

Observe obvious of of of of of of of of of of of of of of of of of of of of of of ofof of of of of of of of of of of of of of of of On on on on on One One one one

only operation operation operation operation operation. other

$\mathrm{L}\mathrm{3}\mathrm{8}\mathrm{1}\mathrm{\right)}\mathrm{2}\mathrm{.}\mathrm{2}\mathrm{2}\mathrm{\right)}\mathit{5}\mathrm{.}\mathit{1}\mathit{0}\mathrm{.}\mathrm{7}\mathrm{.}\mathrm{1}\mathrm{2}\mathrm{7}\mathrm{.}\mathrm{1}\mathrm{2}\mathrm{.}\mathrm{7}\mathrm{.}\mathit{1}\mathit{2}\mathrm{.}\mathrm{7}\mathrm{.}\mathrm{5}\mathrm{7}\mathrm{.}\mathrm{8}$. "nonlinear" eacutetale eacutetale fiber fiber finite finitefirst first (as (by (induced (use

point point point pointed point). preserves previous previous prime prime prime problemproblem, proceeds projection proof Proof Proof property prove prove prove

rational rational rational rational reduced reduced reduction relevant remains required

respect respect Roughly

oame same second separable sequences sets sheaves sheaves shift simplicial sketch sketchsmooth So solve source speaking, stable subquadric suffices

take target that that that that that that that that that The The The the the the thebhe the the the the the the the the the the the the the the the the the the the thethe the the the the the the the the the the the the the the the the the the the theihe the the the the the the the the the the the the the The Then then theorem

fheorem theorem theorem theorem Theorem theorem theory This this this this this thisThus, tivic To To to to to to to to to to to to to to transfer transfer transfers.

hriangulated trivial trivial

Unfortunately, us us us use

vanishes vanishes vanishes vanishes. vanishing

way way, way. We we well. well. with with with with with with with works would

${\mathrm{7}}_{\mathrm{}}\mathit{P}\mathrm{\Gamma }\mathrm{\cap }$ zPro zrro zero