On non-commutative twisting in étale and motivic cohomology

Abstract

It is proved that under a technical condition the ¶etale cohomol-
ogy groups H1(OK[1=S];Hi(¹X ;Qp(j)), where X SpecOK[1=S] is
a smooth, projective scheme, are generated by twists of norm com-
patible units in a tower of number ¯elds associated to Hi(¹X; Zp(j)).
This con¯rms a consequence of the non-abelian Iwasawa main conjec-
ture. Using the -Kato-conjecture" a similar result is proven for
motivic cohomology with ¯nite coe±cients.