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This article confirms a consequence of the non-abelian Iwasawa main conjecture. It is proved that under a technical condition the étale cohomology groups $H^1({𝒪}_K[1/S],H^i(\overline{X},{\mathbb{Q}}_p\left(j\right)))$, where $X\to Spec{𝒪}_K[1/S]$ is a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to $H^i(\overline{X},{\mathbb{Z}}_p\left(j\right))$. Using the “Bloch-Kato-conjecture” a similar result is proven for motivic cohomology with finite coefficients.