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Let $Y$ be a smooth irreducible variety over a field $k$. I know that $\mathcal{O}_Y^\times$, the group of invertible functions on $Y$, embeds in the higher Chow group $\mathrm{CH}^1(Y,1)$ [or the first graded piece of $K_1(Y)$]. Is this always an isomorphism? I know it is when $Y=\mathrm{Spec}(k)$.

If it is always an isomorphism, can you provide a reference? If it is not, can you provide a counterexample?

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See the precise statement in Corollary 4.2 and Theorem 19.1 in Mazza-Voevodsky-Weibel. Lecture notes on Motivic Cohomology.

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