Let $X$ be a smooth k-variety and denote by $K_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K_n(U)$.

The Bloch-Quillen formula says that $CH^n(X) \simeq H^n(X, K_n)$.

In particular, $CH^1(X)\simeq H^1(X, K_1)$.

On the other hand, $CH^1(X)$ is just the Picard group $H^1(X, \mathcal{O}_X^\ast)$.

Question: Is it true that $K_1 \simeq \mathcal{O}_X^\ast$?

I also would be happy if someone could recommend a good reference for this topic.

Thanks!