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!



on the Zariski sheafification of Quillen's K-theory
I'm not sure why you're writing ${\mathbb H}$ instead of $H$. That said, the answer to questions 1 and 2 is yes. The reason is that $K\_1(R)=R^\*$ for any (commutative) local ring, so the map ${\cal O}\_X^\*\rightarrow K\_1$ is stalkwise an isomorphism, hence an isomorphism. To confirm the result for local rings, check that every invertible matrix over a local ring $R$ can be made triangular (hence elementary) via row reduction. (Use the fact that every row and column of an invertible matrix must contain an invertible element.)


