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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!

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  • $\begingroup$ Typo in first line: the presheaf should send $U$ to $K_n(U)$, not $K_n(X)$. $\endgroup$ Jan 27 '14 at 16:46
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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.)

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  • $\begingroup$ thanks for your answer. No need of hypercohomology, I edited the question. Do you have some reference for what you explained? $\endgroup$
    – 1729
    Jan 27 '14 at 16:18
  • $\begingroup$ can you also say how the map $\mathcal{O}_X^\ast \to K_1$ is defined? $\endgroup$
    – 1729
    Jan 27 '14 at 16:20
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    $\begingroup$ @1729: The map is defined by taking a unit to the one-by-one matrix whose only entry is that unit. $\endgroup$ Jan 27 '14 at 16:49
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To answer your reference request, I'd highly recommend Bloch's "Lectures on Algebraic Cycles." It's already a classic, and is uniformly very beautiful. Chapter 4 covers the $K$-theoretic methods to which your question refers. (Ostensibly the chapter is self-contained but does punt proofs of many of the basic properties of $K$-theory, in favor of getting to the cycle theory more rapidly.)

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    $\begingroup$ I'm a big fan of Bloch's lectures on algebraic cycles, but I think it's overkill in the current case, which comes down to the simple linear algebra of row reducibility. $\endgroup$ Jan 27 '14 at 20:55
  • $\begingroup$ I understood the reference request as being about the Bloch-Quillen formula in the second sentence of the question; of course I agree that it is overkill for $K_1$. $\endgroup$ Jan 27 '14 at 22:31

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