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Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence

$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \bigoplus_{x \in X^{(1)}} K_{n - 1}(k(x))$$

where $\mathcal{K}_n$ is the Zariski sheaf associated to $K_n$?

More specifically, I would like the following to be true.

1) The above sequence is exact when $X$ is a (separated noetherian) regular scheme of dimension one.

2) $\mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x))$ is injective for all regular (separated noetherian) schemes.

And I would like these to be true with the Nisnevich sheafification of $K_n$ as opposed to the Zariski one (I'm aware that in many cases the two sheafifications are the same so I guess another part of the question is in which cases $(K_n)_{Zar} = (K_n)_{Nis}$).

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side remark: Adelic resolution for homology sheaves Sergey Gorchinskiy ... In this case it is proven that the adelic complex provides a flasque resolution for the above sheaf and that the natural morphism to the Gersten complex is a quasiisomorphism. The main advantage of the new adelic resolution is that it is contravariant and multiplicative in contrast to the Gersten resolution. – Alexander Chervov Sep 26 '12 at 6:28
up vote 11 down vote accepted

Gersten's conjecture is known for regular local rings containing a field (Panin extended the result of Quillen for smooth schemes). In mixed characteristic, it is known that the statement for a discrete valuation ring implies the statement for smooth local rings over a discrete valuation ring (by work of Gillet-Levine). And the case of a discrete valuation ring is known with finite coefficients (by Gillet away from the residue characteristic, by Geisser-Levine at the characteristic). Unfortunately, rational coefficients are completely unknown as far as I know.

In particular, the answer is "unknown" to both your questions in mixed characteristic.

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You might want to consult Moritz Kerz' ( PhD thesis: On page 4:

"The exactness of the Gersten complex for A regular, equicharacteristic and with infinite residue field, also known as the Gersten conjecture for Milnor K-theory, is of independent geometric interest and one of the further main results of this thesis. For a detailed overview of our results we refer to Sections 3.1 and 4.1."

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Thanks Timo. I'm more interested in Quillen K-theory than Milnor K-theory for the moment though. – name Dec 6 '11 at 17:23

On the arXiv there is a paper of Satoshi Mochizuki on "Gersten’s conjecture for commutative discrete valuation rings" from the year 2007. In this paper Mochizuki claims to prove the Gersten conjecture for all commutative DVRs. But the paper didn't appear in any journal, apperently. Does anyone know if it has been verified?

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