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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: arxiv.org/abs/0705.2597 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
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2 Answers

up vote 7 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' (http://www.mathematik.uni-regensburg.de/kerz/index.html) PhD thesis: http://www.opus-bayern.de/uni-regensburg/volltexte/2008/991/pdf/main.pdf 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
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