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 $(K\_n)\_{Zar} = (K\_n)\_{Nis}$).



State of the art for Gersten's conjecture for K-theory?
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.


