Let $A$ be a one-dimensional, Noetherian, local ring, and let $\mathcal{K}_n$ denote the sheafification of $K_n$ (degree $n$ $K$-theory) on the Nisnevich site of $X:=\mbox{Spec }A$. Then is it true that $H^1(X_{\mbox{Nis}},\mathcal{K}_n))=0$? Since $X_{\mbox{Nis}}$ has cohomological dimension $1$, this is the same as asking whether $\mathcal{K}_n$ is acyclic on $X_{\mbox{Nis}}$.

If $K$-theory satisfies descent on $X$ in the Nisnevich topology (which I think it does if $A$ is regular, but I'd be interested to know if this is true more generally), then the descent spectral sequence $H^p(X_{\mbox{Nis}},\mathcal{K}_q)\Longrightarrow K_{q-p}(X)$ gives short exact sequences

$$0\to H^1(X_{\mbox{Nis}},\mathcal{K}_n)\to K_{n-1}(A)\to H^0(X_{\mbox{Nis}},\mathcal{K_{n-1}})\to 0$$

and so my question becomes equivalent to the following: is $K_{n-1}(A)\to H^0(X_{\mbox{Nis}},\mathcal{K}_{n-1})$ injective (and hence an isomorphism)?

Well, any Nisnevich cover of $X$ must include a point isomorphic to $\mbox{Spec }F$, where $F$ is the field of fractions of $A$ (assume $A$ is a domain for simplicity for a moment), so if $A$ satisfies Gersten's conjecture, i.e. $K$-theory of $A$ embeds into $K$-theory of $F$, then it seems to follow that $K_{n-1}(A)\to H^0(X_{\mbox{Nis}},\mathcal{K}_{n-1})$ is indeed injective. This appears to answer my original question in the affirmative when $A$ is a discrete valuation ring (at least, assuming Gersten for dvrs).

When $A$ is non-regular I therefore seem to be asking for some sort of generalisation of Gersten's conjecture, in the Nisnevich topology. I've asked a few experts but without success, so I'm handing the question to MO, since I'd be amazed if this question had not already been considered! Thank you!



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