The Gersten conjecture for Milnor K-theory, saying that the Gersten complex $$0\rightarrow \mathcal{K}^M_X\rightarrow \oplus_{x\in X^0}i_{x*}(K^M_n(x))\rightarrow \oplus_{x\in X^1}i_{x*}(K^M_{n-1}(x))\rightarrow...$$ is exact, was proved by M. Kerz for $X$ a regular excellent scheme over an infinite field $k$.

Is the same (and in the same form) expected to hold for $k$ finite and for $X$ a regular scheme over a discrete valuation ring $S$ of mixed characteristic? What are the problems that arise here?

If $X$ is a scheme of relative dimension $d$ over a discrete valuation ring $S$ of mixed characteristic, should Bloch's formula (assuming the above conjecture holds in the given form) say that $$H^n(X,\mathcal{K}^M_n)\cong CH^n(X)$$ or that $$H^n(X,\mathcal{K}^M_n)\cong coker(\oplus_{x\in X^{n-1}}i_{x*}(H^{n-1}_x(X,K^M_n(x))\rightarrow \oplus_{x\in X^n}i_{x*}H^n_x(X,K^M_{n}(x))),$$ i.e. does purity hold in this case and does $H^{n+1}_x(X,K^M_{n}(x))$ vanish?

  • $\begingroup$ Did you read all Kerz's papers on the subject? As far as I remember, he proposed "correcting" Milnor's K-theory in the case when residue fields are finite. $\endgroup$ – Mikhail Bondarko Mar 16 '15 at 19:52

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