A restriction isomorphism for zero-cycles with coefficients in Milnor \(K\)-theory (Q668007)

The article studies smooth projective schemes over an excellent henselian discrete valuation ring $\mathcal{O}_K$. The object is to extend an isomorphism result of \textit{M. Kerz} et al. [Camb. J. Math. 4, No. 2, 163--196 (2016; Zbl 1376.14008)], which deals with the restriction map for the Chow group of zero cycles. The author gives here a solution to a conjecture in [loc. cit.], and thus he is able to cover the harder case of higher Chow groups of zero cycles. Specifically, consider a smooth projective scheme $X$ over an excellent henselian discrete valuation ring, assume that the fiber dimension is $d$ and let $X_K$ and $X_0$ be the generic and reduced fiber respectively. As a motivation, recall that for étale cohomology restriction gives an isomorphism $H^{i}_{\text{ét}}(X,\mathbb Z / n\mathbb Z)) \simeq H^{i}_{\text{ét}}X_0, \mathbb Z / n\mathbb Z)) $, where $n$ is prime to the characteristic $p$ of the residue field $k$. The main theorem proved in the present paper deals with higher Chow groups with coefficients in $\Lambda := \mathbb Z / n\mathbb Z$, it reads: The restriction map $\mathrm{res}^{\mathrm{CH}}:\mathrm{CH}^j(X,j-d)_{\Lambda}\rightarrow{} \mathrm{CH}^j(X_0,j-d)_{\Lambda}$ is an isomorphism for all $j$. As a consequence, if the residue field is finite or algebraically closed, then one has that the étale cycle map $\rho_X^{j,j-d}:\mathrm{CH}^j(X,j-d)_{\Lambda} \rightarrow H^{d+j}_{\text{ét}}(X,\Lambda(j))$ is an isomorphism for all $j$. The strategy of proof relies on the use of Kato and Rost complexes, in this way one has at disposal a manageable presentation of the higher Chow groups of zero cycles. Briefly the basic fact is that said complexes make use of the Milnor \(K\)-theory of the function field of the points of given codimension in $X$ in a parallel way as it is done for Quillen complexes. The restriction map with finite coefficients can then be investigated and controlled geometrically, at the core of the author's method is the idea that one can lift zero-cycles from $X_0$ to one cycles on $X$ in a convenient way. The implementation requires careful examination of spectral sequences, it relies on the use of a theorem of Voevodsky, i.e. of the Beilinson-Lichtenbaum conjecture, and also on the Kato conjectures. Several applications are produced, moreover in the final sections results on torsion subgroups are found for the case of non Archimedian local fields and finiteness results are proved over finite fields.