Isomorphism with higher Chow groups with finite coefficients

Prove that the canonical map from the first Zariski cohomology of the finite-coefficient Quillen $K_2$-sheaf on a regular semistable model to the corresponding higher Chow group with finite coefficients is an isomorphism, including in mixed characteristic.

Background

Theorem 1.3 establishes exact sequences describing the cohomology of the finite-coefficient K2K_2-sheaf on a regular three-dimensional semistable model. The resulting Gersten presentation yields a canonical map to a higher Chow group with finite coefficients.

The authors indicate that an argument analogous to Landsburg’s should prove this map is an isomorphism, but they explicitly leave that question unresolved.

References

By assigning a rational function (or a pair of rational functions) its graph, it is not hard to see using Theorem {thm:Main-11} that there exists a canonical map $H1_(, {_{2, }/n) \to 2(, 1; {}/n)$, where the latter is one of the higher Chow groups of $$ with finite coefficients, defined by Levine . Following the strategy of \S~2.3, it should be possible to prove that this map is an isomorphism. We do not pursue this question here.

— Unramified cohomology and Brauer--Manin pairing  (2609.09127 - Krishna et al., 8 Sep 2026) in Section 1, subsection “Bloch's formula for $K_1(Y)$ and applications,” immediately after Theorem 1.3

Question 3.12. Let the notation be as above and d = 2. Is there a geometric proof of the fact that res : CHd+1(X, 1)Z/pr Z → lim←−nHdNis(Xk, KM Xn,d+1/pr ) is an isomorphism which works for OK a henselian discrete valuation ring with arbitrary residue field?

— Higher dimensional local class field theory  (2609.34437 - Lüders, 28 Sep 2026) in Question 3.12, Section 3.5