Torsion freeness of top threefold motivic cohomology

PDF p. 5: Proposition 2.6: torsion freeness

For an affine threefold $X$ over an algebraically closed field $k$, $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{6,3}(X,\mathbb{Z})$ is torsion-free. Threefold means finite type and dimension three, without any smoothness, reducedness or irreducibility assumption.

The article's Proposition 2.6, pp. 5–6, states the full assertion but supplies only a characteristic-zero proof. This is a nonfatal omission in that source. The current argument uses the separate published report Theorem 4 at weight three and EM v1 Lemma 4.16, rather than treating the article's deferred positive-characteristic proof as a premise. The report's separate reducedness hypothesis in Theorem 1 is not imposed in its Theorem 4.

The report Theorem 3(2) prints an étale lower-right corner, while EM v1 Theorem 4.28 prints the éh corner. The printed discrepancy is verified; a theorem-level error in that square is not established here. The proof above uses the precise EM comparison through its declared suppliers. The report's Theorem 4 proof also omits the shift from its description of the comparison cofiber: EM v1 Corollary 4.32 gives $R{\mathrm{\Gamma }}_{\mathrm{c}\mathrm{d}\mathrm{h}}(X,{\tilde{\nu }}_1(3))[-4]$. This confirmed source omission is nonfatal because the report proves vanishing of the obstruction itself.

The report Remark 5 has a separate incorrect duality exponent. With its stated higher-Chow homological indexing, $X={\mathbb{P}}^1$, $j=1$, $i=0$ and invertible prime coefficients give a nonzero left side and a zero degree-three right side. This display is not used above. In the smooth proper prime-to-characteristic setting the compatible degree is $2d-2j+i$; no replacement for the report's full quasiprojective or syntomic scope is asserted.