Let \(k\) be an algebraic number field. Fix an algebraic closure \(\bar k\) of \(k\) and denote by \(G_k\) the absolute Galois group \(\text{Gal}(\bar k/k)\). Let \(X\) be a projective smooth variety over \(k\) and put \(\bar X:=X\otimes_k \bar k\). For any prime \(p\) and integers \(r,m\geq 1\), Bloch and Kato have a conjecture concerning the image of the \(p\)-adic regulator map \[ \text{reg}^{r,m}: \text{CH}^r(X,m)\otimes \mathbb{Q}_p\to H_{\text{cont}}^1(k,H_{\text{ét}}^{2r-m-1}(\bar X,\mathbb{Q}_p(r))) \] from Bloch’s higher Chow group to continuous Galois cohomology of \(G_k\). The paper under review concentrates on the Bloch-Kato conjecture in the special case \((r,m)=(2,1)\). That is
H1: The image of the regulator map \[ \text{reg}^{2,1}: \text{CH}^2(X,1)\otimes \mathbb{Q}_p\to H_{\text{cont}}^1(k,H_{\text{ét}}^{2}(\bar X,\mathbb{Q}_p(2))) \] agrees with \(H_g^1(k,H_{\text{ét}}^{2}(\bar X,\mathbb{Q}_p(2)))\).
In this paper, under the following assumption
(*): There exists a regular scheme \(\mathcal{X}\) which is proper and flat over \(S=\text{Spec}(\mathcal{O}_k)\) and whose generic fiber is \(X\). Moreover, \(\mathcal{X}\) has good or semistable reduction at each closed point of \(S\) of characteristic \(p\),
the authors have shown that H1 is related to the finiteness of two arithmetic objects: one is \(\text{CH}^2(X)_{p\text{-tors}}\), the \(p\)-primary torsion part of the Chow group \(\text{CH}^2(X)\); the other one is \(H^3_{\text{ur}}(K,X;\mathbb{Q}_p/\mathbb{Z}_p(2))\), an unramified cohomology of \(X\) which can be roughly described as follows. Let \(K=k(X)\) be the function field of \(X\), the unramified cohomology group \(H^3_{\text{ur}}(K,\mathbb{Q}_p/\mathbb{Z}_p(2))\) is defined as the subgroup of \(H^3_{\text{ét}}(\text{Spec}(K),\mathbb{Q}_p/\mathbb{Z}_p(2))\) consisting of those elements that are “unramified” along all points \(x\in \mathcal{X}\) of codimension \(1\), and \(H^3_{\text{ur}}(K,X;\mathbb{Q}_p/\mathbb{Z}_p(2))\) is defined as \[ \text{Im}\big(H^3_{\text{ét}}(X,\mathbb{Q}_p/\mathbb{Z}_p(2))\to H^3_{\text{ét}}(\text{Spec}(K),\mathbb{Q}_p/\mathbb{Z}_p(2))\big)\cap H^3_{\text{ur}}(K,\mathbb{Q}_p/\mathbb{Z}_p(2)). \] Actually, the authors consider a variant of H1:
H1*: The image of the regulator map with \(\mathbb{Q}_p/\mathbb{Z}_p\)-coefficients \[ \text{reg}_{\mathbb{Q}_p/\mathbb{Z}_p}: \text{CH}^2(X,1)\otimes\mathbb{Q}_p/\mathbb{Z}_p\to H^1_{\text{Gal}}(k,H_{\text{ét}}^{2}(\bar X,\mathbb{Q}_p/\mathbb{Z}_p(2))) \] agrees with \(H^1_{g}(k,H_{\text{ét}}^{2}(\bar X,\mathbb{Q}_p/\mathbb{Z}_p(2)))_{\text{Div}}\). Here for an abelian group \(M\), \(M_{\text{Div}}\) denotes its maximal divisible subgroup.
They have shown that H1 always implies H1*. The main result of this paper is the following. If (*) holds and \(p\geq5\), then H1* implies that \(\text{CH}^2(X)_{p\text{-tors}}\) and \(H^3_{\text{ur}}(K,X;\mathbb{Q}_p/\mathbb{Z}_p(2))\) are finite. Conversely, assume that the reduced part of every closed fiber of \(\mathcal{X}/S\) has simple normal crossings on \(\mathcal{X}\), and that the Tate conjecture holds in codimension \(1\) for the irreducible components of those fibers, then the finiteness of \(\text{CH}^2(X)_{p\text{-tors}}\) and \(H^3_{\text{ur}}(K,X;\mathbb{Q}_p/\mathbb{Z}_p(2))\) implies H1*.
As an application, the authors have proved an injectivity result on the torsion cycle class map of codimension \(2\) with values in a new \(p\)-adic cohomology of \(\mathcal{X}\) introduced by K. Sato, which is a candidate of the conjectural étale motivic cohomology with finite coefficients of Beilinson-Lichtenbaum.