The Brauer-Manin pairing, class field theory, and motivic homology (Q2840043)

Let \(X\) be a smooth variety over a \(p\)-adic field \(k\). For \(i \in \mathbb{Z}_{\geq 0}\), we write \(X_{(i)}\) for the set of all points of \(X\) of dimension \(i\). We put \(A^{\ast}=\mathrm{Hom}(A,\mathbb{Q}/\mathbb{Z})\) for an abel group \(A\). For any \(x \in X_{(0)}\), local class field theory yields canonical maps \(\psi_x^{\ast}:\mathrm{Br}(x) \cong \mathbb{Q}/\mathbb{Z}\) and \(\rho_x:k(x)^{\times} \to \pi_1^{\mathrm{ab}}(x)\). We define \(\psi_x\) to be the composition of the dual of \(\psi_x^{\ast}\) and canonical inclusion \(\psi_x:\mathbb{Z} \hookrightarrow \hat{\mathbb{Z}} \cong \mathrm{Br}(x)^{\ast}\). We obtain the homomorphisms \(\tilde{\psi}_X:\bigoplus_{x \in X_{(0)}} \mathbb{Z} \to \mathrm{Br}(X)^{\ast}\) and \(\tilde{\rho}_X:\bigoplus_{x \in X_{(0)}} k(x)^{\times} \to \pi_1^{\mathrm{ab}}(X)\) by taking the direct sum of \(\psi_x\) and \(\rho_x\) values. When \(X\) is proper over \(k\), \textit{Yu. I. Manin} [in: Actes Congr. internat. Math. 1970, No.1, 401--411 (1971; Zbl 0239.14010)], \textit{S. Bloch} [Ann. Math. (2) 114, 229--265 (1981; Zbl 0512.14009)] and \textit{S. Saito} [J. Number Theory 21, 44--80 (1985; Zbl 0599.14008)] showed that \(\tilde{\psi}_X\) and \(\tilde{\rho}_X\) factor, respectively, through NEWLINE\[NEWLINE\mathrm{CH}_0(X):=\mathrm{coker}\left[ \bigoplus_{x \in X_{(1)}} k(x)^{\times} \to \bigoplus_{x \in X_{(0)}} \mathbb{Z} \right]NEWLINE\]NEWLINE and NEWLINE\[NEWLINESK_1(X):=\mathrm{coker}\left[ \bigoplus_{x \in X_{(1)}} K_2(k(x)) \to \bigoplus_{x \in X_{(0)}} k(x)^{\times} \right];NEWLINE\]NEWLINE the induced pairing \(\mathrm{CH}_0(X) \times \mathrm{Br}(X) \to \mathbb{Q}/\mathbb{Z}\) and the induced map \(SK_1(X) \to \pi_1^{\mathrm{ab}}(X)\) are, respectively, called the Brauer-Manin pairing and reciprocity map of class field theory, If \(X\) is not proper over \(k\), then these maps do not factor through \(\mathrm{CH}_0(X)\) or \(SK_1(X)\).NEWLINENEWLINEFor a variety over a field \(F\) and \(r \in \mathbb{Z}_{\geq 0}\), let \(C_r(V)\) be Wiesend's tame ideal class group of degree \(r\). If \(V\) is proper over \(F\), then we have \(C_0(V)=\mathrm{CH}_0(V)\) and \(C_1(V)=SK_1(V)\). The author proves that for a smooth variety \(V\) over a perfect field \(F\) and \(r \in \mathbb{Z}_{\geq 0}\), there is a canonical isomorphism \(C_r(V) \cong H_{-r}^M(V,\mathbb{Z}(-r))\), where \(H_{-r}^M(V,\mathbb{Z}(-r))\) is the motivic homology group. He showed that the homomorphisms \(\tilde{\psi}_X\) and \(\tilde{\rho}_X\) induce well-defined homomorphisms \(\psi_X:C_0(X) \to \mathrm{Br}(X)^{\ast}\) and \(\rho_X:C_1(X) \to \pi_1(X)^{\mathrm{ab}}\) for arbitrary smooth (possibly not proper) variety \(X\) over \(k\). In general, these maps are far from being isomorphisms even when \(X\) is proper, but he proves several partial results. For example, he shows that for a curve \(X\), \(\psi_X\) is injective with dense image, the kernel of \(\rho_X\) is the maximal divisible subgroup of \(C_1(X)\), and the cokernel of \(\rho_X\) is a free \(\hat{\mathbb{Z}}\)-module of finite rank. For an open subscheme \(X\) of a smooth projective surface which is geometrically irreducible and rational, the kernal of \(\rho_X\) is the maximal divisible subgroup of \(C_1(X)\), and if the irreducible components of \(\overline{X} \setminus \overline{U}\) generate the Néron-Severi group \(\mathrm{NS}(\overline{X})\), then the left kernel of \(\psi_X\) is the maximal divisible subgroup of \(C_0(X)\).