The local-global exact sequence for Chow groups of zero-cycles (Q2346001)

Let \(k\) be a number field, and \(\Omega_k\) the set of places of \(k\). Let \(X\) be a proper smooth geometrically integral scheme of finite type defined over \(k\). Let \(\text{CH}_0'(X_v)\) denote the usual Chow group for each non-archimedean place \(v\), otherwise set it to be the cokernel of the norm map \(N_{\bar{k}_v/k_v}:\text{CH}_0(X_v\times_{k_v}\bar{k}_v)\rightarrow\text{CH}_0(X_v)\). Furthermore, for any abelian group \(M\), let \(M^{\wedge}=\varprojlim_n M/nM\). Through Colliot-Thélène's extended Brauer-Manin pairing \(\prod_{v\in\Omega_k}\text{CH}_0(X_v)\times\text{Br}(X)\rightarrow\mathbb{Q}/\mathbb{Z}\) (see \textit{J.-L. Colliot-Thélène} [J. Théor. Nombres Bordx. 7, No. 1, 51--73 (1995; Zbl 0870.14002)]), one has the local-global sequence (E): \(\text{CH}_0(X)^{\wedge}\rightarrow\prod_{v\in\Omega_k}\text{CH}_0'(X_v)^{\wedge}\rightarrow\text{Hom}(\text{Br}(X),\mathbb{Q}/\mathbb{Z})\). In this paper the author proves the following two results: (I) Let \(X\) be geometrically rationally connected, and assume that for all finite extensions \(K/k\) which are linearly disjoint from a fixed finite extension \(L\) of \(k\), the Brauer-Manin obstruction is the only obstruction to weak approximation for \(K\)-rational points on \(X_K\). Then the sequence (E) is exact for \(X\). (II) Let \(L/k\) be a finitely generated field extension, in which \(k\) is algebraically closed, and let \(\{L_i\}\) be a finite family of finite extensions of \(L\) such that \([L_i:L]\) have no common factor other than 1. If for each \(i\) the sequence (E) is exact for a proper smooth variety with function field \(L_i\), then (E) is also exact for those varieties with function field \(L\). The first result is a slightly stronger form of the main result of \textit{Y. Liang} [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 35--56 (2013; Zbl 1264.14033)], and the second one is proved by a restriction-corestriction argument.