A finiteness theorem for the Chow group of zero-cycles of a linear algebraic group on a \(p\)-adic field
Let \(X\) be a smooth projective variety over a field \(k\). Let \(\text{CH}_0 (X)\) denote the Chow group of zero cycles modulo rational equivalence, and \(A_0(X)\subset \text{CH}_0(X)\) the subgroup of zero cycles of degree zero. When \(X\) is rationally connected and \(k\) is a \(p\)-adic field, namely a finite extension of \(\mathbb Q_p\), one conjectures that the group \(A_0(X)\) is finite [\textit{J.-L. Colliot-Thélène}, J. Théor. Nombres Bordx. 7, 51--73 (1995; Zbl 0870.14002)]. In this paper the author establishes the following. Let \(G\) be a linear algebraic group over a \(p\)-adic field \(k\) and let \(X\) be a smooth \(k\)-compactification of \(G\). Then the group \(A_0(X)\) is the direct sum of a finite group and a \(p\)-primary torsion group of finite exponent. For the proof, he employs a structure theorem on the group \(G(k)/R\) of \(R\)-equivalence classes of \(G\). When \(G\) is connected reductive \(k\)-group, an exact sequence \(1\to S\to H\to G\to 1\) is said to be a flasque resolution of \(G\), if \(H\) is an extension of a quasi trivial \(k\)-torus by a connected and simply connected semisimple \(k\)-group, and \(S\) is a flasque \(k\)-torus which is central in \(H\). (Here a \(k\)-torus \(T\) is said to be flasque if for any open subgroup \({\mathbf h}\) of the absolute Galois group of \(k\) the cohomology group \(H^1({\mathbf h}, X_* (T))\) vanishes.) A key ingredient mentioned above shows that if \(1\to S \to H\to G\to 1\) is a flasque resolution of \(G\), then the coboundary map \(G(k)\to H^1(k,S)\) induces an isomorphism \(G(k)/R\to H^1(k,S)\) of finite abelian groups. In the final section, he discusses the finiteness of \(A_0(X)\) for rationally connected varieties.
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