A finiteness theorem for the Chow group of zero-cycles of a linear algebraic group on a \(p\)-adic field
type: Item
instance of: scholarly article
title: A finiteness theorem for the Chow group of zero-cycles of a linear algebraic group on a \(p\)-adic field
zbMATH Open document ID: 1080.14012
author: Jean-Louis Colliot-Thélène
published in: Inventiones Mathematicae
publication date: 2005-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0502133
review text: 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.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 2156032
MaRDI profile type: Publication
OpenAlex ID: W1968928059
arXiv ID: math/0502133
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DOI: 10.1007/S00222-004-0393-0
description: scientific article