Colliot-Thélène, Jean-Louis; Saito, Shuji Zero cycles on \(p\)-adic varieties and Brauer group. (Zéro-cycles sur les variétés \(p\)-adiques et groupe de Brauer.) (French) Zbl 0878.14006 Int. Math. Res. Not. 1996, No. 4, 151-160 (1996). The main object of the paper under review is the Chow group \(\text{CH}_0(X)\) of zero-dimensional cycles on a smooth proper variety \(X\) defined over a \(p\)-adic field \(k\). The authors consider the pairing \[ \text{Br}(X)\times \text{CH}_0(X)\to \text{Br}(k)=\mathbb{Q}/\mathbb{Z}. \] They compute the left kernel of this pairing (restricting their attention to the prime-to-\(p\) torsion of \(\text{Br}(X)\)). Under certain purity conditions, this kernel coincides with the Brauer group of a regular proper model \(\mathcal X\) of \(X\) over the ring of integers of \(k\). Another goal is to compute the index of \(X\) (= the greatest common divisor of the degrees of closed points of \(X\)) in terms of \(\text{Br}(X)\) and \(\text{Br}(\mathcal X)\). The results obtained may be viewed as high-dimensional analogues of theorems of Roquette and Lichtenbaum [see S. Lichtenbaum, Invent. Math. 7, 120-136 (1969; Zbl 0186.26402)]. Reviewer: B.Kunyavskii (Ramat Gan) Cited in 2 ReviewsCited in 16 Documents MSC: 14C25 Algebraic cycles 14F22 Brauer groups of schemes 14G20 Local ground fields in algebraic geometry 14C15 (Equivariant) Chow groups and rings; motives Keywords:zero cycle; Chow group; Brauer group; \(p\)-adic field Citations:Zbl 0186.26402 PDFBibTeX XMLCite \textit{J.-L. Colliot-Thélène} and \textit{S. Saito}, Int. Math. Res. Not. 1996, No. 4, 151--160 (1996; Zbl 0878.14006) Full Text: DOI