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Zero cycles on \(p\)-adic varieties and Brauer group. (Zéro-cycles sur les variétés \(p\)-adiques et groupe de Brauer.) (French) Zbl 0878.14006

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)].

MSC:

14C25 Algebraic cycles
14F22 Brauer groups of schemes
14G20 Local ground fields in algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives

Citations:

Zbl 0186.26402
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