On the cycle map for torsion algebraic cycles of codimension two
type: Item
instance of: scholarly article
title: On the cycle map for torsion algebraic cycles of codimension two
zbMATH Open document ID: 0764.14004
DOI: 10.1007/BF01243920
author: Shuji Saito
published in: Inventiones Mathematicae
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/143949
review text: For complete varieties \(X\) over a field \(k\) and positive integers \(n\), conditions that entail the injectivity of the map \(\rho_{n,tor}:CH^ 2(X)_{tor}\to H^ 4(X,\mu_ n^{\otimes 2})\) obtained by composing the inclusion \(CH^ 2(X)_{tor}\hookrightarrow CH^ 2(X)\), the quotient map \(CH^ 2(X)\to CH^ 2(X)/n\) and the cycle map \({\rho_ n:CH^ 2(X)/n\to H^ 4(X,\mu_ n^{\otimes 2})}\) are discussed \((\mu_ n\) is the sheaf of \(n\)-th roots of unity on \(X\) and \(CH^ 2(X)\) is the group of rational classes of codimension 2 cycles on \(X)\). We cannot reproduce here the several sets of subtle conditions that are proved to suffice for the injectivity, but the flavour of them can be appreciated by the following sample case: if \(X\) is projective and smooth over a number field \(k\), if the Picard variety of \(X\) has potentially good reduction, and if \(H^ 2(X_{zar},{\mathcal O}_ X)=0\), then there exists an integer \(N>0\) such that \(\rho_{n,tor}\) is injective for any \(n\) divisible by \(N\). Let us also state one of the theorems that are established with the methods introduced to prove the injectivity results: If \(X\) is smooth over a field \(k\) which is a finitely generated extension of \(\mathbb{Q}\), if \(H^ 2(X_{zar}{\mathcal O}_ X)=H^ 1(X_{zar},{\mathcal O}_ X)=0\), and if \(X\) has a \(k\)-rational point, then \(CH^ 2(X)_{tor}\) is finite.
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 59783
zbMATH Keywords: torsion algebraic cycles
zbMATH Keywords: Chow group
zbMATH Keywords: cycle map
zbMATH Keywords: rational classes of codimension 2
reviewed by: Sebastian Xambó
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59783