Torsion in the Chow group of codimension two: The case of varieties with isolated singularities
type: Item
instance of: scholarly article
title: Torsion in the Chow group of codimension two: The case of varieties with isolated singularities
zbMATH Open document ID: 0589.14007
DOI: 10.1016/0022-4049(84)90033-1
author: Alberto Collino
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let X be a nonsingular quasiprojective algebraic variety over an algebraically closed field k, let \(CH^ 2(X)\) be the group of codimension 2 cycles modulo rational equivalence, and let \({}_ nCH^ 2(X)\) be the kernel of multiplication by n in \(CH^ 2(X)\). From a result of \textit{A. S. Merkur'ev} and \textit{A. A. Suslin} on \(K_ 2\) [cf. Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)] it follows that if n is prime to char k, then \({}_ nCH^ 2(X)\) is finite. The author extends this result to (a suitably defined) Chow group of algebraic varieties with isolated singularities. As an application, the author shows that if Y is a complete surface with isolated singularities, there is an isomorphism \(CH^ 2(Y)(\ell){\tilde \to}Alb(Y')(\ell)\), where Y' is the desingularization of Y and \(CH^ 2(Y)(\ell)\) is the subgroup of elements whose order is a power of \(\ell\).
reviewed by: F. L. Zak
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 3946313
zbMATH Keywords: codimension 2 cycles modulo rational equivalence
zbMATH Keywords: Chow group of algebraic varieties with isolated singularities
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90033-1
OpenAlex ID: W1989122576
cites work: Q3913511
cites work: Q3876042
cites work: Torsion algebraic cycles, K2, and Brauer groups of function fields
cites work: Gersten's conjecture and the homology of schemes
cites work: Quillen's K-theory and algebraic cycles on almost non-singular varieties
cites work: Washnitzer's conjecture and the cohomology of a variety with a single isolated singularity
cites work: Torsion Zero-Cycles on Singular Varieties
cites work: Q3582378
description: scientific article; zbMATH DE number 3946313