Let X be a quasiprojective variety defined over an algebraically closed field k. If X is nonsingular, the vector bundles over X are deeply related with the Chow group of algebraic cycles on X modulo rational equivalence. Namely, H^p(X,K_p) = CH^p(X), where K_p, p \geq 0, are sheaves in the Zariski topology of X defined by Quillen, who was motivated by previous results of Bloch, Gersten, and others. The question arises about the geometrical meaning of H^p(X,K_p) when X is singular. An answer is given in the case of isolated singularities and in the case of a surface X. Also, the kernel of H^p(X,K_p) \to H^p(Y,K_p), where Y !\to X is a desingularization of X, is studied.
Quillen's K-theory and algebraic cycles on singular varieties.inGeometry today (Rome, 1984), 75--85, Progr. Math.,60,Birkhäuser
COLLINO, Alberto
1985-01-01
Abstract
Let X be a quasiprojective variety defined over an algebraically closed field k. If X is nonsingular, the vector bundles over X are deeply related with the Chow group of algebraic cycles on X modulo rational equivalence. Namely, H^p(X,K_p) = CH^p(X), where K_p, p \geq 0, are sheaves in the Zariski topology of X defined by Quillen, who was motivated by previous results of Bloch, Gersten, and others. The question arises about the geometrical meaning of H^p(X,K_p) when X is singular. An answer is given in the case of isolated singularities and in the case of a surface X. Also, the kernel of H^p(X,K_p) \to H^p(Y,K_p), where Y !\to X is a desingularization of X, is studied.
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