K-theory and intersection theory revisited
type: Item
instance of: scholarly article
title: K-theory and intersection theory revisited
zbMATH Open document ID: 0651.14001
DOI: 10.1007/BF00539625
published in: \(K\)-Theory
publication date: 1987-01-01T00:00:00Z
review text: If X is a nonsingular variety over k the Bloch-Quillen isomorphism \(\eta: CH\;p(X)\simeq H\;p(X,{\mathcal K}_ p({\mathcal O}_ X))\) describes the Chow group of codimension p cycles modulo rational equivalence in terms of sheaf cohomology. The following theorem is proved: ``Let X be a smooth variety over k, Y, Z two integral subschemes of codimension p and q, respectively, which intersect properly on X. If \(Y\cdot Z\) is the intersection cycle, then \(\eta (Y\cdot Z)=(-1)^{pq}\eta (Y)\eta (Z)\in H^{p+q}(X,{\mathcal K}_{p+q}({\mathcal O}_ X))9.\) Unlike the former proofs of Grayson and of the author, this proof is essentially a formal argument using natural properties of Quillen's spectral sequence, the K-theory product, cycle classes and the classical intersection product. It is obtained as a corollary that the Bloch-Quillen isomorphism is compatible with inverse images.
reviewed by: Vasile Brînzănescu
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 4061375
zbMATH Keywords: intersection theory
zbMATH Keywords: Bloch-Quillen isomorphism
zbMATH Keywords: Chow group
zbMATH Keywords: K-theory product
zbMATH Keywords: cycle classes
zbMATH Keywords: intersection product
Wikidata QID: Q58947736
author: Henri Gillet
MaRDI profile type: Publication
cites work: Rational equivalence on singular varieties
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cites work: Q3247612
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cites work: On the De Rham cohomology of algebraic varieties
cites work: Higher algebraic K-theory: I
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full work available at URL: https://doi.org/10.1007/bf00539625
OpenAlex ID: W2090500404
description: scientific article; zbMATH DE number 4061375