K-theory of blow-ups and vector bundles on the cone over a surface
type: Item
instance of: scholarly article
title: K-theory of blow-ups and vector bundles on the cone over a surface
zbMATH Open document ID: 0806.14003
DOI: 10.1007/BF00961066
author: Caterina Consani
published in: \(K\)-Theory
publication date: 1995-02-13T00:00:00Z
review text: The \(K\)-theory of varieties with isolated singularities is studied in this paper. Let \(K^ Y_ 0 (X)\) be the zero-th homotopy group of the \(K\)-theory spectrum of Waldhausen's category of perfect complexes on \(X\) which are acyclic on \(Y\) where \(Y\) is a finite subset of the variety \(X\) containing all singularities. This group is computed and shown to contain \(SK_ 1(U)\) as a direct summand for \(U = \text{Spec} ({\mathcal O}_{X,Y}) - Y\). More explicit computations are given in the special case where \(X\) is the cone over a nonsingular projective surface of degree \(d \geq 2\) in \(\mathbb{P}^{d + 1}\) not contained in a hyperplane and \(Y\) is the cone point. In particular the vanishing of the relative Chow groups \(\text{CH}^ i (X,Y)\) is proven for \(i \geq 2\). Finally special cases of a conjecture of Srinivas are verified that for a projective normal nonsingular surface \(T\) over \(\mathbb{C}\) with affine cone \(X\) and cone point \(Y\) the vanishing of \(H^ 2 (T, {\mathcal O}_ T(1))\) is equivalent to the vanishing of \(\text{CH}^ 3 (X,Y)\).
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 19E08
zbMATH DE Number: 488351
zbMATH Keywords: homotopy group of the \(K\)-theory spectrum
zbMATH Keywords: cone
zbMATH Keywords: vanishing of the relative Chow groups
reviewed by: Wolfgang Lück
MaRDI profile type: Publication
cites work: Torsion in the Chow group of codimension two: The case of varieties with isolated singularities
cites work: Quillen's K-theory and algebraic cycles on almost non-singular varieties
cites work: Q3328651
cites work: The localization theorem for protective modules
cites work: Q4143433
cites work: Localization on singular varieties
cites work: Vector bundles over affine surfaces birationally equivalent to a ruled surface
cites work: Q4750775
cites work: Q3356420
description: scientific article; zbMATH DE number 488351