Grothendieck groups of polynomial and Laurent polynomial rings
type: Item
instance of: scholarly article
title: Grothendieck groups of polynomial and Laurent polynomial rings
zbMATH Open document ID: 0615.14009
DOI: 10.1215/S0012-7094-86-05334-2
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: In order to probe nonregularity of noetherian k-schemes X, groups like \(NK_ 0(X)=coker (K_ 0(X)\to K_ 0(X\times {\mathbb{A}}^ 1))\) and \(K_{-1}\) (a similar cokernel) have been defined. Take in particular \(k={\mathbb{C}}\) and for X a normal quasi-projective surface. It is shown that, if R is the semilocal ring of the singular locus, then there is a surjection \(K_{-1}(X)\twoheadrightarrow C1(\hat R)/C1(R)\) where \(\hat R\) is the completion of R and C1 the ideal class group. If, on the other hand, \(Y\to X\) is a resolution of singularities, the reduced exceptional divisor E of which has smooth components and normal crossings, and if, moreover, Pic(nE)\(\twoheadrightarrow Pic(E)\) is not an isomorphism for some \(n>1\), then \(NK_ 0(X)\) has infinite rank. Examples are given.
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13D15
zbMATH DE Number: 3995857
zbMATH Keywords: Laurent polynomial rings
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: \(NK_ 0\)
author: V. Srinivas
reviewed by: J. H. de Boer
MaRDI profile type: Publication
cites work: K\(_2\) and algebraic cycles
cites work: K2of artinian Q-algebras, with application to algebraic cycles
cites work: Q5562621
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: Relative K-theory and vector bundles on certain singular varieties
cites work: A remark on \(K_ 1 \)of an algebraic surface
cites work: Q4135751
cites work: Localization for flat modules in algebraic K-theory
cites work: The K-theory of endomorphisms
cites work: Q3728132
cites work: Torsion Zero-Cycles on Singular Varieties
cites work: $K$-théorie algébrique et représentations de groupes
cites work: Vector bundles over affine surfaces birationally equivalent to a ruled surface
cites work: Q4769130
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: $ K$-COHOMOLOGY OF SEVERI-BRAUER VARIETIES AND THE NORM RESIDUE HOMOMORPHISM
cites work: Q5735274
cites work: Higher algebraic K-theory: I
cites work: Q4750775
cites work: Modules of finite length and Chow groups of surfaces with rational double points
cites work: Q3739257
cites work: Zero cycles on a singular surface. II.
cites work: Residues of differentials on curves
cites work: Algebraic K-theory of generalized free products. II
cites work: K-theory and analytic isomorphisms
cites work: Negative K-theory of varieties with isolated singularities
cites work: Q3947813
cites work: Q3939975
full work available at URL: s0012-7094-86-05334-2
OpenAlex ID: W1530776063
description: scientific article; zbMATH DE number 3995857