Riemann-Roch for algebraic versus topological K-theory
type: Item
instance of: scholarly article
title: Riemann-Roch for algebraic versus topological K-theory
zbMATH Open document ID: 0545.14007
DOI: 10.1016/0022-4049(83)90032-4
published in: Journal of Pure and Applied Algebra
publication date: 1983-01-01T00:00:00Z
review text: The paper extends the work of Dwyer and Friedlander about etale K-theory by defining a mod \(\ell^{\nu}\) topological K-homology theory for quasiprojective schemes X over k (k is a ''nice'' noetherian ring in which the prime \(\ell\) is invertible). There is a natural transformation \(\rho\) (X) from algebraic G-theory to (mod \(\ell^{\nu})\) topological K- homology theory. The Riemann-Roch theorem says that \(\rho\) (X) is natural for projective morphisms. Most part of the paper is concerned with the definition of this topological K-homology theory. After that, the deformation to the normal cone machinery is adapted to prove the theorem. The results are applied to compute the algebraic and topological K-groups of reductive group schemes, homogeneous spaces and complete rational surfaces.
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14C35
zbMATH DE Number: 3867482
zbMATH Keywords: K-homology theory
zbMATH Keywords: quasiprojective schemes
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: normal cone
zbMATH Keywords: K-groups of reductive group schemes
author: Robert W. Thomason
reviewed by: Vasile Brînzănescu
MaRDI profile type: Publication
full work available at URL: 0022-4049(83)90032-4
OpenAlex ID: W2058619990
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Etale homotopy
cites work: Riemann-Roch for singular varieties
cites work: Riemann-Roch and topological K-theory for singular varieties
cites work: Cohomology of line bundles on $G/B$
cites work: Q2546548
cites work: Complements à l'article: ''Groupes reductifs''
cites work: Algebraic K-theory eventually surjects onto topological K-theory
cites work: Etale K-theory. I: Connections with etale cohomology and algebraic vector bundles
cites work: Étale $K$-theory. II. Connections with algebraic $K$-theory
cites work: Riemann-Roch theorems for higher algebraic 𝐾-theory
cites work: Riemann-Roch theorems for higher algebraic K-theory
cites work: On the K-theory of Lie groups
cites work: Q4041681
cites work: Higher algebraic K-theory: I
cites work: Erratum “Algebraic $K$-theory and etale cohomology”
cites work: Q3676259
cites work: Q4100617
description: scientific article; zbMATH DE number 3867482