Adams operations and algebraic \(K\)-theory
type: Item
instance of: scholarly article
title: Adams operations and algebraic \(K\)-theory
zbMATH Open document ID: 0954.19003
DOI: 10.1023/A:1007780931760
author: Florence Lecomte
published in: \(K\)-Theory
publication date: 1998-09-13T00:00:00Z
review text: Let \(X\) be a scheme furnished with a big family of invertible bundles, the author shows in this paper that the graded ring \[ \bigoplus_{i\geq 0}K_i(X)\otimes\mathbb Q \] adquires a structure of \(\lambda\)-ring from the Schechtman operations \[ \Psi_k:K_i(X)\otimes\mathbb Q\rightarrow K_i(X)\otimes\mathbb Q \] for \(i\geq 0\) in Quillen's \(K\)-theory. After a good exposition of bisimplicial schemes, he defines the \(K\)-theory groups of multisimplicial schemes, and for every group \(G\) a morphism \[ R_\mathbb Z(G)\rightarrow K_0(BG) \] where \(R_\mathbb Z\) stands for the Grothendieck ring of the algebraic representations of \(G\), and he computes some particular images for the case \(G=GL(n)\). It is also shown that the group \(K_{-1}(N{\mathcal U}\otimes{\mathbb B}P(N{\mathcal U}))\) is isomorphic to the group \([|{\mathbb B}P(N{\mathcal U})|,K(N{\mathcal U})]\), and finally, that the identities of \(\lambda\)-rings are transported from \(R_\mathbb Z(GL(n))\) to \(K_i(X)\otimes\mathbb Q\).
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19L20
zbMATH DE Number: 1132931
zbMATH Keywords: Adam's operations
zbMATH Keywords: Quillen's \(K\)-theory
zbMATH Keywords: scheme
reviewed by: Juan Antonio Pérez
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1132931