Geometric models for the K-theory of fields
type: Item
instance of: scholarly article
title: Geometric models for the K-theory of fields
zbMATH Open document ID: 0531.18009
DOI: 10.1016/0021-8693(83)90076-5
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: Quillen's space \(BGL^+(k)\) is difficult to study even when k is a field. This paper contains the following model for \(BGL^+(k)\). Denote by \(F_ N(n)\) the Grassmann scheme of subspaces of dimension n in \(k^ N\). It makes sense to go to the limit \(F_{\infty}(n)\). Now consider the simplicial algebra \(k_*\) with n-simplices \(k_ n=k[x_ 0,...,x_ n]/(\sum x_ i-1)\) and operators \(d_ i(x_ j)=x_ j (j<i)\), \(=0 (j=i)\), \(=x_{j-1} (j>i)\), and \(s_ i(x_ j)=x_ j (j<i)\), \(=x_ i+x_{i+1} (j=1)\), \(=x_{j+1} (j>i)\), respectively. Applying \(F_{\infty}(n)\) to \(k_*\) and using a stabilization map \(F_{\infty}(n)(k_*)\to F_{\infty}(n+1)(k_*)\) we get some Kan complex which is homotopically equivalent to \(BGL(k_*)\) and consequently to \(BGL^+(k_*)\) [cf. \textit{S. M. Gersten}, Lect. Notes Math. 341, 3-42 (1973; Zbl 0285.18010)]. This is the main result. As the author remarks it was developed in an attempt to calculate K-theory of an algebraically closed field with finite coefficients. Meanwhile this problem (the Quillen-Lichtenbaum conjecture) was solved by \textit{A. A. Suslin} [Invent. Math. 73, 241-245 (1983; Zbl 0514.18008)].
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 13D15
zbMATH DE Number: 3842106
zbMATH Keywords: fpqc torseurs
zbMATH Keywords: Serre conjecture
zbMATH Keywords: K-theory of field
zbMATH Keywords: Grassmanian variety
zbMATH Keywords: flag variety
zbMATH Keywords: Quillen space
zbMATH Keywords: Quillen-Lichtenbaum conjecture
author: J. F. Jardine
reviewed by: A. N. Parshin
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90076-5
OpenAlex ID: W2068573238
cites work: Homotopy limits, completions and localizations
cites work: Icosahedral Galois representations
cites work: K-theory of tetrahedra
cites work: Q5602064
cites work: Q4771113
cites work: Q4064722
cites work: Algebraic Homotopy Theory
cites work: Fibrations of algebraic groups
cites work: On the homotopy groups of algebraic groups
cites work: Q3928685
cites work: Homotopy types and Sullivan's algebras of O-forms
cites work: $K$-théorie algébrique et représentations de groupes
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Projective modules over polynomial rings
cites work: Q4061677
cites work: Q3676259
description: scientific article; zbMATH DE number 3842106