Representability of the functor \(K_ 0\) \((\tilde K_ 0)\) of the ring of 0-forms
type: Item
instance of: scholarly article
title: Representability of the functor \(K_ 0\) \((\tilde K_ 0)\) of the ring of 0-forms
zbMATH Open document ID: 0581.18006
DOI: 10.1007/BF01141942
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Let R be a regular ring, \(\Omega^ 0_ R: Set^{\Delta^{op}}\to_ R{\mathcal A}\) the cofunctor, which associates with a simplicial set the R- algebra of polynomial 0-forms. \(K_ 0\), \(\tilde K_ 0\) denote the Grothendieck functors. Let Ho Set\(^{\Delta^{op}}\) denote the homotopy category of the category of simplicial sets \(Set^{\Delta^{op}}\). The main theorem states that the functors \(K_ 0 \Omega^ 0_ R\) and \(\tilde K_ 0 \Omega^ 0_ R: Ho Set^{\Delta^{op}}\to {\mathcal A}b\) are representable on finite simplicial sets. The proof essentially uses the representation theorem of \textit{A. Heller}, J. Lond. Math. Soc., II. Ser. 23, 551-567 (1981; Zbl 0477.55013).
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 55P65
zbMATH DE Number: 3929244
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: semisimplicial complexes
zbMATH Keywords: representation of functors
zbMATH Keywords: simplicial set
zbMATH Keywords: polynomial 0-forms
zbMATH Keywords: \(K_ 0\)
zbMATH Keywords: Grothendieck functors
zbMATH Keywords: homotopy category
author: Marek Golasiński
reviewed by: Werner End
MaRDI profile type: Publication
cites work: Cohomology theories
cites work: On the Representability of Homotopy Functors
cites work: Homotopy types and Sullivan's algebras of O-forms
cites work: On a Theorem of Sullivan
cites work: Introduction to Algebraic K-Theory. (AM-72)
full work available at URL: https://doi.org/10.1007/bf01141942
OpenAlex ID: W2016060573
description: scientific article; zbMATH DE number 3929244