Stable \(K\)-theory as a derived functor
type: Item
instance of: scholarly article
title: Stable \(K\)-theory as a derived functor
zbMATH Open document ID: 0812.19002
DOI: 10.1016/0022-4049(94)90101-5
author: Stanislaw Betley
author: Teimuraz Pirashvili
published in: Journal of Pure and Applied Algebra
publication date: 1995-05-07T00:00:00Z
review text: The authors consider stable \(K\)-theory of a ring \(R\) as a functor \(K^ s_ *(R,-)\): \({\mathcal B} \to {\mathcal A}b\), where \({\mathcal B}\) is the abelian category of functors \(T:{\mathcal R}^{op}_ f \times {\mathcal R}_ f \to {\mathcal A}b\), and \({\mathcal R}_ f\) is the additive category of finitely generated, free \(R\)-modules. The category \({\mathcal B}\) has a set of projective generators, \(\{P^{n,m} | n, m \in \mathbb{N} \cup \{0\}\}\), so the situation is amenable for classical homological algebra. The authors' Conjecture A states that the groups \(K^ s_ i(R,P^{n,m})\) vanish for all \(n,m\) and all \(i > 0\). Using basic stability results of \textit{W. van der Kallen} [Homology stability for linear groups, Invent. Math. 60, 269-295 (1980; Zbl 0415.18012)], the authors verify the conjecture for any semi-simple ring and any commutative integral domain of finite Krull dimension. Furthermore, Conjecture A is shown to imply that \(K^ s_ i(R,-)\) is the \(i\)-th derived functor of \(K^ s_ 0(R,-)\). Motivated by the description of \(THH_ * (R,M)\) as the homology groups of the small category \({\mathcal R}_ f\) with coefficients in \(\Hom (-,M \otimes_ R-) \in {\mathcal B}\), the authors define topological Hochschild homology \(THH_ *(R,T) : = H_ *({\mathcal R}_ f,T)\), for any \(T \in {\mathcal B}\). There results a natural transformation \(\nu : K^ s_ * (R,- ) \to THH_ *(R,-)\). The rest of the paper deals with the question: In which cases is \(\nu\) an isomorphism? Basically, I find the approach of this paper quite attractive. However, at least in a couple of spots, more precision in statements and care in proofs would have made for easier reading. First of all, I object to the use (in theorem 2) of ``='' to indicate the existence of a (not even natural) isomorphism between the abelian groups in question. A better statement of theorem 2 appears to be that ``the spectral sequence in question collapses and involves no extension problem.'' In the proof…
reviewed by: Hans Jørgen Munkholm
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 18G40
zbMATH DE Number: 701717
zbMATH Keywords: homological algebra
zbMATH Keywords: derived functor
zbMATH Keywords: topological Hochschild homology
zbMATH Keywords: stable \(K\)-theory
zbMATH Keywords: group homology
zbMATH Keywords: general linear group
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90101-5
OpenAlex ID: W2000341831
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description: scientific article; zbMATH DE number 701717