The upper triangular group and operations in algebraic K-theory
type: Item
instance of: scholarly article
title: The upper triangular group and operations in algebraic K-theory
zbMATH Open document ID: 1008.19003
DOI: 10.1016/S0040-9383(02)00001-0
published in: Topology
publication date: 2002-12-02T00:00:00Z
review text: Let \(bu\) and \(bo\) denote the stable homotopy spectra representing 2-adically completed unitary and orthogonal connective \(K\)-theory respectively. The author considers the group \(\text{Aut}^0 (bu\wedge bo)\), under composition, of homotopy classes of left \(bu\)-module homotopy equivalences of \(bu\wedge bo\) which induces the identity map on \(H_*(bu\wedge bo\); \(\mathbb{Z}/2)\). Let \(U_\infty \mathbb{Z}_2\) denote the group of infinite, invertible upper triangular matrices with entries in 2-adic integers. The main result of the present paper is: There is an isomorphism of the form \(\psi:\text{Aut}^0 (bu \wedge bo)\to U_\infty \mathbb{Z}_2\) (Theorem 1.2). The proof uses essentially the author's prior result concerning a stable decomposition of \(\Omega^2S^3\) such that \(\Omega^2S^3 \simeq\bigvee_{k\geq 1} F_k/F_{k-1}\) where \(F_k\)'s denote an increasing filtration of \(\Omega^2S^3\) with \(F_1=S^1\). Using this splitting the author first gives the following preparatory theorem: There is a 2-local homotopy equivalence of left \(bu\)-module spectra of the form \(\widehat L:\bigvee_{k \geq 0}bu\wedge (F_{4k}/F_{4k-1})\to bu\wedge bo\) (Theorem 2.3, (ii)). Here \(F_0= *\) is replaced by \(S^0\) and \(F_{-1}=*\). The proof of this theorem requires a fair amount of homological technique applied to module spectra in order to construct the map \(\widehat L\). The key to the author's approach is using the graded homotopy-ring structure on \(\bigvee_{k\geq 1} F_k/F_{k-1}\) inherited from \(\Omega^2S^3\). Indeed this induces that on \(\bigvee_{k\geq 0}bu \wedge(F_{4k}/F_{4k-1})\) appropriate to construct a bijection from \(U_\infty (\mathbb{Z}_2)\) to \(\text{Aut}^0 (\bigvee_{k\geq 0} bu\wedge (F_{4k}/F_{4k-1}))\). Thus the discussion of Sections 2 and 3 yields the isomorphism \(\psi\). As an application, in Section 4 the author gives a family of operations on algebraic \(K\)-theory of \(F\)-schemes for an algebraically closed field \(F\) of characteristic differ…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 19L41
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 1838831
zbMATH Keywords: connective \(K\)-theory
zbMATH Keywords: bu
zbMATH Keywords: bo
zbMATH Keywords: stable homotopy spectra
zbMATH Keywords: upper triangular matrices
author: Victor P. Snaith
MaRDI profile type: Publication
full work available at URL: s0040-9383(02)00001-0
OpenAlex ID: W2023394645
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description: scientific article; zbMATH DE number 1838831