Algebraic \(K\)-theory of the two-adic integers
type: Item
instance of: scholarly article
title: Algebraic \(K\)-theory of the two-adic integers
zbMATH Open document ID: 0929.19004
DOI: 10.1016/S0022-4049(97)00156-4
author: John Rognes
published in: Journal of Pure and Applied Algebra
publication date: 1999-11-11T00:00:00Z
review text: In [\textit{J. Rognes}, J. Pure Appl. Algebra 134, No. 3, 219-286 (1999; Zbl 0929.19003); see the paper above] the author calculated the mod two homotopy of the algebraic K-theory of the \(2\)-adic integers \(\widehat{\mathbb{Z}}_2\). In this paper the author uses this to determine the homotopy type and homotopy groups of the two-completed algebraic K-theory spectrum \(K(\widehat{\mathbb{Z}}_2)_2^\wedge\). For \(*>0\) the homotopy groups are given by \[ \pi_*(K(\widehat{\mathbb{Z}}_2)_2^\wedge)=\begin{cases} \widehat{\mathbb{Z}}_2\oplus\mathbb{Z}/2^{v_2(k)+3}& \text{for \(*=4k-1\)}\\ \mathbb{Z}/2^{v_2(k)+3}& \text{for \(*=4k\)}\\ \widehat{\mathbb{Z}}_2\oplus\mathbb{Z}/2& \text{for \(*=4k+1\)}\\ \mathbb{Z}/2& \text{for \(*=4k+2\)}\end{cases} \] where \(v_2(k)\) is the two-adic valuation of \(k\). These homotopy groups are the direct sum of the homotopy groups of \(\text{Im } J {\mathbb{C}}_2^\wedge\) (the complex image of \(J\)), i.e., the fiber of \(\mathbb{Z}\times BU@>{\psi^3-1}>>BU\), \(B \text{ Im } J {\mathbb{C}}_2^\wedge\) and \(BBU_2^\wedge\), but as opposed to the odd-primary case the homotopy type is more complicated. More precisely, \(K(\widehat{\mathbb{Z}}_2)_2^\wedge\) fits into fiber sequences built out of these spectra, but the sequences do not split. The extension problems are analyzed and identified, determining the infinite loop space structure of \(K(\widehat{\mathbb{Z}}_2)_2^\wedge\) up to homotopy equivalence.
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 11S70
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55Q35
Mathematics Subject Classification ID: 55Q50
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 19F99
zbMATH DE Number: 1308293
zbMATH Keywords: cyclotomic trace map
zbMATH Keywords: Galois reduction map
zbMATH Keywords: image of \(J\)
zbMATH Keywords: invertible spectra
zbMATH Keywords: Adams operations
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00156-4
OpenAlex ID: W1970477031
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description: scientific article; zbMATH DE number 1308293