Continuous \(K\)-theory
type: Item
instance of: scholarly article
title: Continuous \(K\)-theory
zbMATH Open document ID: 0833.19003
DOI: 10.1007/BF00961470
author: Grzegorz Banaszak
author: Piotr M. Zelewski
published in: \(K\)-Theory
publication date: 1996-02-26T00:00:00Z
review text: For a scheme \(X\) the continuous \(K\)-theory \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell)\) is defined and its properties are derived. This parallels Jannsen's construction of continuous étale cohomology as a derived functor theory. For a scheme \(X\) let \(BQP (X)\) denote the classifying space of Quillen's \(Q\)-construction on the category \(P(X)\) of finite rank locally free \({\mathcal O}_X\)-modules. Let \(Y^n_{\ell^k}\) be the cofiber of \(S^{n - 1} @>\ell^k>> S^{n - 1} \to Y^n_{\ell^k}\) and write \(Y^n_{\ell^\infty}=\varprojlim Y^n_{\ell^k}\) (where the limit is taken over \(k)\). Then, by definition, \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell) :=[Y^{n + 1}_{\ell^\infty}, BQP (X)]\) for \(n \geq 1\). \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell)\) fits in an exact sequence: \[ 0 \to \varprojlim{^1 K_{n+ 1}}(X; \mathbb{Z}/\ell^k) \to K_n^{\text{cts}} (X; \mathbb{Z}_\ell) \to K_n (X; \mathbb{Z}_\ell) \to 0, \] where \(K_n (X; \mathbb{Z}_\ell)=\varprojlim K_n (X; \mathbb{Z}/\ell^k)\). One has a Bockstein exact sequence relating the various \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell)\) and \(K_n (X; \mathbb{Z}/\ell^k)\). As a first result one has, under the assumption that \(K_n (X)_\ell\) contains no non-trivial divisible subgroup for \(n > 0\), isomorphisms \(K_n (X) [\ell^k] @>\sim>> K_n^{\text{cts}} (X; \mathbb{Z}_\ell) [\ell^k]\) for \(n > 0\) and \(K_n (X)/ \ell^k @>\sim>> K_n^{\text{cts}} (X; \mathbb{Z}_\ell)/ \ell^k\) for \(n > 1\). As a corollary one obtains a canonical isomorphism \(c :K_n (X)_\ell @>\sim>> K_n^{\text{cts}} (X; \mathbb{Z}_\ell)_\ell\) which restricts to an isomorphism on the divisible elements of both groups. For number fields one gets more precise results on divisibility. Several special cases are described: (i) For \(X=\text{Spec} \mathbb{F}_q\) one has \(K_n^{\text{cts}} (\mathbb{F}_q; \mathbb{Z}_\ell)=K_n (\mathbb{F}_q) \otimes \mathbb{Z}_\ell\); (ii) similarly for the ring of integers in a number field; (iii) for a field \(F\) with \(\mu_{\ell^\i…
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 19E08
zbMATH DE Number: 800397
zbMATH Keywords: Quillen \(K\)-theory
zbMATH Keywords: étale \(K\)-theory
zbMATH Keywords: continuous \(K\)-theory
zbMATH Keywords: continuous cohomology
zbMATH Keywords: étale cohomology
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00961470
OpenAlex ID: W4247703499
description: scientific article; zbMATH DE number 800397