Inverting the motivic Bott element
type: Item
instance of: scholarly article
title: Inverting the motivic Bott element
zbMATH Open document ID: 0948.19003
DOI: 10.1023/A:1007874218371
published in: \(K\)-Theory
publication date: 2000-11-06T00:00:00Z
review text: The main theorem of this paper is an analog for motivic cohomology of Thomason's descent theorem for Bott periodic algebraic \(K\)-theory [\textit{R. W. Thomason}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437---552 (1985, Zbl 0596.14012)]. Suppose that \(k\) is a field containing a primitive \(n\)th root of unity where \((n,\text{char}(k)) = 1\), and let \(X\) be a localization of a smooth \(k\)-scheme of finite type. Then the author's main result (theorem 4.5, which could be called a descent theorem for Bott periodic motivic cohomology) says that the motivic cohomology complex \(\mathcal{Z}(X,\ast)/n[\beta_n^{[-1]}]\) with the motivic Bott element inverted satisfies Galois cohomological descent over the base field. This complex is weakly equivalent to the object \(\bigoplus _{q=-\infty}^{\infty} \mu_{n}^{\otimes q}\) in the étale derived category (lemma 5.1), so Bott periodic motivic cohomology is isomorphic to étale cohomology for such schemes \(X\) (theorem 1.1). The proof is essentially a reprise of that given by Thomason, except that it deals with chain complexes rather than spectra and all transfer maps are abstractly defined, so that the homotopy coherence issues faced in the original proof disappear.
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 19E20
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 1423565
zbMATH Keywords: Bott element
zbMATH Keywords: motivic cohomology
zbMATH Keywords: descent theorem
zbMATH Keywords: étale cohomology
author: Marc Levine
MaRDI profile type: Publication
full work available at URL: a:1007874218371
OpenAlex ID: W1992937392
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description: scientific article; zbMATH DE number 1423565