Inverting the motivic Bott element
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.
scientific article; zbMATH DE number 1423565
scientific article