On the \(K\)-theory spectrum of a ring of algebraic integers
The authors study the algebraic \(K\)-theory spectrum \(KR_F\), where \(R_F= {\mathcal O}_F [{1\over \ell}]\) is the ring of algebraic \(\ell\)-integers in a number field \(F\). The study involves methods of algebraic topology, homotopy theory in particular. Let \(\widehat{\mathcal K}\) denote the \(\ell\)-adic completion of the topological \(K\)-theory spectrum. The cohomology \(\widehat{\mathcal K}^* (KR_F)\) as a module over the ring of cohomology operations in \(\widehat{\mathcal K}^*\) has been determined. The result (theorem 1.7) is given in terms of the Iwasawa module of a number field, and this shows that \(\ell\)-adic complex \(K\)-theory \(\widehat {\mathcal K} (KR_F)\) carries significant number theoretic information. The theorem is proven via calculations involving Thomason's spectral sequence [\textit{R. W. Thomason}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)]. For any spectrum \(E\) or rather homology theory it defines there exists an \(E_*\)-localization functor \(L_E\). This is the content of a well known theorem of \textit{A. K. Bousfield} [Topology 18, 257-281 (1979; Zbl 0417.55007)]. The computation of \(\widehat{\mathcal K}(KR_F)\) allows the authors to determine \(\widehat L(KR_F)\), where \(\widehat L\) is the Bousfield localization with respect to the spectrum \({\mathcal K} \wedge {\mathcal M}_\ell\). Here \({\mathcal K}\) denotes the topological \(K\)-theory spectrum and \({\mathcal M}_\ell\) is the Moore spectrum of type \((\mathbb{Z}/ \ell,0)\). In fact the authors obtain the wedge decomposition \(\widehat L(KR_F) \simeq \widehat L(K\mathbb{F}) \vee \widehat L(K^{red} F)\) (example 9.7) and a theorem describing further possible decomposition of \(\widehat L(K^{red}F)\) (theorem 9.2). \(\mathbb{F}\) denotes \(R_F/{\mathfrak P}\) for a suitably choose prime \({\mathfrak P}\) of \(R_F\) and \(K^{red}F\) is the homotopy fiber of the natural map \(KR_F\to K\mathbb{F}\). Section 12 of the paper contains nice examples of such wedge decompositions for particular fields \(F\). In section 10 there is a calculation of the \(\mathbb{Z}/ \ell\)-cohomology of the space \(\Omega_0^\infty \widehat L(K^{red}R_F)\) (theorem 10.11) and the conjectural (relying upon the validity of the Lichtenbaum-Quillen conjecture) description of the \(\mathbb{Z}/\ell\) cohomology \(H^*(BGL(R_F), \mathbb{Z}/ \ell)\). The beginning sections contain necessary material from number theory, étale cohomology and homotopy theory, what makes the exposition clear and well organized.
scientific article; zbMATH DE number 1197099