\(K\)-theory hypercohomology spectra of number rings at the prime 2 (Q2707511)

Let \(F\) be a number field with ring of integers \(O_F\), let \(\ell\) be a prime and \(R=O_F[1/\ell]\). In a joint paper [\textit{W. G. Dwyer} and \textit{S. A. Mitchell}, ``On the \(K\)-theory spectrum of a ring of algebraic integers'', \(K\)-Theory 14, No.~3, 201-263 (1998; Zbl 0905.19003)] the authors computed the \(\ell\)-completed periodic complex \(K\)-theory spectrum \(\widehat{\mathcal K}\) of the algebraic \(K\)-theory spectrum \(KR\) of \(R\) for \(\ell\) odd or \(\sqrt{-1} \in F\) in terms of Iwasawa theory. In these cases the \(K\)-theory spectrum \(KR\) coincides with the Thomason--Jardine étale hypercohomology spectrum \(\mathbf H^{\cdot}_{\acute et}(Y;\underline{K})\) for \(Y=\text{spec} R\). The author uses similar methods as \textit{J. Rognes} and \textit{C. Weibel} [``Two-primary algebraic \(K\)-theory of rings of integers in number fields'', J. Am. Math. Soc. 13, No.~1, 1-54 (2000; Zbl 0934.19001)], which in turn are based on Voevodsky's proof of the Milnor conjecture, to deal with the case that \(\ell=2\) and \(F\) admits a real embedding. The additional technical problems arising from the fact that \(Y\) has infinite cohomological \(2\)-dimension are solved by looking at a modified étale site introduced by \textit{Th. Zink} [``Étale cohomology and duality in number fields'', appendix to: ``Galois cohomology of algebraic number fields'' by \textit{K. Haberland}, Berlin (1978; Zbl 0418.12004)]. NEWLINENEWLINENEWLINEThe main result of the paper can be summarized as follows: For any number field \(F\) and any prime \(\ell\): NEWLINE\[NEWLINE\widehat{\mathcal K}^p\mathbf H^{\cdot}_{\acute et}(Y;\underline{K}) \cong \begin{cases} \Lambda' \otimes_{\Lambda'_F} \mathbb Z_l \text{if} p=0\\ \Lambda' \otimes_{\Lambda'_F} M_{\infty} \text{if} p=-1. \end{cases} NEWLINE\]NEWLINE Here \(\Lambda'\) is the ring of operations \([\widehat{\mathcal K},\widehat{\mathcal K}]\), \(\Lambda_F'\) is the Iwasawa-algebra of the cyclotomic \(\mathbb Z_\ell\)-extension \(F_\infty\) of \(F\), and \(M_\infty\) is the basic Iwasawa-module, i.e., the Galois group of the maximal abelian pro-\(\ell\)-extension of \(F_\infty\), which is unramified outside \(\ell\) and \(\infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].