On 2-adic cyclotomic elements in \(K\)-theory and étale cohomology of the ring of integers
In this paper, the authors define, for odd integers \(n\), groups \({\mathcal C}^{\text{ét}}(n-1) \subset H^1_{\text{ét}}(\mathbb{Z}[1/2]; \widehat \mathbb{Z}_2(n))\) and \({\mathcal C}(n-1)\subset K_{2n-1}(\mathbb{Z})\otimes \widehat\mathbb{Z}_2\) of 2-adic cyclotomic elements. In both cases and for all odd \(n\) it is shown that these groups are free \(\mathbb{Z}_2\)-modules of rank one. They prove that for any odd integer \(n\) there is a natural isomorphisms of \(\widehat\mathbb{Z}_2\)-modules \[ H^1_{ \text{ét}}\bigl (\mathbb{Z}[1/2]; \widehat\mathbb{Z}_2(n) \bigr)\cong{\mathcal C}^{\text{ét}} (n-1)\oplus \mathbb{Z}/2 \] (theorem 3.5) and a natural isomorphism \[ K_{2n-1}(\mathbb{Z}) \otimes\widehat \mathbb{Z}_2\cong \begin{cases}{\mathcal C}(n-1)\quad &\text{if }n \equiv 3\bmod 4\\ {\mathcal C}(n-1) \oplus\mathbb{Z}/2 \quad & \text{if }n\equiv 1 \bmod 4\end{cases} \] (theorem 5.14). The \(K\)-theoretic and cohomological 2-adic cyclotomic elements are compared via étale \(K\)-theory. However, since there is not a well-behaved Dwyer-Friedlander spectral sequence at the prime 2 for number fields admitting a real embedding, they introduce a relative version of étale \(K\)-theory (with \(\mathbb{Z}/\ell^k\) coefficients) for this purpose. They prove that (for odd \(n)\) there is a natural isomorphism \[ K_{2n-1}(X,Y; \widehat \mathbb{Z}_2) \cong{\mathcal C}(n-1) \subset K_{2n-1}(\mathbb{Z}) \otimes\widehat \mathbb{Z}_2 \] where \(X=\text{Spec} \mathbb{Z} [1/2]\) and \(Y=\text{Spec} \mathbb{R}\), and that the composite \[ K_{2n-1} (X,Y;\widehat \mathbb{Z}_2)\to K^{\text{ét}}_{2n-1}(X,Y; \widehat \mathbb{Z}_2)\to H^1_{\text{ét}} (X,Y;\widehat \mathbb{Z}_2(n))\to H^1_{\text{ét}} \bigl(\mathbb{Z}[1/2]; \widehat\mathbb{Z}_2(n)\bigr) \] induces an isomorphism \[ {\mathcal C}(n-1) \cong{\mathcal C}^{\text{ét}}(n-1). \] The authors also prove that certain products of 2-adic cyclotomic elements in the ring \(K_*(\mathbb{Z}) \otimes \widehat \mathbb{Z}_2\) are zero. In the course of their computations they use the calculation of the groups \(K_*(\mathbb{Z})\otimes \widehat\mathbb{Z}_2\) by Rognes and Weibel.
scientific article; zbMATH DE number 1466186
scientific article