A note on generators of even-numbered \(K\)-groups of rings of integers
The aim of this paper is to give explicit generators for the étale \(K\)-groups \(K_{2n}^{\text{ét}}(\mathcal{O}_F[1/\ell])\) in the particular case where \(\mathcal{O}_F\) is the ring of integers of some cyclotomic field \(F\) and \(\ell\) is an odd prime number. The above groups were introduced by \textit{W. G. Dwyer} and \textit{E. M. Friedlander} [Trans. Am. Math. Soc. 292, 247--280 (1985; Zbl 0581.14012)]: they are related to both Quillen's \(K\)-theory and étale cohomology. In fact, throughout the paper the authors identify \(K_{2n}^{\text{ét}}(\mathcal{O}_F[1/\ell])\) with \(H^2(\mathcal{O}_F[1/\ell],\,\mathbb{Z}_\ell(n+1))\). Set now \(F=\mathbb{Q}(\mu_\ell)\) and \(E=\mathbb{Q}(\mu_{\ell^k})\) for some \(k\geq 1\). The main tool of the paper is a map \[ \alpha:\mathcal{O}_{E,\,S}^\times \otimes \mathbb{Z}/\ell^k(n)\longrightarrow K_{2n}^{\text{ét}}(\mathcal{O}_{F,\,S})[\ell^k] \] where \(S\) is a set of primes of \(E\) containing those above \(\ell\). For \(n=1\), this map is simply given by \(\alpha(x\otimes \zeta)=\text{Tr}_{E/F}\{x,\,\zeta\}\) where we identify étale \(K\)-theory with \(K\)-theory and the transfer \(\text{Tr}_{E/F}\) is applied to the symbol \(\{x,\,\zeta\}\). The general definition of \(\alpha\) is given by \[ \alpha:\mathcal{O}_{E,\,S}^\times\otimes \mathbb{Z}/\ell^k(n)\to H^1(\mathcal{O}_{E,\,S},\,\mathbb{Z}_\ell(1))\otimes H^0(\mathcal{O}_{E,\,S},\,\mathbb{Z}/\ell^k(n))\to H^2(\mathcal{O}_{E,\,S},\,\mathbb{Z}_\ell(n+1))[\ell^k] \] where the first map is the Kummer map in the first factor and trivial in the other, while the second map is the cup product followed by the canonical epimorphism \(H^1(\mathcal{O}_{E,\,S},\,\mathbb{Z}/\ell^k(n+1))\to H^2(\mathcal{O}_{E,\,S},\,\mathbb{Z}_\ell(n+1))[\ell^k]\). This map has also been studied by \textit{G. Banaszak} and \textit{W. Gajda} [Contemp. Math. 199, 7--18 (1996; Zbl 0866.19004)]. The first part of the paper is devoted to a special case, namely the case where the minus part of the class group of \(F\) is cyclic as a \(\mathbb{Z}_\ell[\text{Gal} (F/\mathbb{Q})]\)-module (this is predicted by various conjectures, such as Vandiver's conjecture). This hypothesis implies that the Stickelberger ideal is the precise annihilator of the class group. Using this fact, the authors show that a generator for \(K_{2n}^{\text{ét}}(\mathbb{Z}[1/\ell])\) is \(\text{Tr}_{F/\mathbb{Q}}(\alpha(x\otimes 1))\) where \(x\in \mathcal{O}_{E,\,S}^\times\) has the following property: \[ (x)=\mathfrak{q}^{\Theta_*} \] where \(\Theta_*\) is an appropriate multiple of the Stickelberger element and \(\mathfrak{q}\) is a generator of the \(\ell\)-part of the class group of \(E\). In the rest of the paper the hypothesis on the cyclicity of the minus part of the class group is no longer assumed. A result analogous to the one in the previous part is obtained, using similar techniques (Fitting ideals replace annihilators).
scientific article; zbMATH DE number 5707684