\(\ell\)-adic and \({\mathbb{Z}}/\ell ^{\infty}\)-algebraic and topological K-theory. (English) Zbl 0547.18003

Snaith, Victor

Proc. Edinb. Math. Soc., II. Ser. 28, 73-90 (1985).

The author shows that if A is a commutative \({\mathbb{Z}}[1/\ell,\xi_{\ell^{\infty}}]\)-algebra, that commutative diagrams exist which generalize the one constructed in [Algebraic Topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 128-155 (1984; Zbl 0551.18004)]. Furthermore, these diagrams fit together coherently as one takes \(\lim_{\leftarrow_{\nu}}\) or \(\lim_{\to_{\nu}}\) of them. The author explains how ker \(\rho_{\nu}\) is expected to be zero (for reasonable A) and that these diagrams serve to characterize \(\ker(\lim_{\leftarrow_{\nu}} \rho_{\nu})\) and \(\ker(\lim_{\to_{\nu}} \rho_{\nu})\) in terms of the KU-theory Hurewicz homomorphism.

MSC:

18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55N15	Topological \(K\)-theory

Keywords:

\(\ell\)-adic K-theory; \({\mathbb{Z}}/\ell^{\infty}\)-algebraic K-theory; topological K-theory; kernels of limits; \(K_ 2\); KU-theory; Hurewicz homomorphism

Citations:

Zbl 0551.18004

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