On Bott-periodic algebraic $K$-theory

Let $K_*(A;{\bold Z}/\ell^n)$ denote the mod-$\ell^n$ algebraic $K$-theory of a ${\bold Z}[1/\ell]$-algebra $A$. Snaith \cite{14}, \cite{15}, \cite{16}, has studied Bott-periodic algebraic $K$-theory $K_i(A; {\bold Z}/\ell^n)[1/\beta_n]$, a localized version of $K_*(A;{\bold Z}/\ell^n)$ obtained by inverting a Bott element $\beta_n$. For $\ell$ an odd prime, Snaith has given a description of $K_*(A;{\bold Z}/\ell^n)[1/\beta_n]$ using Adams maps between Moore spectra. These constructions are interesting, in particular, for their connections with the Lichtenbaum-Quillen conjecture \cite{16}.

In this paper we obtain a description of $K_*(A;{\bold Z}/2^n)[1/\beta_n]$, $n\geq 2$, for an algebra $A$ with $1/2\in A$ and $\sqrt{-1}\in A$. We approach this problem using low dimensional computations of the stable homotopy groups of $B{\bold Z}/4$, and transfer arguments to show that a power of the mod-4 Bott element is induced by an Adams map.