On the spectrum representing algebraic \(K\)-theory for a finite field (Q1070569)

Let \({\mathbb Z}\times \text{BGL}{\mathbb F}^+_ r\) be the representing space for the algebraic \(K\)-theory of a field with \(r\) elements. Let \(p\) be an odd prime such that \(r\) generates \(({\mathbb Z}/p^ 2)^*\). Let \(A\) denote the spectrum obtained by \(p\)-localisation from \(K{\mathbb F}_ r\). The author computes \(H^*(A; {\mathbb Z}/p)\) and \(H_*(A; {\mathbb Z}/p).\) Let \(G\) be the ring spectra resulting from Adam's splitting \[ bu_{(p)}=\bigvee^{p- 1}_{j=1}\Sigma^{2(j-1)} G. \] The \(G\)-module action on \(A\) and Quillen's fibre sequence \(A\to^{n}G\to^{\theta}G\) are used to analyze the ring \(A_*({\mathbb C}P^{\infty})\).