Quillen famously constructed a fibration sequence $BGL(\mathbb{F}\_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^\ell-1$ for $\ell$ a generator of $\mathbb{Z}\_q^\times$. Does this lift to the level of the K-theory spectra?


I don't know much about delooping, but I would guess that if we had a fibration sequence $BGL(\mathbb{F}\_q)^+ \to BU\times \mathbb{Z} \to BU\times \mathbb{Z}$ of $E\_\infty$-spaces, this would then deloop to a fibration sequence of the associated connective $\Omega$-spectra. However, this seems out of reach, since Quillen's method proceeds via the Atiyah-Segal completion theorem and only produces a homotopy class of map.



Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?
The infinite loop space/spectrum level statements were written down in


May, Quinn, Ray, Tornehave:
"$E\_\infty$ Ring Spaces and $E\_\infty$ Ring Spectra" (1977) [http://www.math.uchicago.edu/~may/BOOKS/e\_infty.pdf](https://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf) ,


see Theorem VIII.3.2 and its proof on pages 224-225. For a prime $p$ and a prime power $r=q$ generating the $p$-adic units (take $r=3$ for $p=2$), May writes $j^\delta\_p$ for $K(\mathbb{F}\_r)$ completed at $p$, viewed as a discrete model for the connective complex image-of-$J$ spectrum, and argues that this spectrum is equivalent to the homotopy fiber $F\psi^r$ of $\psi^r-1 \colon kU \to bu$, after $p$-completion. Here $kU = ku$ is the connective complex $K$-theory spectrum and $bu$ is its $1$-connected cover. See also Section 19 of the survey paper


May: "Infinite Loop Space Theory" (1977)
<http://www.math.uchicago.edu/~may/PAPERS/18.pdf> ,


which may be easier to read.


