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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 $\psi^q-1$. 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.

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    $\begingroup$ As far as I know, such a fiber sequence of spectra (with $ku$ in place of $KU$ for obvious connectivity reasons), exists only when you $l$-complete where $l$ is a prime different from $p$. In fact I'm not even sure the Brauer lift is a map of H-spaces if you do not complete away from the characteristic $\endgroup$ – Denis Nardin Dec 6 '18 at 22:57
  • $\begingroup$ is there a reference where this is all written down? $\endgroup$ – xir Dec 7 '18 at 1:36
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    $\begingroup$ First, as Denis said, you need connective K-theory. If you're fine with p-completion (where p is different from char(F_q)), then there is a fiber sequence L_K(1) K(F_q) -> KU_p -> KU_p. So you'd need to show that the p-completion of K(F_q) is the connective cover of its K(1)-localization. This seems to be true, just by inspection. I don't know if there is an integral version. $\endgroup$ – skd Dec 7 '18 at 1:43
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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 ,

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.

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  • $\begingroup$ thanks, this is very helpful! is there any way to see how the brauer lift fails to be good enough before completion? (e.g. fails to even be a map of h-spaces, as denis said?) $\endgroup$ – xir Dec 30 '18 at 0:41

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