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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    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 – Denis Nardin Dec 6 at 22:57
  • is there a reference where this is all written down? – xir Dec 7 at 1:36
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    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. – skd Dec 7 at 1:43

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