# Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?

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.

• 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 Dec 6, 2018 at 22:57
• is there a reference where this is all written down?
– xir
Dec 7, 2018 at 1:36
• 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, 2018 at 1:43

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.

• 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?)
– xir
Dec 30, 2018 at 0:41