# A modern interpretation of Quillen's computation of the K theory of finite fields

In his beautiful paper On the cohomology and K theory of the general linear group over a finite field, Quillen constructs (if I understand correctly) an isomorphism on connected components of K-theory $K(\mathbb{F}_q)\mid_{\ge 1} \cong \text{Fib}(1-\Psi^q:ku\to ku).$ His construction, while very cool, seems to be based on a miracle: that the Brauer lift produces complex representations from mod-$p$ ones. It also uses the +-construction for defining K-theory (though perhaps it's not that hard to update his construction to work with Segal formalism)?

I'm curious whether there is a more modern perspective on this equivalence of spectra, maybe using cyclic structure on topological Hochschild homology.

• For what it's worth, many years ago (2006?) I posed this question (more or less) to an expert on trace methods, and the response seemed to be that no such proof of Quillen's result is known. It's a little uncler whether Quillen was aware of Segal's K-theory construction and the Q-construction before he wrote the paper in question. My impression is that invented the plus-construction specifically for this calculation. One could probably determine the dates by looking through Quillen's notebooks. – Dan Ramras Aug 8 '16 at 18:42
• Quillen's famous Letter to Segal (where he talks about "freeing myself from the shackes of the simplicial way of thinking" and discovering the category Q) is from July 1972 - he says he discovered Q in Spring '72 - and the Annals paper in question was published in November 1972. There was an earlier preprint of the Annals paper, though, which is on the K-theory archive (math.uiuc.edu/K-theory/1006), but it's undated. – Dan Ramras Aug 8 '16 at 18:43
• Didn't Dwyer say something about this at the 2012 WCATSS? – Sean Tilson Aug 10 '16 at 12:17
• @SeanTilson I don't recall anything about this from WCATSS. – Dan Ramras Aug 13 '16 at 16:48
• I do. I will write you a message. – Sean Tilson Aug 23 '16 at 10:11

Firstly, there is a proof using the motivic spectral sequence (the Atiyah-Hirzerbruch style spectral sequence from motivic cohomology to algebraic $$K$$-theory). This is written in the master's thesis of Gabe Angelini-Knoll.