Reference request: K-theory of finite fields by Q construction

Question

I have read Quillen's Q construction and Waldhausen's S construction of K-groups of an exact category. Most references for K theory, calculate K groups for finite fields by the + construction and show the + construction is equivalent to the Q construction. Is there a reference for calculating K groups of finite fields directly from the Q or S construction?

Answer 1

I don't know of one. It might be hard to do, since the main point of Quillen's computation is to compare the K-groups to the homotopy groups of $BU$. Since $BU$ is realized by a simplicial space with U as the space of 1-simplices, it makes sense to use a parallel construction of K-theory that has the invertible matrices as its 1-simplices.

Quillen remarked to me (about 1975) that the Q-construction seems adapted to proving general theorems, whereas the +-construction seems adapted to making computations. At its essence, the reason for that is this: If $X$ is a space and you have an explicit description of $\Omega ^ n X$, the $n$-fold loop space, then you probably have a presentation of $\pi_0 \Omega^n X$, which is $\pi_n X$. In other words, computation of $\pi_n X$ may succeed by first "computing" $\Omega^n X$. You may even have a presentation of $\pi_1 \Omega^n X$, but it's rare to know $\pi_2$ of a space explicitly, even if the space is known "explicitly". (An example of a space whose $\pi_0$ and $\pi_1$ are both known is $BG$ for a group $G$.)

If you want to try to make such a proof, one thing to try would be to consider the S-construction applied to the topological category of finite dimensional $C$-vector spaces. It would give a delooping of $BU$.