$K$-theory as an Eilenberg-Maclane spectrum

TL;DR

This paper constructs a chain complex model for the $K$-theory spectrum of an additive Waldhausen category over a finite field, using elementary homological algebra without advanced spectral notions.

Contribution

It provides a simple, direct construction of the $K$-theory spectrum as an Eilenberg-MacLane spectrum, avoiding complex spectral machinery.

Findings

Constructs a chain complex model for $K$-theory spectrum

Works well in families and is elementary in nature

Applicable to finite fields of characteristic p

Abstract

For an additive Waldhausen category linear over a ring $k$, the corresponding $K$-theory spectrum is a module spectrum over the $K$-theory spectrum of $k$. Thus if $k$ is a finite field of characteristic $p$, then after localization at $p$, we obtain an Eilenberg-Maclane spectrum -- in other words, a chain complex. We propose an elementary and direct construction of this chain complex that behaves well in families and uses only method of homological algebra (in particular, the notions of a ring spectrum and a module spectrum are not used).