The spectrum of units of algebraic $K$-theory

TL;DR

This paper investigates the structure of units in algebraic $K$-theory, providing a splitting result for the $[0,1]$-truncation and calculating strict units for specific spectra, revealing that units do not split as a presheaf.

Contribution

It extends known splitting results from topological to algebraic $K$-theory and computes the spectrum of strict units for $K( extbf{Z})$ and $K( extbf{F}_ extbf{ell})$, showing units do not split as a presheaf.

Findings

Splitting for the $[0,1]$-truncation of algebraic $K$-theory units.

Complete calculation of strict units for $K( extbf{Z})$ and $K( extbf{F}_ extbf{ell})$.

Units of algebraic $K$-theory do not split as a presheaf or pointwise.

Abstract

It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\glone \KO$ and $\glone \KU$, respectively, are split, and that the splitting is provided by the ($\Z/2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\glone K(\Z)$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\Z)$ and $K(\F_\ell)$ for a prime $\ell$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise.