A note on K-theory and triangulated derivators

TL;DR

This paper demonstrates that derivator K-theory can coincide for different algebras while Waldhausen K-theory distinguishes them, and shows that certain conjectures about derivator K-theory cannot both hold.

Contribution

It provides a counterexample illustrating the difference between derivator and Waldhausen K-theory and disproves the simultaneous validity of two major conjectures.

Findings

Derivator K-theory can be identical for non-isomorphic algebras.

Waldhausen K-theory can distinguish between these algebras.

Maltsiniotis's conjectures cannot both be true at the same time.

Abstract

In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis's comparison and localization conjectures for derivator K-theory cannot be simultaneously true.