A counterexample to vanishing conjectures for negative $K$-theory

TL;DR

This paper disproves two conjectures claiming the vanishing of negative K-theory for certain categories, showing that these negative K-theories can be non-zero, thus challenging previous assumptions in algebraic K-theory.

Contribution

The paper provides counterexamples to longstanding conjectures about the vanishing of negative K-theory in abelian and t-structured categories.

Findings

Negative K-theory can be non-zero in some abelian categories.

Counterexamples disprove the vanishing conjectures from 2006 and 2019.

Challenges existing beliefs about negative K-theory behavior.

Abstract

In a 2006 article Schlichting conjectured that the negative {\it K--}theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative {\it K--}theory of any category with a bounded {\it t--}structure must vanish. Both conjectures will be shown to be false.