A remark on the invariance of $K$-theory under duality

TL;DR

This paper discusses the formal invariance of algebraic K-theory under duality and provides a counterexample showing that this invariance does not extend to all localizing invariants.

Contribution

It clarifies the formal nature of K-theory invariance under duality and presents a counterexample to the invariance of the universal localizing invariant under opposite categories.

Findings

K-theory invariance under duality is purely formal in specific cases.

Counterexample to the invariance of the universal localizing invariant under opposite categories.

Abstract

In this short remark, we explain that two examples of invariance under duality for a localizing invariant $F$ hold purely formally when $F$ is $K$-theory, whereas the general statement for arbitrary localizing invariants does not reduce to a formal statement. We record a counterexample to the claim that the universal localizing invariant is invariant under the operation of taking opposite categories, originally due to Tabuada.