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

Georg Lehner

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.

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

Abstract

In this short remark, we explain that two examples of invariance under duality for a localizing invariant hold purely formally when is -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.
Paper Structure (1 section, 5 theorems, 14 equations)

This paper contains 1 section, 5 theorems, 14 equations.

Table of Contents

  1. A counterexample

Key Result

Theorem 3

There exists a natural equivalence of functors $\mathrm{Cat}^\mathrm{perf} \rightarrow \mathrm{Sp}$,

Theorems & Definitions (13)

  • Example 1
  • Example 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Example 5
  • Theorem 6
  • Theorem 8: Albert--Brauer--Hasse--Noether, NSW2008
  • Proposition 9
  • ...and 3 more