TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Duality induces an anti-isomorphism on the Grothendieck ring

Statement

For a tensor category, $[X] \mapsto [X^{\lor}]$ induces an additive anti-isomorphism of $K_{0} (C)$ with its opposite ring. It need not be an involution: its inverse is induced by right duals.

Facts & Assumptions

Given: A tensor category $C$ and chosen left duals.

[F1]

Dualization is exact (Dualization in a multitensor category is exact).

[F2]

$(X \otimes Y)^{\lor} ≅ Y^{\lor} \otimes X^{\lor}$ (Left duality is a contravariant antimonoidal functor).

[F3]

The Grothendieck group is defined by exact-sequence relations (The Grothendieck ring of a tensor category).

Proof

technique · direct

1.1F1F3given

By [F1], dualization takes each short exact relation in [F3] to a short exact relation, so $[X] \mapsto [X^{\lor}]$ is a well-defined additive map.

2.1step 1.1F2given∎

By [F2], it reverses products: $([X] [Y])^{*} = [Y]^{*} [X]^{*}$ . Chosen right duals give the inverse map on isomorphism classes and hence on $K_{0}$ . Therefore this is an anti-isomorphism; no identification of $X$ with $X^{\lor \lor}$ is asserted.

Depends on

Used by

The Grothendieck ring of a tensor category is always commutative False statement

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Section 4.5 (standard reference, not scraped)