LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact equivalences preserve triangle K zero

PDF p. 12: Bundle classes in spectral K zero

Statement

An exact equivalence $F : T \to U$ of essentially small triangulated categories induces an isomorphism $K_{0}^{t r i} (T) \to K_{0}^{t r i} (U)$ , sending $[X]$ to $[F (X)]$ . Here $K_{0}^{t r i}$ is generated by isomorphism classes with $[Y] = [X] + [Z]$ for each distinguished triangle $X \to Y \to Z \to X [1]$ . Exact means that $F$ commutes with translation up to its specified natural isomorphism and sends distinguished triangles to distinguished triangles.

Facts & Assumptions

Given: An exact equivalence $F$ and an exact quasi-inverse $G$ , with unit and counit natural isomorphisms.

Goal: Prove that the induced maps on triangle Grothendieck groups are inverse.

Proof

technique · Use the presentations and the quasi-inverse

1.1givenconstruct

The functor $F$ sends isomorphic objects to isomorphic objects and each defining triangle relation in $T$ to a defining relation in $U$ . It consequently induces a homomorphism $F_{0}$ on the presented groups. The same argument gives $G_{0}$ .

2.1givenstep 1.1

The unit and counit identify $G F (X)$ with $X$ and $F G (Y)$ with $Y$ . Hence $G_{0} F_{0}$ and $F_{0} G_{0}$ fix every generator of their respective groups, and so are identity homomorphisms.

3.1step 2.1∎

Thus $F_{0}$ is the claimed isomorphism, with inverse $G_{0}$ .

Used by · 1

Affine perfect triangle classes and bundle Euler classes