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.

Additive equivalences preserve split K zero

PDF p. 12: Bundle classes in spectral K zero

Statement

An additive equivalence $F : A \to B$ of essentially small additive categories induces an isomorphism of their split Grothendieck groups. The split group is generated by isomorphism classes $[M]$ , with relations $[M \oplus N] = [M] + [N]$ . An additive equivalence is an equivalence preserving finite direct sums up to isomorphism.

Facts & Assumptions

Given: An additive equivalence $F$ with additive quasi-inverse $G$ .

Goal: Identify the two split Grothendieck groups by their presentations.

Proof

technique · Transport generators and direct-sum relations

1.1givenconstruct

Sending $[M]$ to $[F (M)]$ respects isomorphism and the defining relation, since $F (M \oplus N)$ is isomorphic to $F (M) \oplus F (N)$ . It therefore defines a homomorphism $F_{0}$ . The same construction gives $G_{0}$ .

2.1givenstep 1.1

The unit and counit identify $G F (M)$ with $M$ and $F G (N)$ with $N$ . Thus the two composite homomorphisms fix all generators, hence are identities.

3.1step 2.1∎

Therefore $F_{0}$ is an isomorphism with inverse $G_{0}$ .

Used by · 1

Affine bundle K zero is the projective split group