PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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.

The iterated free-module resolution is canonical in ZF

Statement

For every left $R$ -module $M$ , repeatedly taking the canonical free cover of the current kernel yields a functorial exact augmented complex of free modules $\dots \to F_{2} \to F_{1} \to F_{0} \to M \to 0.$ This construction is available in ZF because it uses only underlying sets and canonical free-module maps. Under the previously recorded choice boundary for free modules, the same complex is a projective resolution.

Facts & Assumptions

Given: A left $R$ -module $M$ .

[L1]

Every module has a canonical free cover $R^{(X)} ↠ X$ on its underlying set (Every module is a quotient of a free module).

[L2]

Free modules are projective with the previously recorded choice boundary (Free modules are projective, with the exact choice boundary).

[L3]

A chosen surjection onto the current kernel extends an exact augmented complex by one degree (One-step extension of a partial projective resolution).

Proof

technique · constructive

1.1L1construct

Put $F_{0} : = R^{(M)}$ and let $ε_{0} : F_{0} ↠ M$ be the canonical map from [L1]. Having defined $K_{n}$ as the current kernel, set $F_{n + 1} : = R^{(K_{n})}$ and let $ε_{n + 1} : F_{n + 1} ↠ K_{n}$ be its canonical free cover from [L1]. These assignments are functorial because they depend only on the underlying-set construction in [L1].

2.1L2L3step 1.1construct

Each $F_{n}$ is free, hence projective under the recorded boundary [L2]. By applying [L3] successively to the canonical surjections of step 1.1, one obtains an exact augmented complex $\dots \to F_{2} \to F_{1} \to F_{0} \to M \to 0.$

3.1L2step 2.1discharge-construct∎

Therefore the iterated free-cover construction gives a functorial exact free resolution in ZF. No basis choice or arbitrary lift is used at any stage; projectivity enters only through [L2].

The iterated free-module resolution is canonical in ZF

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources