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

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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

1.1

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].

L1construct

2.1

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 $\dots \to F_{2} \to F_{1} \to F_{0} \to M \to 0.$

L2L3step 1.1construct

3.1

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].

L2step 2.1discharge-construct∎

Depends on

Used by

Dependency tree · two levels

11 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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.28: Projectives (standard reference, not scraped)