CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Under the Axiom of Choice, every module admits a projective resolution

Statement

Assume the Axiom of Choice. Then every left module over a unital ring admits a projective resolution.

Facts & Assumptions

Given: A unital ring $R$ and a left $R$ -module $M$ .

[L1]

The canonical iterated free-cover construction gives an exact augmented free resolution in ZF (The iterated free-module resolution is canonical in ZF).

[L2]

Under the Axiom of Choice, every free module is projective (Free modules are projective, with the exact choice boundary).

[L3]

A projective resolution is an exact augmented complex of projectives (Projective resolutions in an abelian category).

Proof

technique · direct

1.1

By [L1], the module $M$ has a canonical exact augmented complex of free $R$ -modules ending in $M$ . Because AC is assumed, [L2] makes every term of that complex projective.

L1L2

2.1

By [L3], that exact augmented complex is a projective resolution of $M$ , including the case $M = 0$ .

L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)