PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Modules over a field are projective, flat, and injective

Statement

Assume the Axiom of Choice. Every module over a field $k$ is free, hence projective and flat, and is also injective.

Proof

Given: a $k$ -vector space $V$ , under Choice.

1.1

By Every vector space has a basis, Choice supplies a basis of $V$ . It identifies $V$ with a direct sum of copies of $k$ , so $V$ is free and therefore projective and flat.

given

2.1

Under Choice a subspace has a vector-space complement. Therefore every map from a subspace into $V$ extends across an inclusion, which is the injectivity criterion.

step 1.1construct∎

Depends on

Used by

Cohomology over a field is dual to homology for finite-dimensional complexes Corollary
Kunneth over a field Corollary

Dependency tree · two levels

18 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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)