False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Flat modules need not have projective dimension zero

Statement

False claim: every flat module has projective dimension zero.

Refutation

Given: the $Z$ -module $Q$ .

1.1

The group $Q$ is torsion-free, hence flat over the PID $Z$ .

given

2.1

If $Q$ were projective over $Z$ , it would be free; every nonzero free abelian group has a nonzero map to $Z$ , while ${Hom}_{Z} (Q, Z) = 0$ .

step 1.1algebra

3.1

Thus $Q$ is flat but not projective, so its projective dimension is not zero.

step 2.1algebra∎

Depends on

Used by

A flat nonprojective module Example

Dependency tree · two levels

6 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)