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

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FALSE: projective dimension is the length of any projective resolution

Statement

Facts & Assumptions

Given: $M = Z / n Z$ with $n \geq 2$ as a $Z$ -module.

Refutation

technique · direct

1.1givenalgebra

The two-term resolution $0 \to Z \overset{n}{\to} Z \to M \to 0$ has length one, and ${Ext}^{1} (M, Z) ≅ Z / n Z \neq 0$ , so the projective dimension of $M$ is exactly one.

2.1step 1.1algebra∎

Yet $0 \to Z \overset{c \mapsto (0, c)}{\to} Z^{2} \overset{(a, b) \mapsto n a}{\to} Z \to M \to 0$ is also exact and consists of projectives, but has length two. Hence projective dimension is the least possible length, not the length of every resolution.

Depends on

Used by

Nothing in the library uses this result yet.

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, Chapters 3–4 (standard reference, not scraped)