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

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Projective dimension at most n iff the nth syzygy is projective

Statement

Let $A$ be an abelian category with enough projectives, fix a projective resolution $P_{∙} \to M$ , and let $n \geq 1$ . Then $pd (M) \leq n ⟺ Ω_{P}^{n} (M) is projective .$ In particular, the condition is independent of the chosen projective resolution.

Facts & Assumptions

Given: The displayed hypotheses and a fixed projective resolution of $M$ .

[L1]

The $n$ th syzygy relative to a resolution is the kernel at its $n$ th stage (Syzygies and cosyzygies relative to a chosen resolution).

[L2]

Projective dimension at most $n$ means the existence of a projective resolution of length at most $n$ (Projective dimension of an object).

[L3]

Schanuel's lemma compares kernels of two projective presentations (Schanuel's lemma in an abelian category).

Proof

technique · direct

1.1L1L2givenconstruct

If $Ω_{P}^{n} (M)$ is projective, truncate the fixed resolution after that object. The resulting length- $n$ projective resolution proves $pd (M) \leq n$ .

2.1L2L3givenalgebra∎

Conversely, compare the fixed resolution with a projective resolution of length at most $n$ . Iterating [L3] through their first $n$ projective presentations shows that $Ω_{P}^{n} (M)$ plus a finite direct sum of projectives is isomorphic to the terminal projective of the short resolution plus another finite direct sum of projectives. Hence $Ω_{P}^{n} (M)$ is a direct summand of a projective object and is projective. This also proves independence of the chosen resolution.

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