TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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

L1L2givenconstruct

2.1

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.

L2L3givenalgebra∎

Depends on

Used by

FALSE: projective dimension is the length of any projective resolution False statement
Projective dimension at most n iff higher Ext vanishes Theorem

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)