PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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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.

A projective object has a length-zero projective resolution

Statement

Every projective object admits a length-zero projective resolution.

Facts & Assumptions

Given: A projective object $P$ .

[L1]

A projective resolution is an exact augmented complex of projectives (Projective resolutions in an abelian category).

[L2]

Length at most zero means that all higher terms vanish (The length of a resolution).

[L3]

Projectivity is the standing hypothesis on the degree-zero term (Projective object).

Proof

technique · direct

1.1L1L3construct

Consider the augmented complex $0 \to P \overset{1_{P}}{\to} P \to 0,$ with $P$ placed in degree zero. It is exact because the augmentation is the identity, and its only nonzero term is projective by [L3].

2.1L1L2step 1.1∎

By [L2], this exact augmented complex has length zero. Therefore [L1] identifies it as a length-zero projective resolution of $P$ , including the case $P = 0$ .

A projective object has a length-zero projective resolution

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources