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

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.

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

1.1

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].

L1L3construct

2.1

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

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)