DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Projective dimension of an object

Definition

Assume projective resolutions are supplied or exist in the relevant class. The projective dimension of $M$ is $pd (M) = \inf {d \geq 0 : M has a projective resolution of length d},$ with value $\infty$ if this set is empty. A length-zero projective resolution exists exactly when $M$ is projective.

Depends on

Used by

Global dimension of an abelian category Definition
Left and right global dimension of a ring Definition
FALSE: projective dimension is the length of any projective resolution False statement
Flat modules need not have projective dimension zero False statement
Hereditary rings have global dimension at most one Proposition
Projective dimension at most n iff the nth syzygy is projective Theorem

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)