DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

The left derived map relative to supplied resolution data

Definition

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum on a class $D$ in an abelian category $A$ , let $F : A \to B$ be an additive functor to an abelian category $B$ , let $A, B \in D$ , and let $n \in Z$ . For a morphism $u : A \to B$ , choose any comparison lift ${\tilde{u}}_{∙} : P_{∙} (A) \to P_{∙} (B) .$

The left derived map of $u$ in degree $n$ relative to $P$ is the induced map on homology $L_{n}^{P} F (u) : = H_{n} ⁣ (F ({\tilde{u}}_{∙})) : L_{n}^{P} F (A) \to L_{n}^{P} F (B) .$

By A morphism has a comparison lift between the supplied projective resolutions such a lift exists, and by The induced homology map is independent of the chosen comparison lift the result does not depend on which lift was chosen.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors`