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

Used by

FALSE: the definition of a derived map may depend on the chosen comparison lift False statement
A natural transformation induces natural transformations of left derived functors Proposition
Left derived maps preserve composition Proposition
Left derived maps preserve identities Proposition
Left derived functors relative to supplied data are additive functors Theorem

Dependency tree · two levels

13 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, Chapter 2 `Derived Functors` (standard reference, not scraped)