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 right derived map relative to supplied resolution data

Definition

Assume the Axiom of Dependent Choice.

Let $I$ be a supplied injective 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 injective comparison extension ${\tilde{u}}^{∙} : I^{∙} (A) \to I^{∙} (B) .$

The right derived map of $u$ in degree $n$ relative to $I$ is the induced map on cohomology $R_{I}^{n} F (u) : = H^{n} ⁣ (F ({\tilde{u}}^{∙})) : R_{I}^{n} F (A) \to R_{I}^{n} F (B) .$

Existence of ${\tilde{u}}^{∙}$ comes from A morphism has a comparison extension between the supplied injective resolutions, and independence of the chosen extension comes from The induced cohomology map is independent of the chosen injective comparison extension.

Depends on

Dependency tree · 2 levels

Sources

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