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.

Right derived objects relative to supplied injective resolution data

Definition

Let $I$ be a supplied injective resolution datum on a class $D$ of objects in an abelian category $A$ , and let $F : A \to B$ be an additive functor to an abelian category $B$ .

For $A \in D$ and $n \in Z$ , the $n$ th right derived object of $F$ relative to $I$ at $A$ is $R_{I}^{n} F (A) : = H^{n} ⁣ (F (I (A)_{d e l})),$ where $I (A)_{d e l}$ is the deleted injective resolution from Deleted resolutions.

The superscript records the supplied injective datum. This page keeps that data visible instead of suppressing it into an unstated global choice.

Depends on

Used by

An acyclic object for a left exact functor Definition
The right derived map relative to supplied resolution data Definition
The induced cohomology map is independent of the chosen injective comparison extension Lemma
A bifunctor can be derived in either variable when the relevant resolution data are supplied Proposition
An exact functor has vanishing positive derived functors Proposition
Contravariant derived functors are derived on the opposite category Proposition
Negative derived degrees vanish for one-sided resolutions Proposition
Positive right derived functors vanish on injective objects Proposition

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)