PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Positive right derived functors are effaceable by injectives

Statement

Assume the Axiom of Dependent Choice.

Let $A$ be an abelian category with enough injectives, let $I$ be supplied injective resolution data on all objects of $A$ . Let $F : A \to B$ be an additive left exact functor. Then the cohomological delta functor $(R_{I}^{n} F)$ is effaceable in positive degrees by injectives.

Facts & Assumptions

Given: An object $A \in A$ and an integer $n > 0$ .

[L1]

Enough injectives gives a monomorphism $u : A ↣ J$ with $J$ injective (A category with enough projectives and with enough injectives).

[L2]

Positive right derived functors vanish on injective objects (Positive right derived functors vanish on injective objects).

[L3]

Effaceability in positive degrees means killing the induced map into some injective object (Effaceable cohomological delta functor in positive degrees).

Proof

technique · direct

1.1L1L2givenconstruct

By [L1], choose a monomorphism $u : A ↣ J$ with $J$ injective. The datum $I$ is defined on all of $A$ , so $R_{I}^{n} F (J)$ is defined; since $J$ is injective and $n > 0$ , [L2] gives $R_{I}^{n} F (J) = 0$ .

2.1L3step 1.1∎

The induced map $R_{I}^{n} F (u) : R_{I}^{n} F (A) \to R_{I}^{n} F (J)$ is therefore zero. By [L3], this is exactly the required positive-degree effacement.

Depends on

Used by

Restriction and corestriction in group cohomology Definition
Derived functors are universal delta functors Theorem

Dependency tree · two levels

17 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

The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)