PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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 $R_{I}^{n} F (J) = 0$ .

L1L2givenconstruct

2.1

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.

L3step 1.1∎

Depends on

Used by

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)