TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)

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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.

Right derived functors form a cohomological delta functor

Statement

Assume the Axiom of Dependent Choice.

Let $A$ and $B$ be abelian categories, let $I$ be supplied injective resolution data on all objects of $A$ , and let $F : A \to B$ be an additive left exact functor. Then the additive functors $R_{I}^{n} F : A \to B, n \geq 0,$ admit connecting maps that make them into a cohomological delta functor on $A$ , and $R_{I}^{0} F$ is naturally isomorphic to $F$ .

Facts & Assumptions

Given: A short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ and an integer $n \geq 0$ .

[L1]

Each $R_{I}^{n} F$ is an additive functor (Right derived functors relative to supplied data are additive functors).

[L2]

The injective horseshoe is obtained by dualizing the projective horseshoe; the latter fits into a degreewise split short exact sequence of augmented complexes. Consequently the injective horseshoe also carries the dual degreewise split short exact sequence of cochain complexes (The horseshoe lemma for injective resolutions, The horseshoe lemma for projective resolutions).

[L3]

Applying $F$ to a short exact sequence of injective resolution complexes produces a long exact sequence in cohomology, natural under morphisms of such sequences (The long exact sequence in cohomology, Naturality of the cohomology connecting morphism).

[L4]

Replacing the supplied injective resolution datum at one object changes the derived functor only by natural isomorphism (Two supplied injective resolution data define naturally isomorphic right derived functors).

[L5]

The zeroth right derived functor of a left exact functor recovers the original functor (The zero-th right derived functor of a left exact functor recovers the functor).

[L6]

A cohomological delta functor is exactly the data listed in Cohomological delta functor.

[L7]

Different injective comparison extensions of the same object morphism induce the same maps on cohomology (The induced cohomology map is independent of the chosen injective comparison extension).

Proof

technique · direct

1.1

Choose the injective horseshoe from [L2]. Its degreewise split short exact sequence remains split exact after applying the additive functor $F$ , so it becomes a short exact sequence of cochain complexes in $B$ . Now [L3] gives its long exact cohomology sequence, and [L4] transports the middle cohomology groups to the fixed datum $I$ . This defines connecting maps $\partial^{n} : R_{I}^{n} F (A^{''}) \to R_{I}^{n + 1} F (A^{'}) .$

L2L3L4givenconstruct

2.1

Given a morphism of short exact sequences, use the degreewise biproduct form of the two injective horseshoes and construct compatible comparison maps by the dual comparison induction: injectivity extends the off-diagonal correction at each degree. By [L7], different comparison extensions induce the same maps on cohomology. Naturality of the cohomology connecting morphism in [L3] then makes the connecting squares commute, and [L1] supplies additivity. Therefore [L6] identifies $(R_{I}^{n} F, \partial^{n})$ as a cohomological delta functor on $A$ .

L1L2L3L4L6L7step 1.1construct

3.1

The degree-zero term is naturally isomorphic to $F$ by [L5].

L5step 2.1∎

Depends on

Used by

The derived long exact sequence Corollary
One dimension shift along an injective copresentation Example
Natural transformations of base functors give morphisms of derived delta functors Proposition
Positive right derived functors are effaceable by injectives Proposition
Satellites give the first derived functor Proposition
Derived functors are universal delta functors Theorem

Dependency tree · two levels

33 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)
The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)