TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04‡ 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.

The acyclic-resolution theorem for right derived functors

Statement

Assume the Axiom of Dependent Choice.

Let $I$ be a supplied injective resolution datum on a class $D$ , let $F : A \to B$ be an additive left exact functor, and let $0 \to A \to J^{0} \to J^{1} \to J^{2} \to \dots$ be an $F$ -acyclic resolution of $A$ relative to $I$ . Assume moreover that $A \in D$ and that, for $Z^{0} : = A$ and $0 \to Z^{q} \to J^{q} \to Z^{q + 1} \to 0 (q \geq 0),$ each $Z^{q}$ lies in $D$ . Then for every $n \geq 0$ there is a canonical isomorphism $R_{I}^{n} F (A) \tilde{\to} H^{n} (F (J_{d e l}^{∙})) .$

Facts & Assumptions

Given: An $F$ -acyclic resolution $0 \to A \to J^{0} \to J^{1} \to \dots$ of $A$ relative to $I$ , the associated objects $Z^{q} \in D$ , and an integer $n \geq 0$ .

[L1]

An $F$ -acyclic resolution is an exact coaugmented complex whose terms are $F$ -acyclic objects (An F-acyclic resolution, An acyclic object for a left exact functor).

[L2]

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

[L3]

Change of supplied injective resolution data produces natural isomorphisms of right derived functors (Two supplied injective resolution data define naturally isomorphic right derived functors).

[L4]

Passing to the opposite abelian category and applying the projective horseshoe lemma produces injective resolutions of a short exact sequence in a degreewise split short exact sequence of cochain complexes (The opposite of an abelian category is abelian, The horseshoe lemma for projective resolutions).

[L5]

A short exact sequence of cochain complexes yields a long exact sequence in cohomology (The long exact sequence in cohomology).

[L6]

An additive functor preserves finite biproducts (An additive functor preserves finite biproducts), and therefore preserves split short exact sequences.

Proof

technique · direct

1.1

By exactness in [L1], let $Z^{0} = A$ and for each $q \geq 0$ let $Z^{q + 1}$ fit into a short exact sequence $0 \to Z^{q} \to J^{q} \to Z^{q + 1} \to 0.$ Every $J^{q}$ is $F$ -acyclic by [L1].

L1givenconstruct

2.1

Apply [L4] to each short exact sequence from step 1.1 using the supplied injective resolutions of $Z^{q}$ and $Z^{q + 1}$ , which exist by the domain hypothesis in the statement. The result is a degreewise split short exact sequence of injective resolutions. By [L6], applying $F$ preserves its degreewise exactness, so [L5] gives a long exact cohomology sequence. The middle injective resolution supplied by the horseshoe construction may differ from the one fixed for $J^{q}$ , but [L3] identifies their right derived objects. Its higher cohomology therefore vanishes because $J^{q}$ is $F$ -acyclic. Using [L2] for degree $0$ , we obtain exact sequences $0 \to F (Z^{q}) \to F (J^{q}) \to F (Z^{q + 1}) \to R_{I}^{1} F (Z^{q}) \to 0$ and isomorphisms $R_{I}^{m + 1} F (Z^{q}) \tilde{\to} R_{I}^{m} F (Z^{q + 1}) (m > 0) .$

L2L3L4L5L6step 1.1algebra

3.1

Repeatedly applying the isomorphisms from step 2.1 gives $R_{I}^{n} F (A) = R_{I}^{n} F (Z^{0}) \tilde{\to} R_{I}^{1} F (Z^{n - 1}) (n > 0) .$

step 2.1algebra

4.1

For $n > 0$ , the exact sequence $0 \to F (Z^{n}) \to F (J^{n}) \to F (Z^{n + 1})$ from step 2.1 shows that the $n$ th cohomology of $F (J_{d e l}^{∙})$ is the cokernel of $F (J^{n - 1}) \to F (Z^{n})$ . The same step identifies that cokernel with $R_{I}^{1} F (Z^{n - 1})$ , so step 3.1 gives $H^{n} (F (J_{d e l}^{∙})) ≅ R_{I}^{n} F (A) .$

step 2.1step 3.1algebra

5.1

For $n = 0$ , step 2.1 with $q = 0$ gives $0 \to F (A) \to F (J^{0}) \to F (Z^{1})$ exact, so $H^{0} (F (J_{d e l}^{∙})) ≅ F (A)$ . By [L2], $F (A) ≅ R_{I}^{0} F (A)$ . Together with step 4.1, this proves the theorem for all $n \geq 0$ .

L2step 2.1step 4.1∎

Depends on

Used by

Adapted classes compute derived functors Corollary

Dependency tree · two levels

32 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)