CorollaryStatement: 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.

Adapted classes compute derived functors

Statement

Assume the Axiom of Dependent Choice.

Let $I$ be a supplied injective resolution datum on a class $D$ , and let $F$ be an additive left exact functor. Suppose $C \subseteq D$ is made of $F$ -acyclic objects, is closed under cokernels of monomorphisms between objects of $C$ , and every object of $D$ admits a monomorphism into an object of $C$ . Then any coaugmented resolution of an object $A \in D$ obtained by iterating monomorphisms $Z^{q} ↪ C^{q}, C^{q} \in C, Z^{q + 1} : = coker (Z^{q} ↪ C^{q}) \in D,$ computes $R_{I}^{n} F (A)$ .
Dually, let $P$ be a supplied projective resolution datum on a class $D$ , let $F$ be additive and right exact, and suppose $C \subseteq D$ is made of $F$ -acyclic objects, is closed under kernels of epimorphisms between objects of $C$ , and every object of $D$ admits an epimorphism from an object of $C$ . Then any augmented resolution of an object $A \in D$ obtained by iterating epimorphisms $C_{q} ↠ Z_{q}, C_{q} \in C, Z_{q + 1} : = \ker (C_{q} ↠ Z_{q}) \in D,$ computes $L_{n}^{P} F (A)$ .

Facts & Assumptions

Given: One of the two clause-wise hypotheses from the statement.

[L1]

Once the relevant supplied datum is fixed, an $F$ -acyclic resolution is exactly a resolution whose terms are $F$ -acyclic and whose orientation matches the side being derived (An F-acyclic resolution).

[L2]

Such resolutions compute right derived functors (The acyclic-resolution theorem for right derived functors).

[L3]

Such resolutions compute left derived functors (The acyclic-resolution theorem for left derived functors).

Proof

technique · direct

1.1

In the left exact case, start with $Z^{0} = A$ . By hypothesis, every object of $D$ admits a monomorphism into an object of $C$ , so we may choose monomorphisms $Z^{q} ↪ C^{q}$ with $C^{q} \in C$ and define $Z^{q + 1}$ to be the cokernel, still in $D$ . This produces an exact coaugmented resolution by objects of $C$ . Because every object of $C$ is $F$ -acyclic, [L1] identifies the result as an $F$ -acyclic resolution relative to $I$ .

L1givenconstruct

1.2

The right exact case is dual: start with $Z_{0} = A$ , repeatedly choose epimorphisms $C_{q} ↠ Z_{q}$ with $C_{q} \in C$ , and define $Z_{q + 1}$ to be the kernel, still in $D$ . The resulting exact augmented resolution has all terms in $C$ , hence is an $F$ -acyclic resolution relative to $P$ by [L1].

L1givenconstruct

2.1

Apply [L2] to the resolution from step 1.1 and [L3] to the resolution from step 1.2. This proves both clauses.

L2L3step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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)