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

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.

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

1.1L1givenconstruct

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 $I$ .

1.2L1givenconstruct

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].

2.1L2L3step 1.1step 1.2∎

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

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)