TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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 zero-th right derived functor of a left exact functor recovers the functor

Statement

Assume the Axiom of Dependent Choice.

Let $I$ be a supplied injective resolution datum on a class $D$ , and let $F : A \to B$ be an additive left exact functor between abelian categories. Then for every $A \in D$ there is a canonical isomorphism $R_{I}^{0} F (A) \tilde{\to} F (A),$ natural in $A$ .

Facts & Assumptions

Given: An object $A \in D$ .

[L1]

The chosen injective resolution of $A$ is an exact coaugmented complex $0 \to A \to I^{0} (A) \to I^{1} (A) \to \dots$ (Injective resolutions in an abelian category).

[L2]

The zeroth cohomology object is the quotient of the kernel of $d^{0} : F (I^{0} (A)) \to F (I^{1} (A))$ by the zero-th coboundary, which is $0$ (Cohomology object of a cochain complex).

[L3]

Left exactness means that $F$ preserves the kernel at the beginning of the displayed injective resolution (Left exact and right exact functors).

[L4]

The assignments $A \mapsto R_{I}^{0} F (A)$ are already functorial (Right derived functors relative to supplied data are additive functors).

Proof

technique · direct

1.1

By [L1], the morphism $0 \to A \to I^{0} (A) \to I^{1} (A)$ is exact. Applying $F$ and using [L3] gives an exact sequence $0 \to F (A) \to F (I^{0} (A)) \to F (I^{1} (A)) .$ Therefore $F (A) = \ker (F (I^{0} (A)) \to F (I^{1} (A)))$ .

L1L3givenalgebra

2.1

By [L2], the zeroth cohomology of $F (I (A)_{d e l})$ is that kernel, because the zero-th coboundary object is $0$ . Hence $R_{I}^{0} F (A) = H^{0} (F (I (A)_{d e l})) ≅ F (A)$ .

L2step 1.1

3.1

Step 2.1 is natural in $A$ , and [L4] records the functoriality of $R_{I}^{0} F$ . Therefore the displayed isomorphism is natural.

L4step 2.1∎

Depends on

Used by

A balanced derived bifunctor Definition
The right derived functors of Hom from a fixed object Example
The acyclic-resolution theorem for right derived functors Theorem

Dependency tree · two levels

20 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

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)