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 left derived functor of a right exact functor recovers the functor

Statement

Assume the Axiom of Dependent Choice.

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

Facts & Assumptions

Given: An object $A \in D$ .

[L1]

The chosen projective resolution of $A$ is an exact augmented complex $\dots \to P_{1} (A) \to P_{0} (A) \to A \to 0$ (Projective resolutions in an abelian category).

[L2]

The zeroth homology of the deleted complex is the cokernel of the boundary map into degree $0$ (Homology object of a chain complex).

[L3]

Right exactness means that $F$ preserves the cokernel appearing at the end of the displayed augmented resolution (Left exact and right exact functors).

[L4]

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

Proof

technique · direct

1.1

By [L1], the morphism $P_{1} (A) \to P_{0} (A) \to A \to 0$ is exact. Applying $F$ and using [L3] gives an exact sequence $F (P_{1} (A)) \to F (P_{0} (A)) \to F (A) \to 0.$ Hence $F (A)$ is the cokernel of $F (P_{1} (A)) \to F (P_{0} (A))$ .

L1L3givenalgebra

2.1

By [L2], that same cokernel is exactly $H_{0} (F (P (A)_{d e l})) = L_{0}^{P} F (A)$ . Therefore there is a canonical isomorphism $L_{0}^{P} F (A) \tilde{\to} F (A)$ .

L2step 1.1

3.1

The construction in steps 1.1 and 2.1 is functorial in $A$ , and [L4] already supplies the functoriality of $L_{0}^{P} F$ . Thus the isomorphism is natural in $A$ .

L4step 2.1∎

Depends on

Used by

L₀ of a non-right-exact functor need not recover the functor Counterexample
A balanced derived bifunctor Definition
The left derived functors of an exact functor Example
FALSE: every additive functor has L₀ naturally isomorphic to itself False statement
The acyclic-resolution theorem for left derived functors Theorem

Dependency tree · two levels

21 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)