PropositionStatement: 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.

An exact functor has vanishing positive derived functors

Statement

Let $P$ be a supplied projective resolution datum on a class $D_{P}$ , let $I$ be a supplied injective resolution datum on a class $D_{I}$ , and let $F : A \to B$ be an exact functor between abelian categories. Then for every $A \in D_{P}$ and every $n > 0$ , $L_{n}^{P} F (A) = 0,$ and for every $B \in D_{I}$ and every $n > 0$ , $R_{I}^{n} F (B) = 0.$

Facts & Assumptions

Given: An integer $n > 0$ , an object $A \in D_{P}$ , and an object $B \in D_{I}$ .

[L1]

The left and right derived objects are the homology or cohomology of the deleted chosen resolutions after applying $F$ (Left derived objects relative to supplied projective resolution data, Right derived objects relative to supplied injective resolution data).

[L2]

Exact functors commute with homology of chain complexes (An exact functor commutes with homology).

[L3]

Exactness means that $F$ is exact on both the projective and injective resolution complexes (Exact functor between abelian categories).

Proof

technique · direct

1.1

The deleted projective resolution of $A$ is exact in every positive degree. By [L3], applying $F$ preserves that exactness, so the resulting chain complex has zero homology in every positive degree. Using [L1], this says $L_{n}^{P} F (A) = 0$ for $n > 0$ .

L1L3givenalgebra

1.2

Read the deleted injective resolution of $B$ as a reindexed chain complex. It is exact in every positive cohomological degree, and [L3] preserves that exactness after applying $F$ . By [L2], the resulting homology, hence cohomology, is zero in every positive degree. Therefore $R_{I}^{n} F (B) = 0$ for $n > 0$ .

L1L2L3algebra

2.1

Steps 1.1 and 1.2 prove the claimed vanishing on both sides.

step 1.1step 1.2∎

Depends on

Used by

The left derived functors of an exact functor Example
FALSE: an acyclic resolution is the same thing as an injective resolution False statement

Dependency tree · two levels

14 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)