False statementConstruction: Literature-sourcedVerification: 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.

FALSE: an acyclic resolution is the same thing as an injective resolution

Statement

An acyclic resolution is the same thing as an injective resolution.

Facts & Assumptions

Given: The identity functor on abelian groups, a supplied projective resolution datum $P$ on a class containing the free abelian groups, and the standard free resolution $\dots \to 0 \to Z \overset{\times 2}{\to} Z \to Z / 2 Z \to 0.$

[L1]

An $F$ -acyclic resolution only requires a correctly oriented exact resolution by $F$ -acyclic objects (An F-acyclic resolution).

[L2]

Exact functors have vanishing positive derived functors on every object, so every object is acyclic for the identity functor (An exact functor has vanishing positive derived functors).

[L3]

Injective objects are characterized by an extension property across monomorphisms (Injective object).

Refutation

technique · direct

1.1

The identity functor is exact, so [L2] makes every object in the domain of $P$ Id-acyclic. The terms of the displayed free resolution are free abelian groups, hence lie in that domain. Therefore [L1] identifies the displayed free resolution of $Z / 2$ as an Id-acyclic resolution relative to $P$ .

L1L2given

2.1

The term $Z$ in that resolution is not injective: the inclusion $2 Z ↪ Z$ and the map $2 n \mapsto n$ into $Z$ admit no extension $Z \to Z$ . Thus [L3] fails. So an $F$ -acyclic resolution need not be an injective resolution, and the displayed claim is false.

L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)