ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passaudited 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.

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An acyclic resolution that is not an injective resolution

Example

For the identity functor on abelian groups and a supplied projective resolution datum $P$ on a class containing the free abelian groups, the standard free resolution $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 Z \to 0$ is an $F$ -acyclic resolution relative to $P$ , but it is not 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 displayed free resolution.

[L1]

Projective objects are acyclic for left derived functors (Positive left derived functors vanish on projective objects).

[L2]

An $F$ -acyclic resolution is an exact augmented resolution by $F$ -acyclic objects (An F-acyclic resolution).

[L3]

Projective and injective objects are defined by distinct lifting and extension properties (Projective object, Injective object).

Verification

technique · direct

1.1L1L2given

The terms of the displayed resolution are free abelian groups, hence projective and in the domain of $P$ . By [L1], they are acyclic for the identity functor, so [L2] identifies the displayed exact sequence as an acyclic resolution relative to $P$ .

2.1L3step 1.1algebra∎

The term $Z$ is not injective, because the map $2 Z \to Z$ , $2 n \mapsto n$ , does not extend across $2 Z ↪ Z$ . By [L3], the resolution is therefore not an injective resolution.

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)