False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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

1.1L1L2given

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 $P$ .

2.1L3step 1.1algebra∎

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.

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)