False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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FALSE: objectwise projective-resolution choices uniquely determine a resolution functor

Statement

False. Once one projective resolution has been chosen for each object in a category with enough projectives, those objectwise choices uniquely determine comparison maps and hence a projective-resolution functor.

Facts & Assumptions

Given: The category of abelian groups and the standard projective resolution of $Z / 2 Z$ .

[L1]

Enough projectives gives projective resolutions only after choosing successive projective epimorphisms for each fixed object (A chosen chain of projective epimorphisms gives a projective resolution).

[L2]

Finite-rank free modules are projective without any infinite choice (Free modules are projective, with the exact choice boundary).

Refutation

technique · direct

1.1

The proof of [L1] is objectwise: it chooses terms and differentials but supplies no unique lift of a morphism between resolved objects.

1.2

The exact row $0 \to Z \overset{2}{\to} Z \to Z / 2 Z \to 0$ is a projective resolution by [L2]. On two copies of it, multiplication by $1$ in both degrees and multiplication by $3$ in both degrees are distinct chain maps lifting the identity of $Z / 2 Z$ : both commute with multiplication by $2$ , and $3 \equiv 1 (m o d 2)$ . Thus the chosen objectwise resolution does not uniquely determine the map assigned to the identity morphism.

L2algebra

2.1

Therefore objectwise resolution choices do not uniquely determine comparison maps or a resolution functor; additional coherent choices or a separate functorial construction are required.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.28: Projectives (standard reference, not scraped)