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

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FALSE: a comparison map between resolutions is unique as a chain map

Statement

False. A comparison map between two projective resolutions is unique as a > chain map.

Facts & Assumptions

Given: The standard projective resolution $0 \to Z \overset{2}{\to} Z \to Z / 2 Z \to 0$ of $Z / 2 Z$ .

[L1]

Comparison maps lifting the same object morphism are unique only up to chain homotopy (Projective comparison maps are unique up to chain homotopy).

Refutation

technique · direct

1.1

On two copies of the displayed resolution, multiplication by $1$ in both degrees and multiplication by $3$ in both degrees are distinct augmentation-preserving chain maps lifting $1_{Z / 2 Z}$ , because $3 \equiv 1 (m o d 2)$ and $2 \cdot 3 = 3 \cdot 2$ .

givenalgebra

2.1

Step 1.1 exhibits two different comparison maps lifting the same object morphism. By [L1], the positive theorem only identifies them up to homotopy, so uniqueness as an actual chain map is false.

L1step 1.1∎

Depends on

Projective comparison maps are unique up to chain homotopy

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)