False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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FALSE: chain-homotopic maps are equal as chain maps

Statement

Every pair of chain-homotopic chain maps is equal as chain maps.

Facts & Assumptions

Given: The two-term complex $C_{∙}$ in $A b$ with $C_{1} = C_{0} = Z$ , differential $d_{1} = 1_{Z}$ , and all other terms zero.

[A1]

The statement refuted is: every pair of chain-homotopic chain maps is equal as chain maps.

[L1]

A chain homotopy satisfies $f - g = d s + s d$ (A chain homotopy).

[L2]

Passing to the homotopy category remembers only homotopy classes of maps (The homotopy category of chain complexes).

Refutation

technique · direct

1.1

Let $f = 1_{C}$ and let $g = 0 : C_{∙} \to C_{∙}$ . Define a degree- $1$ map $s$ by $s_{0} = 1_{Z}$ and all other components zero. A direct calculation gives $d s + s d = 1_{C},$ so [L1] yields $f ≃ g$ .

L1givenalgebra

2.1

The maps $f$ and $g$ are not equal, because $f_{0} = 1_{Z}$ while $g_{0} = 0$ . Thus [A1] is false. This is exactly why [L2] passes to homotopy classes instead of identifying homotopic maps inside $Ch (A)$ itself.

A1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)