False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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FALSE: every acyclic complex of projective objects is contractible

Statement

False. Every acyclic complex of projective objects is contractible.

Facts & Assumptions

Given: The bi-infinite chain complex over $R = Z / 4 Z$ with $C_{n} = R$ for every $n$ and differential $d_{n}$ equal to multiplication by $2$ .

[L1]

Contractibility is the existence of a homotopy from the identity to zero (A contractible complex).

[L2]

A bounded-below acyclic complex of projectives is contractible once its cycle epimorphisms split (A bounded below acyclic complex of projective objects is contractible when its cycle epimorphisms split).

Refutation

technique · direct

1.1givenalgebra

Since $2^{2} = 4 = 0$ in $R$ , the displayed differentials satisfy $d_{n - 1} d_{n} = 0$ . Also $\ker (d_{n}) = 2 R = im (d_{n + 1}),$ so the complex is acyclic. Each term $R$ is a free rank-one $R$ -module and hence projective.

2.1L1step 1.1algebra

If a contracting homotopy existed, then for each $n$ one would have $1_{R} = d_{n + 1} s_{n} + s_{n - 1} d_{n} = 2 s_{n} + 2 s_{n - 1} .$ But every endomorphism of the free rank-one module $R$ is multiplication by an element of $R$ , and the right-hand side is always even while $1_{R}$ is not. Contradiction.

3.1L2step 2.1∎

Therefore the complex is acyclic and degreewise projective but not contractible. The positive theorem [L2] does not apply because this standard counterexample is not bounded below with the required splitting data.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)