False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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FALSE: the mapping-cone differential needs no minus sign

Statement

The mapping-cone differential still squares to zero if one removes the minus sign from the shifted summand.

Facts & Assumptions

Given: The identity map $1_{C}$ on the two-term complex $0 \to Z \overset{1}{\to} Z \to 0,$ placed in degrees $1$ and $0$ .

[A1]

The statement refuted is: the mapping-cone differential still squares to zero if one removes the minus sign from the shifted summand.

[L1]

With the minus sign present, the mapping-cone differential squares to zero (The mapping-cone differential squares to zero).

[L2]

The actual cone differential is $d (y, x) = (d (y) + f (x), - d (x))$ (The mapping cone of a chain map).

Refutation

technique · direct

1.1

Let $x = 1$ be the generator of the copy of $Z$ in $C_{1}$ . If the minus sign were removed, then for the displayed identity map the modified square on $(0, x)$ would have first component $d_{C} (x) + d_{C} (x) = 2 d_{C} (x) = 2 \neq 0$ in the copy of $Z$ in degree $0$ . So the modified differential does not square to zero on this cone.

A1givenalgebra

2.1

This contradicts the claim in [A1]. The actual definition [L2] and the verified lemma [L1] show that the minus sign is exactly what cancels the mixed terms.

A1L1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Romyar Sharifi, Homological Algebra (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)