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

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FALSE: the shift of a complex keeps the same differential with no sign

Statement

The shift of a chain complex keeps the same differential, with no sign change.

Facts & Assumptions

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

[A1]

The statement refuted is: the shift of a chain complex keeps the same differential, with no sign change.

[L1]

The adopted shift convention is $d_{n}^{C [1]} = - d_{n - 1}^{C}$ (The shift of a chain complex).

[L2]

With the shifted differential from [L1], the shifted complex is again a chain complex (The shifted differential squares to zero).

Refutation

technique · direct

1.1

By [L1], the differential of $C [1]$ from degree $2$ to degree $1$ is $d_{2}^{C [1]} = - d_{1}^{C} = - 1_{Z} .$ It is therefore not the unchanged differential $1_{Z}$ .

givenL1algebra

2.1

The explicit complex in step 1.1 contradicts [A1]. The adopted definition is [L1], and [L2] confirms that it gives a legitimate shifted complex. Hence [A1] is false.

A1L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)
The Stacks Project, Section 12.14: Homotopy and the shift functor (standard reference, not scraped)