False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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FALSE: any sequence of functors with long exact sequences is a delta functor

Statement

False. Any family of functors that sends every short exact sequence to a long exact sequence is automatically a delta functor.

Facts & Assumptions

Given: The homology delta functor on complexes and the canonical cone sequence of the identity on the stalk complex $Z [0]$ .

[L1]

A delta functor requires naturality of the connecting maps, not only exactness of the long sequence (Homological delta functor, Cohomological delta functor).

[L2]

Homology of complexes is a genuine homological delta functor (Homology of complexes satisfies the delta-functor naturality and exactness laws).

[L3]

The canonical mapping-cone sequence is degreewise split short exact makes the cone sequence of $1_{Z [0]}$ short exact, and The cone connecting map agrees with the shifted identity up to the declared sign makes its connecting map $H_{1} (Z [1]) \to H_{0} (Z [0])$ equal to $\pm 1$ on $Z$ , hence nonzero.

Refutation

technique · direct

1.1L2L3givenconstruct

Start from the homology delta functor of [L2]. Keep all functors $H_{n}$ and all connecting maps unchanged except on one chosen short exact sequence with nonzero connector from [L3], where replace the connecting map by its negative. Each long exact sequence stays exact, because negating one map does not change its image or kernel.

2.1L1step 1.1algebra∎

By construction, the modified family still has long exact sequences, but on an isomorphism between the altered sequence and an unaltered copy the connecting square no longer commutes: one side uses $\partial$ and the other uses $- \partial$ , which are different because $\partial \neq 0$ . Thus [L1] fails, so the modified family is not a delta functor.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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, Chapter 2 `Derived Functors` (standard reference, not scraped)