CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedaudited 2026-09-07

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A three-term zero-composite diagram that is not distinguished

Statement refuted

In $K (Z -Mod)$ , the diagram $S^{0} Z ⟶ 0 ⟶ S^{0} Z [1] ⟶ S^{0} Z [1]$ with all three displayed maps zero has zero consecutive composites, but it is not distinguished.

Counterexample

Given: The displayed data.

1.1

All consecutive composites are zero because every displayed map is zero.

given

2.1

If it were distinguished, it would be isomorphic to the standard cone triangle of $0 : S^{0} Z \to 0$ , whose final map is the identity of $S^{0} Z [1]$ ; no triangle isomorphism can turn that nonzero map into zero.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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 10, Exercise 10.2.4 (standard reference, not scraped)