CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Nonuniqueness of a TR3 completion

Statement refuted

TR3 completions need not be unique. In $K (Z -Mod)$ , take the standard cone triangle of the zero map $S^{0} Z \to S^{0} Z [1]$ . The zero square on its first two terms has two distinct completions.

Counterexample

Given: The displayed data.

1.1

The cone is $S^{0} Z [1] \oplus S^{0} Z [1]$ . Besides the zero third component, take the chain map whose matrix has the identity from the second summand to the first and zero in all other entries.

given

2.1

Both third components kill the cone inclusion and are killed by the cone projection, so each completes the same zero square; they differ on the second summand and hence are distinct in $K (Z -Mod)$ .

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

The Stacks Project, Derived Categories, Remark 13.9.11 (standard reference, not scraped)