PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A map is zero exactly when the corresponding representable map vanishes

Statement

For $a : X \to Y$ in a triangulated category, $a = 0$ if and only if the natural transformation $T (-, a) : T (-, X) \to T (-, Y)$ vanishes.

Facts & Assumptions

Given: A morphism $a : X \to Y$ .

Proof

1.1

If $a = 0$ , composition with $a$ is zero at every object, so the natural transformation vanishes.

given

2.1

Conversely its component at $X$ sends $1_{X}$ to $a$ ; if that component is zero then $a = 0$ .

step 1.1given∎

Depends on

Used by

A distinguished triangle with zero first map is split Proposition

Dependency tree · two levels

11 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

Amnon Yekutieli, A Course on Derived Categories, Proposition 8.2.6 (standard reference, not scraped)