TheoremStatement: 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.

Representable Hom functors on a triangulated category are homological or cohomological

Statement

For each $W \in T$ , $T (W, -)$ is homological and $T (-, W)$ is cohomological, both valued in abelian groups.

Facts & Assumptions

Given: A distinguished triangle $X \overset{f}{\to} Y \overset{g}{\to} Z \overset{h}{\to} X [1]$ and an object $W$ .

Proof

1.1given

TR1 and TR3 first give $g f = 0$ . If $u : W \to Y$ satisfies $g u = 0$ , compare the appropriately rotated identity triangle of $W$ with the given triangle; TR3 supplies $v : W \to X$ with $f v = u$ . Rotating the given triangle gives the same factorisation at every translated term.

2.1step 1.1given∎

Thus $T (W, -)$ is homological. The dual rotated-identity-triangle argument gives exactness for $T (-, W)$ with the arrows reversed, so it is cohomological.

Depends on

Used by

Long exact Hom sequences of a distinguished triangle Theorem

Dependency tree · two levels

18 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, Lemma 13.4.2 (standard reference, not scraped)