Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.
  • 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.

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Representable Hom functors on a triangulated category are homological or cohomological

Statement

For each W∈T, T(W,−) is homological and T(−,W) is cohomological, both valued in abelian groups.

Facts & Assumptions

Given: A distinguished triangle X→fY→gZ→hX[1] and an object W.

Proof

1.1

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

given
2.1

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

step 1.1given∎

Depends on

Used by

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