PropositionStatement: AI-adaptedProof: 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.

Cohomology factors through the derived category

Statement

For every integer $n$ , $H^{n} : K (A) \to A$ factors uniquely through $Q : D (A)$ . More precisely, there is a unique ${\overline{H}}^{n} : D (A) \to A$ with $H^{n} = {\overline{H}}^{n} Q$ , and ${\overline{H}}^{n} (s, f) = H^{n} (f) H^{n} (s)^{- 1}$ .

Facts & Assumptions

Given: For every integer $n$ , $H^{n} : K (A) \to A$ factors uniquely through $Q : D (A)$ . More precisely, there is a unique ${\overline{H}}^{n} : D (A) \to A$ with $H^{n} = {\overline{H}}^{n} Q$ , and ${\overline{H}}^{n} (s, f) = H^{n} (f) H^{n} (s)^{- 1}$ .

[F1]

The derived category is localization at quasi-isomorphisms (Derived category of an abelian category).

[F2]

Homology is a homological functor on the homotopy category (Homology is a homological functor on the homotopy category).

Proof

1.1

Cochain reindexing of homology gives a functor $H^{n}$ on $K (A)$ , and it inverts every quasi-isomorphism by definition. In particular it sends the zero complex to zero.

F1F2

2.1

The localization property therefore gives the unique factorization. On a roof $Q (f) Q (s)^{- 1}$ functoriality forces the displayed value. No triangulation of $D (A)$ is needed for this assertion.

F1step 1.1∎

Depends on

Used by

Canonical t structure on a derived category Definition
fs-localization-identifies-a-quasi-isomorphism-with-an-identity-morphism.md False statement
fs-two-roofs-are-equal-whenever-their-right-hand-arrows-are-equal.md False statement
A complex is zero in the derived category exactly when it is acyclic Proposition
The derived category inherits a triangulated structure Theorem

Dependency tree · two levels

10 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

13.11.1–13.11.6 (standard reference, not scraped)