TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 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.

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The derived category inherits a triangulated structure

Statement

The derived category, with shift $[1]$ and distinguished triangles the isomorphic images of cone triangles, is triangulated. The localization functor $Q$ is exact. Every such triangle gives a long exact cohomology sequence $\dots \to H^{n} (X) \to H^{n} (Y) \to H^{n} (Z) \to H^{n + 1} (X) \to \dots$ .

Facts & Assumptions

Given: An abelian category $A$ , the localization $Q : K (A) \to D (A)$ under the standing size convention, and the declared class of triangles isomorphic to localized cone triangles.

[F1]

Localized cone triangles satisfy TR1–TR3, with descended shift (Localized cone triangles satisfy tr one through tr three).

[F2]

Localized cone triangles satisfy the octahedral axiom (Localized cone triangles satisfy the octahedral axiom).

[F3]

An exact functor is additive, has a specified shift isomorphism, and sends distinguished triangles to distinguished triangles (Exact functor between triangulated categories).

[F4]

Each $H^{n} : K (A) \to A$ factors uniquely through the derived localization (Cohomology factors through the derived category).

[F5]

Every cone triangle in the homotopy category carries the cone long exact homology sequence, and cochain reindexing gives the corresponding long exact cohomology sequence (The cone long exact sequence).

[F6]

The roof localization is additive, with bilinear composition and preservation of zero objects and finite biproducts (Addition of roofs makes an additive localization).

Proof

1.1

The additive structure, shift, TR1, signed TR2 and TR3 have been established, including identity triangles and zero objects. TR4 has also been established. These are precisely the triangulated-category axioms.

F1F2F6

2.1

The functor $Q$ is additive, its shift comparison is the identity in the roof model, and it takes every cone triangle to a distinguished triangle by definition. It is therefore exact. By [F4], the cohomology functors are defined on $D (A)$ , and on a localized cone triangle their maps are the maps in the cone long exact sequence [F5]. Transporting this sequence by a triangle isomorphism preserves exactness, proving the assertion for every distinguished triangle.

F1F3F4F5F6algebra∎

Depends on

Used by

Canonical t structure on a derived category Definition
Bounded derived localizations embed fully faithfully Proposition
Total derived functors send distinguished triangles to distinguished triangles Proposition
Canonical truncations fit a distinguished triangle Theorem
The derived category is the verdier quotient by acyclic complexes Theorem

Dependency tree · two levels

23 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

13.5.5–13.5.6, including all TR1–TR4 proof paragraphs (standard reference, not scraped)