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

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Total derived functors send distinguished triangles to distinguished triangles

Statement

The bounded total derived functors $L F$ and $R F$ are exact functors of triangulated categories, with shift comparisons transported through their model equivalences. If $F$ is exact as a functor of abelian categories, its termwise functor already descends to the derived category and the respective derived augmentation or coaugmentation is an isomorphism.

Facts & Assumptions

Given: The bounded total derived functors $L F$ and $R F$ are exact functors of triangulated categories, with shift comparisons transported through their model equivalences. If $F$ is exact as a functor of abelian categories, its termwise functor already descends to the derived category and the respective derived augmentation or coaugmentation is an isomorphism.

[F1]

The left derived functor is the composite through the bounded projective model and $K (F)$ (Existence of the bounded above left total derived functor).

[F2]

The right derived functor is the composite through the bounded injective model and $K (F)$ (Existence of the bounded below right total derived functor).

[F3]

The localization is exact for the derived cone triangulation (The derived category inherits a triangulated structure).

Proof

1.1

In either model, shifts and cones stay in the bounded projective or injective subcategory. Additive $F$ preserves their finite biproduct formulas and signs, hence their cone triangles and shift identifications. The model equivalence and the target localization are exact, so their composite sends every distinguished triangle to a distinguished triangle. This includes split triangles and zero objects.

F1F2F3

2.1

If $F$ is exact, it preserves the kernel-image sequences defining cohomology, giving $H^{n} (F X) ≅ F (H^{n} X)$ . Consequently it preserves quasi-isomorphisms, so termwise $F$ descends by localization. In particular each $F (p_{X})$ or $F (j_{X})$ is a quasi-isomorphism, making the derived comparison invertible. This verifies the asserted comparison without an extra exactness hypothesis in the existence theorem.

F1F2step 1.1algebra∎

Depends on

Used by

Classical derived functors are the cohomology objects of the total derived functor Proposition

Dependency tree · two levels

14 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

10.5.1–10.5.8, pp. 391–393; restrict to supplied replacement data (standard reference, not scraped)