PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

An exact functor carries mapping-cone sequences to mapping-cone sequences

Statement

Let $F : A \to B$ be an exact functor between abelian categories. For every chain map $f : C_{∙} \to D_{∙}$ , there is a natural chain isomorphism $F (Cone (f)) ≅ Cone (F (f)),$ compatible with the canonical inclusion and projection maps, so $F$ carries the mapping-cone sequence of $f$ to the mapping-cone sequence of $F (f)$ .

Facts & Assumptions

Given: An exact functor $F : A \to B$ and a chain map $f : C_{∙} \to D_{∙}$ .

[L1]

Exact functors are additive (A left or right exact functor between abelian categories is automatically additive).

[L2]

Additive functors apply degreewise to complexes and chain maps (An additive functor applies degreewise to complexes and chain maps).

[L3]

The cone differential is $d (y, x) = (d^{D} (y) + f (x), - d^{C} (x))$ (The mapping cone of a chain map).

[L4]

The canonical inclusion and projection are the obvious coordinate maps (The canonical inclusion and projection for a mapping cone).

Proof

technique · direct

1.1L1L2L3givenalgebra

By [L1] and [L2], $F$ preserves direct sums and the scalar $- 1$ . Therefore $F (Cone (f))_{n}$ is $F (D_{n} \oplus C_{n - 1}) ≅ F (D_{n}) \oplus F (C_{n - 1}),$ and under this identification the differential is $(F (d^{D}), F (f), - F (d^{C})) .$

2.1L3L4step 1.1algebra∎

The displayed differential is exactly the cone differential of $F (f)$ from [L3], so the degreewise biproduct identification is a chain isomorphism $F (Cone (f)) ≅ Cone (F (f))$ . The coordinate maps in [L4] are preserved under the same identification, so the full cone sequence is transported naturally.

Depends on

Used by

Nothing in the library uses this result yet.

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

The Stacks Project, Section 12.7: Additive functors (standard reference, not scraped)
The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)