PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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 $(F (d^{D}), F (f), - F (d^{C})) .$

L1L2L3givenalgebra

2.1

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.

L3L4step 1.1algebra∎

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)