TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ 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 additive functor is exact exactly when it preserves kernels and cokernels

Statement

Let $F : A \to B$ be an additive functor between abelian categories. Then $F$ is exact if and only if it preserves kernels and cokernels.

Facts & Assumptions

Given: An additive functor $F : A \to B$ between abelian categories.

[L1]

An additive functor between additive categories is left exact exactly when it preserves kernels (An additive functor is left exact exactly when it preserves kernels).

[L2]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

[L3]

Exact means additive, left exact, and right exact (Exact functor between abelian categories).

Proof

technique · direct

1.1

If $F$ is exact, then [L3] says it is left exact and right exact. The left-exact half and [L1] show that $F$ preserves kernels. Applying the same argument to $F^{o p} : A^{o p} \to B^{o p}$ and using [L2] shows that $F$ preserves cokernels as well.

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2.1

Conversely, assume $F$ preserves kernels and cokernels. By [L1], kernel preservation makes $F$ left exact. Passing to opposites and using [L2], cokernel preservation makes $F$ right exact. Since $F$ is additive by hypothesis, [L3] shows that $F$ is exact.

L1L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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, Lemma 12.7.2 (standard reference, not scraped)