DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Exact functor between abelian categories

Definition

Let $A$ and $B$ be abelian categories. A functor $F : A \to B$ is exact when it is additive (Additive functor) and both left exact and right exact (Left exact and right exact functors).

Thus exactness is a two-sided preservation condition: $F$ preserves the finite limits and finite colimits that exist in its source, and it preserves the additive structure carried by the hom-sets.

Depends on

Used by

An exact functor need not be faithful Counterexample
Localization of modules gives an exact functor between module categories Example
An additive functor is exact exactly when it preserves kernels and cokernels Theorem
An equivalence between abelian categories is exact Theorem

Dependency tree · two levels

6 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 (standard reference, not scraped)
Gautam Tamme, Algebra II Lecture 10, §10.4 (standard reference, not scraped)