TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27‡ 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 left exact exactly when it preserves kernels

Statement

Let $F : C \to D$ be an additive functor between additive categories. Then $F$ is left exact if and only if it preserves kernels.

Facts & Assumptions

Given: An additive functor $F : C \to D$ .

[L1]

Additive categories are preadditive with finite biproducts (Additive category).

[L2]

In a preadditive category, equalizers are kernels of differences (In a preadditive category, the equalizer of a parallel pair is the kernel of their difference).

[L3]

An additive functor preserves finite biproducts, hence finite products (An additive functor preserves finite biproducts).

[L4]

A functor preserves finite limits exactly when it preserves finite products and equalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).

Proof

technique · direct

1.1

If $F$ is left exact, then it preserves all finite limits by definition, so in particular it preserves kernels because a kernel is a finite limit.

1.2

Conversely, assume $F$ preserves kernels. By [L1] the source and target are preadditive, and by [L3] the functor preserves finite products. If $e : E \to A$ equalizes $f, g : A ⇉ B$ , then [L2] identifies $e$ as a kernel of $f - g$ . Since $F$ is additive, $F (f - g) = F f - F g$ , so the image of $e$ is a kernel of $F f - F g$ , hence again an equalizer of $F f$ and $F g$ by [L2]. Therefore $F$ preserves equalizers.

L1L2L3

2.1

Now [L4] applied to step 1.2 shows that $F$ preserves all finite limits. So $F$ is left exact.

L4step 1.2

3.1

Thus left exactness and kernel preservation are equivalent for additive functors.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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)