TheoremStatement: Literature-sourcedProof: AI-adaptedSession-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.

Left exactness, right exactness, and exactness are characterized by short exact sequences

Statement

Let $F : A \to B$ be a functor between abelian categories.

$F$ is left exact if and only if for every short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ , the sequence $0 \to F (A^{'}) \to F (A) \to F (A^{''})$ is exact.
$F$ is right exact if and only if for every short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ , the sequence $F (A^{'}) \to F (A) \to F (A^{''}) \to 0$ is exact.
$F$ is exact if and only if it carries every short exact sequence in $A$ to a short exact sequence in $B$ .

Facts & Assumptions

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

[L1]

Left exactness or right exactness already forces additivity (A left or right exact functor between abelian categories is automatically additive).

[L2]

A functor between additive categories is additive exactly when it preserves finite biproducts (A functor between additive categories is additive exactly when it preserves finite biproducts).

[L3]

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

[L4]

Abelian categories remain abelian after passing to the opposite (The opposite of an abelian category is abelian).

[L5]

In an abelian category every monomorphism is the kernel of its cokernel (Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel).

Proof

technique · direct

1.1

If $F$ is left exact, then [L1] makes it additive, and [L3] says that it preserves kernels. Therefore whenever $0 \to K \overset{k}{\to} A \overset{c}{\to} Q \to 0$ is short exact, the map $F (k)$ is a kernel of $F (c)$ , which is exactly the left-exact short-sequence criterion. The right-exact half is the dual statement applied in opposite categories using [L4].

L1L3L4

1.2

Conversely, assume $F$ carries every short exact sequence to one exact through the middle. Applying that to the two split short exact sequences $0 \to A \to A \oplus B \to B \to 0$ and $0 \to B \to A \oplus B \to A \to 0$ shows that $F (A \oplus B)$ is a biproduct of $F (A)$ and $F (B)$ , so $F$ preserves finite biproducts and is additive by [L2]. Now every kernel $k : K \to A$ fits into a short exact sequence $0 \to K \to A \to A / K \to 0$ by [L5], so the hypothesis makes $F (k)$ a kernel. Then [L3] gives left exactness. The right-exact converse is the dual argument in opposite categories using [L4].

L2L3L4L5

2.1

Clause 3 is exactly the conjunction of the first two clauses: a short exact sequence stays short exact precisely when the transformed sequence is both left exact and right exact.

step 1.1step 1.2∎

Depends on

Used by

A left exact functor preserves monomorphisms and a right exact functor preserves epimorphisms Theorem

Dependency tree · two levels

22 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(2)-(4) (standard reference, not scraped)
Gautam Tamme, Algebra II Lecture 10, §10.4 (standard reference, not scraped)