Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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  • 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.
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An additive functor is exact exactly when it preserves kernels and cokernels

Statement

Let F:A→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→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
1.1L1L2L3

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 Fop:Aop→Bop and using [L2] shows that F preserves cokernels as well.

2.1L1L2L3∎

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.

Depends on

Dependency tree · 2 levels

Sources