TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Exactness is self-dual

Statement

A composable pair $A \overset{f}{\to} B \overset{g}{\to} C$ is exact at $B$ in an abelian category $A$ if and only if the opposite pair $C \overset{g^{o p}}{\to} B \overset{f^{o p}}{\to} A$ is exact at $B$ in $A^{o p}$ .

Facts & Assumptions

Given: A composable pair $A \overset{f}{\to} B \overset{g}{\to} C$ in an abelian category $A$ .

[L1]

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

[L2]

Exactness at $B$ means either $[im (f)] = [\ker (g)]$ or, equivalently, $[coker (f)] = [coim (g)]$ (Exactness at a node, Image and coimage in a category with kernels and cokernels).

Proof

technique · direct

1.1L1L2

By [L1], the opposite category $A^{o p}$ is again abelian, so the definition [L2] applies there. Passing to the opposite exchanges kernels with cokernels and images with coimages.

2.1L2step 1.1∎

Therefore the equality $[im (f)] = [\ker (g)]$ in $A$ is exactly the equality $[im (g^{o p})] = [\ker (f^{o p})]$ in $A^{o p}$ . By [L2], those are the two exactness assertions.

Exactness is self-dual

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources