TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Sharp five lemma in an abelian category

Statement

In a commutative diagram with exact rows

the following hold:

if $f_{1}$ is epic and $f_{2}, f_{4}$ are monic, then $f_{3}$ is monic;
if $f_{2}, f_{4}$ are epic and $f_{5}$ is monic, then $f_{3}$ is epic.

Facts & Assumptions

Given: The commutative exact-row diagram in the statement.

[L1]

The four lemma gives the monic and epic conclusions on any four-column window with exact rows (Four lemma in an abelian category).

Proof

technique · direct

1.1L1assume-hyp

For the monic clause, apply [L1] to the left four columns $A_{1} \to A_{2} \to A_{3} \to A_{4},$ $B_{1} \to B_{2} \to B_{3} \to B_{4} .$ The hypotheses there are exactly that $f_{1}$ is epic and that $f_{2}$ and $f_{4}$ are monic, so the four lemma gives that $f_{3}$ is monic.

1.2L1assume-hyp

For the epic clause, apply [L1] to the right four columns $A_{2} \to A_{3} \to A_{4} \to A_{5},$ $B_{2} \to B_{3} \to B_{4} \to B_{5} .$ The hypotheses there are exactly that $f_{2}$ and $f_{4}$ are epic and that $f_{5}$ is monic, so the four lemma gives that $f_{3}$ is epic.

2.1step 1.1step 1.2∎

Therefore the sharp five lemma holds in both halves.

Depends on

Four lemma in an abelian category

Sharp five lemma in an abelian category

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources