A diagram-chasing lemma stating that in a commutative diagram of two exact rows with five objects each, if four of the five vertical maps are isomorphisms (or under the standard mono/epi hypotheses) then the remaining middle vertical map is also an isomorphism.

Principle

Exactness in the rows and commutativity force kernels and cokernels to match; when enough adjacent vertical maps are isomorphisms the induced maps on kernels and cokernels are isomorphisms and hence the central map must be an isomorphism by the five-term diagram chase.

Demonstration

In the category of abelian groups consider a commutative diagram A1→A2→A3→A4→A5 with exact rows and vertical maps f_i. If f1,f2,f4,f5 are isomorphisms then, by chasing kernels and cokernels across the rows, f3 is an isomorphism. A common variant assumes f1 and f5 are isomorphisms, f2 is surjective and f4 is injective to conclude f3 is an isomorphism.

Misapplication

Using the Five Lemma where rows are not exact, the diagram does not commute, or in categories lacking kernels/cokernels can lead to incorrect conclusions; treating the lemma as purely formal without checking mono/epi hypotheses in non-abelian contexts is a common error.

Consequence

Allows one to transfer isomorphism information across a long exact diagram and is a key tool for proving that induced maps between derived objects are isomorphisms when most surrounding maps are known to be isomorphisms.

Reversal

The converse—inferring isomorphisms of adjacent maps from an isomorphism of the center—does not follow without additional information; failure of exactness or one missing isomorphism can break the chain.

Boundary

Requires two commutative rows that are exact and a category with well-defined kernels and cokernels (typically abelian categories). It does not apply verbatim in purely nonabelian diagram settings without reinterpretation.

Semantic Tension

Close relatives include the Four Lemma and Nine Lemma; the tension is between variants that require explicit mono/epi conditions and the simplified statement that 'four isomorphisms imply the fifth', which can hide necessary side hypotheses in some texts.

Synthesis

The Five Lemma is a diagram-chasing criterion in exact sequences: when a commutative diagram has exact rows and sufficient surrounding maps are isomorphisms (or satisfy the mono/epi variants), the central map is forced to be an isomorphism, enabling transfer of isomorphism data through the diagram.