##  [Five Lemma](/five-lemma-1) 

 Definition

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

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

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

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

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

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

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

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

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.