##  [Five Lemma](/five-lemma-0) 

 Definition

A diagram-chasing result in homological algebra: given a commutative diagram of two exact rows A1→A2→A3→A4→A5 and B1→B2→B3→B4→B5 with vertical maps f_i: A_i→B_i, if f1, f2, f4, f5 are isomorphisms and the middle maps satisfy the appropriate surjectivity/injectivity hypotheses, then f3 is an isomorphism.

 

 

 

 

 

 





## Principle

Principle

Exactness plus commutativity allow local information about four surrounding morphisms to force invertibility of the central morphism by chasing images and kernels across the diagram.

 

 

 

 

 





## Demonstration

Demonstration

In algebraic topology, compare the long exact sequences of a pair (X,A) and (Y,B) induced by a map (X,A)→(Y,B). If the outer four induced maps on homology groups are isomorphisms and the required surjectivity/injectivity holds at adjacent degrees, the Five Lemma yields that the induced map in the middle degree is an isomorphism.

 

 

 

 

## Misapplication

Misapplication

Applying the lemma when rows are not exact, when the diagram fails to commute, or when the assumed maps are only bijections on underlying sets (not group/module homomorphisms) — any of these invalidates the kernel/image chase and can lead to false conclusions.

 

 

 

 

 





## Consequence

Consequence

One can promote local isomorphisms to a global isomorphism in the middle of an exact sequence, simplifying proofs that certain derived or homology objects coincide under a map.

 

 

 

 

## Reversal

Reversal

The converse is false in general: having the middle map an isomorphism does not imply the outer four maps are isomorphisms. A reversal that does hold in related situations is the Four Lemma, which gives injectivity or surjectivity of one map from injectivity/surjectivity of others but not full isomorphism.

 

 

 

 

 





## Boundary

Boundary

Requires abelian-category-style exactness (or at least exact sequences in groups/modules), strict commutativity of the diagram, and the specific surjectivity/injectivity hypotheses; it does not apply verbatim in non-abelian settings without extra hypotheses.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Closely connected to the Snake Lemma and the Nine Lemma; the Five Lemma targets isomorphism in the middle, while the Snake Lemma produces connecting homomorphisms and exact sequences from kernels and cokernels — they overlap in technique but answer different structural questions.

 

 

 

 

 





## Synthesis

Synthesis

The Five Lemma is a diagram-chasing instrument: under exactness and commutativity, the pattern of isomorphisms and one-sided maps around a central morphism forces that central morphism to be an isomorphism, letting one deduce equivalence of middle objects from equivalences at the boundaries.