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

  ##  [Five Lemma](https://mathlogic.quantumdictionary.io/five-lemma-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

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.

 

 

 

 

 





 

 



 ##  [Five Lemma](https://algebra.quantumdictionary.io/five-lemma-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

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.