##  [Minimal Free Resolution](/index.php/minimal-free-resolution) 

  ##  [Minimal Free Resolution](https://algebra.quantumdictionary.io/minimal-free-resolution-0) 

  

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

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A free resolution of a module M over a ring R is minimal if no free direct summand can be removed at any degree; equivalently, over a local ring (R,m) the differentials in a minimal free resolution have images contained in m times the next module, so the ranks of the free modules (Betti numbers) are as small as possible and are invariant.