##  [Derived Functor](/derived-functor) 

  ##  [Derived Functor](https://mathlogic.quantumdictionary.io/derived-functor-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 construction that assigns to a (typically non-exact) additive functor F between abelian categories a sequence of functors R^iF or L_iF (right or left derived functors) measuring the failure of exactness, obtained by applying F to an injective or projective resolution and taking cohomology or homology, or equivalently by passing to the total derived functor in the derived category.

 

 

 

 

 





 

 



 ##  [Derived Functor](https://algebra.quantumdictionary.io/derived-functor-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 construction that associates to a (left or right) exact functor between abelian (or suitably derived) categories a sequence of higher functors (R^iF or L_iF) obtained by applying F to resolutions (injective or projective) and taking cohomology; it measures how far F fails to be exact.