##  [Derived Functor](/derived-functor-1) 

 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.

 

 

 

 

 

 





## Principle

Principle

Derived functors convert failure of exactness into computable homological invariants: by resolving objects and applying the original functor, cohomology groups record obstructions and extension data that the underived functor misses.

 

 

 

 

 





## Demonstration

Demonstration

Example: Ext^i(-,-) are the right derived functors R^i Hom(−,−), and Tor_i(-,-) are the left derived functors L_i(− ⊗ −); these arise by applying Hom or tensor to injective or projective resolutions and taking cohomology.

 

 

 

 

## Misapplication

Misapplication

Attempting to compute derived functors without appropriate resolutions, using non-resolving classes when the category lacks enough projectives/injectives, or treating derived functors as objectwise equalities rather than cohomological constructions.

 

 

 

 

 





## Consequence

Consequence

Derived functors produce long exact sequences, spectral sequences, and obstruction classes that link short exact sequences in the source category to homological data in the target; they systematize cohomological computations and invariants.

 

 

 

 

## Reversal

Reversal

The reversal is the underived functor acting on projective or injective objects where higher derived functors vanish: in that restricted setting the derived construction collapses to the original functor and no new cohomology appears.

 

 

 

 

 





## Boundary

Boundary

Derived functors require abelian or derived settings and typically enough projectives or injectives (or model/derived enhancements); in non-abelian contexts one must use homotopical replacements (derived functor in homotopy category) rather than naive resolutions.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Derived functor notion competes with homotopy-invariant or total derived constructions: the classical R^i/L_i picture suffices in many abelian cases, but in homotopical or ∞-categorical contexts one prefers total derived functors or derived functors of the homotopy category for better invariance.

 

 

 

 

 





## Synthesis

Synthesis

A derived functor extends an exactness-limited functor to a graded family measuring its failure to be exact: through resolutions and cohomology it encodes extension, obstruction and higher-order information that is crucial to homological algebra.