(derived functors)
Let \(\mca\) be an abelian category with enough injectives and \(F: \mca \to \mcb\) an additive covariant left exact functor. The right derived functors \(R^iF: \mca \to \mcb\) are \[ R^iF(A) = h^i(F(I^\bullet)) , \] where \(A \to I^\bullet\) is any injective resolution of \(A\).
If instead \(F\) is right exact and \(\mca\) has enough projectives, the left derived functors are \[ L^iF(A) = h^i(F(P^\bullet)) , \] where \(P^\bullet \to A\) is any projective resolution.
For contravariant \(F\), regard \(F\) as a covariant functor on \(\opcat{\mca}\).
\(R^0F \cong F\) by left exactness, and \(R^iF = 0\) for all \(i>0\) if and only if \(F\) is exact. \(R^iF(A)\) is independent of the injective resolution up to canonical isomorphism: any two injective resolutions are homotopy equivalent, and homotopic maps agree on cohomology (D-DEFCPLX).
Any resolution of \(A\) by \(F\)-acyclic objects computes \(R^iF(A)\) (D-DEFACYC); so flasque resolutions compute sheaf cohomology and free resolutions compute \(\Tor\).
Dependencies
D-DEFEXACTExactness, and left, right, and exact functorsusesD-DEFINJOBInjective objects, injective resolutions, and enough injectivesusesD-DEFPROJOProjective objects and projective modulesuses
Backlinks
D-DEFDELTA$\delta$-functors, universality, and effaceabilityusesD-DEFACYCAcyclic objects, and computing derived functors without injectivesusesD-DEFEXTThe $\Ext$ functors, in either variableusesD-DEFTORThe $\Tor$ functors, and flatness as $\Tor$-vanishinguses