Derived Functors

If in an abelian category sufficiently many injective or projective objects are available (i.e. any object can be imbeddedin an injective object, respectively is quotient of a projective object), the derived functors Ext (-,-) of Hom(-,-) can be defined, and they are isomorphic (in two ways, cf. GP, 3 5, proposition 10) with E (-,-). 

Such a pair (RF, C) (respectively (LF, )) is called a right-derived (respectively left-derived) functor of F. 

Example 2.1.3. If F J —> E transforms quasi-isomorphisms into isomorphisms then F = Fog for a unique F Dj —> E and (F, identity) is both a right-derived and a left-derived functor of F. 

A similar statement holds for left-derived functors when there exists a family as in (2.2). 

This checking is left to the reader, as is the proof for left-derived functors. 

Corollary 2.2.7. Let A, A be abelian categories, let 3 C K(.4), J C K(.A ) be A-subcategories with canonical Junctors Q 3 —> Dj, Q 3 —> Dj/ to their respective derived categories, and let F 3 —> J and G J —> E be A-Junctors. Assume that G has a right-derived functor RG and that every complex X J admits a quasi-isomorphism into a right- Q F)-acydic complex Ax such that F Ax) is right-G-acyclic. Then Q F and GF have right-derived functors, denoted RF and R GF), and there is a unique A-Junctorial isomorphism... 

We leave the corresponding statements for left-F-acyclic complexes and left-derived functors to the reader. 

Corollary 2.3.2.3. Suppose that there exists a family of q-injective resolutions ipx Ix X J), i.e., for each X, is a quasi-isomorphism and Ix is 3-q-injective. Then any A-functor F J —> E has a right-derived functor (RF, C) with... 

The variables A,B are treated quite differently in the above definition of RHom. But there is a more synometric characterization of this derived functor, analogous to the one in (2.1.1). This is given in (2.4.4), after the necessary preparation. 

---