Right-Derived Functors via Injective Resolutions

The basic example of a family (v x) in (2.2.6) arises when A has enough injectives, i.e., every object of A admits a monomorphism into an injective object. Then every complex X K (.A) admits a quasi-isomorphism iPx- X Ix into a bounded-below complex of injectives (see (1.8.2)) and by (2.3.4) and (2.3.2.1) below, this Ix is right-F-acyclic for every A-functor F K ( ) — E, whence F is right-derivable. 

Later on, however, it will become important for us to be able to deal with unbounded complexes and for this purpose the following more general injectivity notion is, via (2.3.5), essential. 

Definition 2.3.1. Let be an abelian category, and let J be a A-subcategory of K( ). A complex / G J is said to be q-injective in J (or J-q-injective) if for every diagram Y X I in ii with s a quasi-isomorphism, there exists 

1) take X = I and / = identity to see that if / is q-injective then the quasi-isomorphism s has a left inverse. Conversely, by (1.6.3) any 

Proof If any quasi-isomorphism I Y has a left inverse, then setting X = I in (2.2.5) we see at once that I is right-F-acyclic. Conversely, if I is right-F-acyclic for the identity functor J — J, then every quasi-isomorphism I — Y has a left inverse. Q.E.D. 

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