Derived Homomorphism Functors

For fixed B and variable A, Horn (A, Ib) is a contravariant A-functor from K(A to K(2lb) (see 1.5.3), which takes quasi-isomorphisms in K( ) to quasi-isomorphisms in K(2lb) ((2.3.8)(iv) ) and hence—after composition with the natural functor Q K(2lb) — D(2lb)—to isomorphisms in D(2lb). So by (1.5.1)—and the exercise preceding it—there results a A-functor D(A° D(2lb). Thus we obtain a functor of two variables 

For example, let L C K = K( ) be as above, with respective derived categories Dl C D, and consider the functor 

As in the exercise preceding (1.5.1), we can consider the opposite category K°P to be triangulated, with translation inverse to that in K, in such a way that the canonical contravariant functor K — K°p and its inverse, together with 0 = identity, are both A-functors. This being so, one checks then that Horn is a A-fiuictor (see (1.5.3)). 

Taking note of the following Lemma, we can proceed as above to derive a A-functor 

Lemma 2.4.5.1. If I is a q-injective complex in K(.4) then the functor Hom —, I) takes quasi-isomorphisms to quasi-isomorphisms. 

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