Bounded Functors Way-Out Lemma

Many of the main results in subsequent chapters will be to the effect that some natural map or other is a functorial isomorphism. So well need isomorphism criteria. In (1.11.3) we review some commonly used ones ( Lemma on way-out functors, [H, p. 68, Prop. 7.1]). 

For a subcategory E of D( ), E (resp. E ) will denote the full subcategory of E whose objects are those complexes A such that = 0 

The upper dimension dim+ and lower dimension dim of these functors are  

The functor F is hounded above (resp. bounded below) if dim+F oo (resp. dim F oo) and similarly for F. F (resp. F ) is bounded if it is both 

Here is a typical proof we deal with dim F when E = Dj(A). 

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