, For s ? S, Q S (s) is an isomorphism in C S

, Let C be another category and F : C ? C a functor that sends morphisms of S to isomorphisms in C . Then F factors uniquely through Q S

, A morphism f : X ? Y in C(C ) is said to be a quasi-isomorphism (qiso for short) if H n (f ) is an isomorphism for all n. The definition generalizes readily for morphisms in K(C)

K. S-=-{f-?-hom,

, (C )), we define the derived categories D b (C ) (resp. D + (C ), resp. D ? (C )). Observe that by proposition A

, (C ))) is equivalent to the full subcategory of D(C ) consisting of objects X such that H n (X) = 0 for |n| >> 0

, The composition of functors C ? K(C ) ? D(C ) is fully faithfull. Therefore, C is equivalent to the full subcategory of D(C ) consisting of objects such that H n (X) = 0 for n = 0

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