Truncation Functors

One checks that r (respectively t ) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D D, see 1.2. In fact if D n (resp. D ) is the full subcategory of D whose objects are the complexes A such that H A ) = 0 for m n (resp. m n) then we have additive functors 

Proposition 1.10.1. The preceding maps i, j induce functorial isomorphisms 

Proof Bijectivity of (1.10.1.1) means that any map ip B with B G D factors uniquely via 

To prove that (1.10.1.1) is also injective, we assume that or v5 = 0 and deduce that = 0. As in 1.2, the assumption means that there is a commutative diagram in K(.4) 

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The truncation functors of 1.10 and the way-out lemmas of 1.11 supply repeatedly useful techniques for working with derived categories and functors. These two sections may well be skipped until needed. 

Indeed, one can apply any such (—)cts for A = Ax to the just-constructed truncated Godement resolution, to produce a resolution with all the desired properties. (For this, condition (4.1.5)(iii) is needed only when D = identity functor.)... 

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