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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 [Pg.37]

Proposition 1.10.1. The preceding maps i, j induce functorial isomorphisms [Pg.37]

Proof Bijectivity of (1.10.1.1) means that any map ip B with B G D factors uniquely via [Pg.37]

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) [Pg.38]


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. [Pg.12]

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.)... [Pg.164]


See other pages where Truncation Functors is mentioned: [Pg.36]    [Pg.37]    [Pg.96]    [Pg.36]    [Pg.37]    [Pg.96]    [Pg.165]   


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