The Derived Category

The derived category D = D( ) is the category whose objects are the same as those of K, but in which each morphism A — B is the equivalence class f/s of a pair (s, /) 

In particular, with (s,/) as above and c the homotopy class of the identity map of C, we have 

Moreover [H, p. 33, Prop. 3.4] any morphism Q — Q2 of such functors extends uniquely to a morphism Q[ — Q. In other words, composition with Q gives, for any category E, an isomorphism of the functor category Hom(D, E) onto the full subcategory of Hom(K, E) whose objects are the functors K — E which transform quasi-isomorphisms in K into isomorphisms in E. 

One checks that the category D supports a unique additive structure such that the canonical functor Q K — D is additive and accordingly we will always regard D as an additive category. If the category E and the above functor Q K E are both additive, then so is Q.  

Remark 1.2.1. The homology functors B K — . A defined in (1.1) transform quasi-isomorphisms into isomorphisms, and hence may be regarded as functors on D. 

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Proposition (3. 3.4) Consider an exact sequence of finite locally-free S groups 0 —> G —> G —> G" —> 0. Then there is an exact triangle in the derived category, D(S) ... 

Now let KJ" be the full subcategory of K" " whose objects are the bounded-below I-complexes. Since the additive subcategory I C is closed under finite direct sums, one sees that Kj" is a A-subcategory of K. According to (1.7.2)°P, the derived category DJ" of KJ" can be identified with a A-subcategory of D , and the above family [Pg.34]

Corollary 2.3.2.2. The 3-q-injective complexes are the objects of a localizing subcategory I. Every quasi-isomorphism in I is an isomorphism, so the pair (I, identity) has the universal property of the derived category Dj ( 1.2), and therefore I = Dj can he identified with a A-subcategory of Dj. 

More explicitly (details in 3.4, 3.5), if / X F is a map of ringed spaces, then the derived categories D( x), D( Ay) have natural structures of symmetric monoidal closed categories, given by [Pg.83]

Scholium. Let S be the category of ringed spaces. For each object X S S, set X = Xsi< = D(X) the derived category of the category of Ox-modules), a closed -category with product unit Ox, and internal horn RT om. 

With the definitions of and rj in 3.2, and the fact that the direct image of a flasque sheaf is stiU flasque, it is a straightforward exercise to verify that the map 9cj F) is isomorphic to the derived category map given by the natural composition... 

The discussion on derived categories of categories of sheaves over diagrams of schemes are interpreted to the derived categories of the categories of (G, Gx)-modules. 

CompliAbShifY)) with respect to quasi-isomorphisms is called the derived category of chain complexes of sheaves on T and denoted by D(/4i5 Az (T)). 

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