By Fuchs J., et al. (eds.)

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We define the localizing functor from the homotopy category to the derived category ????A : ????∗ (A) → ????∗ (A) in the following way: it is the identity on the set of the objects and, for any ???? morphism in ????∗ (A), we define ????A (????) to be ???? d dd dd ???? dd ???? c???? . ???? Observe that ????A (????) is an isomorphism in ????∗ (A) for every ???? quasi-isomorphism in ????∗ (A). In fact, the idea of derived category is to identify an object of an Abelian category A with all its resolutions; to do this we consider a category, ????(A), whose objects are Derived categories and derived functors | 45 all the complexes of objects in A and the morphisms are defined in such way that two quasi-isomorphic complexes are isomorphic in ????(A).

A triangle in C is a sextuple (????, ????, ????, ????, ????, ????) of objects ????, ????, ???? in C and morphisms ???? : ???? → ????, ???? : ???? → ????, ???? : ???? → ????(????). It is often denoted ???? ???? ???? ???? ????→ ???? ????→ ???? ????→ ????(????). A morphism of triangles is a commutative diagram ???? ???? ???????? ???? ???????? G???? ???? G ???????? ???? ???????? G???? ℎ G ???????? ???? ???????? G ????(????) ????(????) G ????(???????? ) . Definition. We say that an additive category C equipped with an additive automorphism ???? and with a family of triangles, called distinguished triangles, is a triangulated category if the following axioms hold: (1) Every triangle isomorphic to a distinguished triangle is a distinguished triangle.

Another way is the following: if the rank of ???? is ????, we define – ???????? (????) to be the Poincaré dual of the zero locus of a general ????∞ section ???? of ????; – ????????−1 (????) to be the Poincaré dual of the zero locus of ????1 ∧ ????2 where ????1 , ????2 are general ????∞ section of ????; – more generally, ???????? (????) to be the Poincaré dual of the zero locus of ????1 ∧ ⋅ ⋅ ⋅ ∧ ????????−????+1 , where ????1 , . . , ????????−????+1 are general ????∞ sections of ????. Finally we want to mention the definition of Chern character. Let ????(????)(????) = ∏(1 + ???????? ????) ???? be the “formal factorization” of the Chern polynomial ????(????)(????) of a complex bundle ????; the ???????? are called the Chern roots.