Chain Complexes and Stable Categories

Under suitable assumptions, we extend the inclusion of an additive subcategory X ⊂ A ( = stable category of an exact category with enough injectives) to an S-functor [15] H0]X → A , where H0]X is the homotopy category of chain complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ’Morita theory for Derived categories‘ [17] and a sharpening of a theorem of Happel [12, 10.10] on the ’module-theoretic description‘ of the derived category of a finite-dimensional algebra. 1. Notation and Results 1.1 Let B be an additive category. We denote by • CB the category of chain complexes K = (. . .→ Kn+1 d n+1 → Kn dn → Kn−1 → . . .) , Kn ∈ B , n ∈ Z , • HB the homotopy catgory CB/N , where N is the ideal of morphisms homotopic to 0, endowed with the suspension functor S : HB → HB , K 7→ SK , (SK)n = Kn−1 , d SK = −d and with the triangles X → Y → Z → SX furnished by the pointwise split exact sequences of CB (cf. [19]), • C+B, CbB, C0] B and C b 0] B (resp. H+B, HbB, H0] B and H b 0] B) the full subcategories of CB (resp. HB) consisting of the right bounded (Kn = 0 ∀n 0), the right and left bounded (Kn = 0 ∀n 0 and ∀n 0), the positive (Kn = 0 ∀n < 0) and the bounded positive (Kn = 0 ∀n < 0 and ∀n 0) chain complexes, respectively. We denote the homotopy class of a morphism of complexes f by f . We identify B with the full subcategory of HB consisting of the complexes K with Kn = 0 ∀n 6= 0.