Assume that A is abelian. Show that a short exact sequence 0 ? A ? B ? C ? 0 of complexes does give.

Assume that A is abelian. Show that a short exact sequence 0 ? A ? B ? C ? 0 of complexes does give rise to a distinguished triangle in K(A) if the sequence splits termwise; i.e. for all i, the exact sequence 0 ? A i ? B i ? C i ? 0 splits. In this exercise, you will show that the natural functor is an equivalence of triangulated categories. 1. Observe that it is fully faithful. 2. Use induction along the Bruhat order to show that, for every w ? W, the complex consisting of Bw in cohomological degree 0 lies in the essential image. 3. Look up the definition of a triangulated functor (also

Assume that A is abelian. Show that a short exact sequence 0 ? A ? B ? C ? 0 of complexes does give rise to a distinguished triangle in K(A) if the sequence splits termwise; i.e. for all i, the exact sequence 0 ? A i ? B i ? C i ? 0 splits. In this exercise, you will show that the natural functor is an equivalence of triangulated categories. 1. Observe that it is fully faithful. 2. Use induction along the Bruhat order to show that, for every w ? W, the complex consisting of Bw in cohomological degree 0 lies in the essential image. 3. Look up the definition of a triangulated functor (also called an exact functor) and of an equivalence of triangulated categories. Any functor between additive categories induces a triangulated functor of the homotopy categories. Deduce that the essential image of the functor K b BSBim ? K b SBim is closed under taking cones. 4. Deduce that the functor above is an equivalence of triangulated categories.

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