Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) is equivalent to $K^b(_AP)$ (bounded homotopy category of complexes made with left projective modules) ?
On the bounded derived category of a finite dimensional algebra with finite global dimension
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abstract-algebra
category-theory
homological-algebra
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1I'm a bit puzzled about what is missing: Fact 2) from [your other question](http://math.stackexchange.com/q/197370) shows that the inclusion $K^b({}_AP) \to D^b(A)$ is fully faithful and to show that it is essentially surjective (and hence an equivalence of categories) you need only prove that every bounded complex of f.g. modules is quasi-isomorphic to a bounded complex of **f.g.** projectives (I assume that's what you mean). Two possible proofs of that latter fact were mentioned in the comments there. Could you elaborate on what is unclear? – 2012-11-08