# Generators of the derived category

For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective modules, in the sense that every object is isomorphic to a direct sum of them. Is there a similar statement for the bounded-above derived category $\mathcal{D}^-_R=\mathcal{K}^-\left(\mathcal{P}_R\right)$? I want to say that $\mathcal{D}^-_R$ is generated by projective resolutions of the irreducible modules. Is that right?

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No, it's wrong. If that were so then we would expect to be able to write the indecomposable projectiles as a direct sum of the projective resolutions of the simple modules so that taking 0th homology we'd get that they were decomposable. –  Eitan Chatav Feb 29 '12 at 2:07