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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In a triangulated or dg-category, this is not the usual notion of "generators." One definition is that there is no smaller triangulated subcategory containing the objects. In this sense, the indecomposable projectives do generate the homotopy category over all projectives. This is also true for the projective resolutions of the simples; since the algebra is finite dimensional, there's a finite iterated cone of the resolutions of the simples which is quasi-isomorphic to any indecomposable projective, even if the algebra doesn't have finite global dimension. |
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