Locally Well Generated Homotopy Categories of Complexes

We show that the homotopy category of complexes $\mathbf{K}(\mathcal{B})$ over any finitely accessible additive category $\mathcal{B}$ is locally well generated. That is, any localizing subcategory $\mathcal{L}$ in $\mathbf{K}(\mathcal{B})$ which is generated by a set is well generated in the sense of Neeman. We also show that $\mathbf{K}(\mathcal{B})$ itself being well generated is equivalent to $\mathcal{B}$ being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring $R$ for $\mathcal{B}= \textrm{Mod-}R$.

2010 Mathematics Subject Classification: 18G35 (Primary) 18E30, 18E35, 16D90 (Secondary)

Keywords and Phrases: compactly and well generated triangulated categories, complexes, pure semisimplicity

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