Finiteness conditions and relative singularity categories. (English) Zbl 1516.16004
Summary: We introduce the \(n\)-pure projective (resp., injective) dimension of complexes in \(n\)-pure derived categories, and give some criteria for computing these dimensions in terms of the \(n\)-pure projective (resp., injective) resolutions (resp., coresolutions) and \(n\)-pure derived functors. As a consequence, we get some equivalent characterizations for the finiteness of \(n\)-pure global dimension of rings. Finally, we study Verdier quotient of bounded \(n\)-pure derived category modulo the bounded homotopy category of \(n\)-pure projective modules, which is called an \(n\)-pure singularity category since it can reflect the finiteness of \(n\)-pure global dimension of rings.
MSC:
| 16E35 | Derived categories and associative algebras |
| 16E10 | Homological dimension in associative algebras |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
Keywords:
\(n\)-pure projective dimension; \(n\)-pure injective dimension; \(n\)-pure global dimension; \(n\)-pure derived category; \(n\)-pure singularity categoryReferences:
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