On contravariant finiteness of subcategories of modules of projective dimension \(\leq i\). (English) Zbl 0856.16006
Summary: Let \(\Lambda\) be an Artin algebra. This paper presents a sufficient condition for the subcategory \({\mathcal P}^i(\Lambda)\) of \(\text{mod }\Lambda\) to be contravariantly finite in \(\text{mod }\Lambda\), where \({\mathcal P}^i(\Lambda)\) is the subcategory of \(\text{mod }\Lambda\) consisting of \(\Lambda\)-modules of projective dimension less than or equal to \(i\). As an application of this condition it is shown that \({\mathcal P}^i(\Lambda)\) is contravariantly finite in \(\text{mod }\Lambda\) for each \(i\) when \(\Lambda\) is stably equivalent to a hereditary algebra.
MSC:
| 16G10 | Representations of associative Artinian rings |
| 16E10 | Homological dimension in associative algebras |
| 16D90 | Module categories in associative algebras |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 18G20 | Homological dimension (category-theoretic aspects) |
Keywords:
contravariantly finite subcategories; Artin algebras; projective dimension; stably equivalent algebras; hereditary algebrasReferences:
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