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Left triangulated categories arising from contravariantly finite subcategories. (English) Zbl 0811.18005

Assume that \(A\) is an Artin algebra. Given a full subcategory \(\mathcal A\) of \(\text{mod}(A)\) we denote by \(\underline{\text{mod}}_{\mathcal A}(A)\) the factor category of \(\text{mod}(A)\) modulo the ideal consisting of all \(A\)-homomorphisms having a factorization through homomorphisms in the category \({\mathcal A}\). It is proved in the paper that any contravariantly finite (resp. covariantly finite) subcategory \(\mathcal A\) of \(\text{mod}(A)\) induces a left (resp. right) triangulated structure on the category \(\underline{\text{mod}}_{\mathcal A}(A)\). The following special case is studied in the paper. Assume that \(F\) is an additive subfunctor of \(\text{Ext}^ 1_ A(-,-)\). Let \({\mathcal P}(F)\) and \({\mathcal I}(F)\) be the full subcategory of \(\text{mod}(A)\) formed by the \(F\)-projective and the \(F\)-injective objects, respectively. One of the main results of the paper asserts that if there are enough \(F\)-projectives and enough \(F\)- injectives then the category \({\mathcal P}(F)\) induces on \(\underline{\text{mod}}_{\mathcal P(F)}(A)\) a left triangulated structure, and the category \({\mathcal I}(F)\) induces on \(\underline{\text{mod}}_{\mathcal I(F)}(A)\) a right triangulated structure.
Reviewer: D.Simson (Toruń)

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
16G10 Representations of associative Artinian rings
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
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