Relative singularity categories. (English) Zbl 1417.18006
Summary: We study the properties of the relative derived category \(D_{\mathcal{C}}^b(\mathcal{A})\) of an abelian category \(\mathcal{A}\) relative to a full and additive subcategory \(\mathcal{C}\). In particular, when \(\mathcal{A} = A\)-mod for a finite-dimensional algebra \(A\) over a field and \(\mathcal{C}\) is a contravariantly finite subcategory of \(A\)-mod which is admissible and closed under direct summands, the \(\mathcal{C}\)-singularity category \(D_{\mathcal{C} - \mathrm{sg}}(\mathcal{A}) = D_{\mathcal{C}}^b(\mathcal{A}) / K^b(\mathcal{C})\) is studied. We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category \(\mathcal{G}(\mathcal{C})\) of \(\mathcal{C}\).
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16E35 | Derived categories and associative algebras |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
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