Contravariantly finite subcategories closed under predecessors.
In this paper the following two main results are proved: Theorem A. Let \(A\) be an Artin algebra and \(\mathcal C\) be a full skeleton subcategory of the category of all indecomposable finitely generated right \(A\)-modules, closed under predecessors. Let \(E\) denote the direct sum of ext-injective modules in \(\mathcal C\). Then the following conditions are equivalent: 1) \(\mathrm{add\,}\mathcal C\) is contravariantly finite. 2) \(\mathcal C^\perp\) is contravariantly finite. 3) \(E\) is a cotilting module over the support algebra of \(\mathcal C\). 4) \(\mathrm{add\,}\mathcal C\) consists of all \(M\) such that \(\Hom_A(M,E)\neq 0\). 5) Any morphism \(f\colon L\to M\) where \(L\in\mathcal C\) and \(M\) is indecomposable not in \(\mathcal C\) factors through \(\mathrm{add\,}E\). Theorem B. Let \(A\) and \(\mathcal C\) be as in Theorem A and assume that \(\mathrm{pgd\,}\mathcal C<\infty\). Let \(F\) be the direct sum of all indecomposable projective \(A\)-modules not lying in \(\mathcal C\). Then the following conditions are equivalent: (i) \(\mathrm{add\,}\mathcal C\) is contravariantly finite. (ii) \(E\) is a tilting module over the support algebra of \(\mathcal C\). (iii) \(E\oplus F\) is a tilting \(A\)-module. Moreover, in this case \(\mathcal C^\perp=E^\perp\) and \(\mathcal C\) consists of all predecessors of \(E\).
scientific article; zbMATH DE number 5611752
scientific article