Tilting modules and Auslander's Gorenstein property.
Some parts of the theory of generalized tilting modules developed by \textit{T. Wakamatsu} for Artin algebras [in J. Algebra 114, No. 1, 106-114 (1988; Zbl 0646.16025), ibid. 134, No. 2, 298-325 (1990; Zbl 0726.16009) and in Finite dimensional algebras and related topics, NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 361-390 (1994; Zbl 0814.16009)] are generalized to arbitrary rings. Given a ring \(R\), a right \(R\)-module \(T_R\) is called tilting if it has a projective resolution consisting of finitely generated modules, \(\text{Ext}_R^n(T_R,T_R)=0\) for \(n\geq 1\) and there exists an exact sequence \(0\to R\to T_0\to T_1\to\cdots\) of right \(R\)-modules with \(T_i\in\text{add}(T_R)\) and the sequence remains exact after applying \(\Hom_R(-,T_R)\). Let \(S=\text{End}_R(T_R)\). Under the assumptions that \(S\) is left Noetherian, \(R\) is right Noetherian and the injective dimensions \(\text{id}({_ST})\) and \(\text{id}(T_R)\) are finite the author obtains some results on approximations of finitely generated \(R\)-modules. Under the same assumptions it is shown that \(\text{Ext}^l_R(-,T_R)\) induces a duality between the class of left finitely generated \(S\)-modules \(L\) with \(\text{Ext}^n_S(L,{_ST})=0\) for \(n\neq l\), \(n\geq 0\), and the class of the right finitely generated \(R\)-modules \(M\) with \(\text{Ext}^n_R(M,T_R)=0\) for \(n\neq l\), \(n\geq 0\). If moreover the \(S\)-\(R\)-bimodule \(_ST_R\) has Auslander's \(l\)-Gorenstein property with \(\text{id}({_ST})=\text{id}(T_R)=l<\infty\) then the modules in the above classes are Artinian.
scientific article; zbMATH DE number 2105785
scientific article