Tilting modules and Auslander’s Gorenstein property. (English) Zbl 1076.16006

Some parts of the theory of generalized tilting modules developed by 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 \(\operatorname{Hom}_R(-,T_R)\).
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.


16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16D90 Module categories in associative algebras
16E10 Homological dimension in associative algebras
16E05 Syzygies, resolutions, complexes in associative algebras
Full Text: DOI


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