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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.

MSC:

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
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References:

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