×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Auslander, M.; Buchweitz, R., The homological theory of maximal Cohen-Macaulay approximations, Mem. Soc. Math. France, 38, 5-37 (1989) · Zbl 0697.13005
[2] Auslander, M.; Reiten, I., \(k\)-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra, 92, 1-27 (1994) · Zbl 0803.16016
[3] Borel, A., Algebraic \(D\)-Modules (1987), Academic Press · Zbl 0642.32001
[4] Bongartz, K., Tilted algebras, (Springer Lecture Notes in Math., vol. 903 (1981)), 26-38
[5] Brenner, S.; Butler, M., Generalization of Bernstein-Gelfand-Ponomarev reflection functors, (Springer Lecture Notes in Math., vol. 832 (1980)), 103-169
[6] Björk, J., Rings of Differential Operators (1979), North-Holland
[7] Cartan, H.; Eilenberg, S., Homological Algebra (1956), Princeton Univ. Press · Zbl 0075.24305
[8] Fossum, R.; Griffith, P.; Reiten, I., Trivial Extensions of Abelian Categories, Springer Lecture Notes in Math., vol. 456 (1975) · Zbl 0255.16014
[9] Miyachi, J., Duality for derived categories and cotilting bimodules, J. Algebra, 185, 583-603 (1996) · Zbl 0873.16011
[10] Miyashita, Y., Tilting modules of finite projective dimension, Math. Z., 193, 113-146 (1986) · Zbl 0578.16015
[11] H. Sato, Holonomic modules over Gorenstein rings, preprint, 1995; H. Sato, Holonomic modules over Gorenstein rings, preprint, 1995
[12] Tachikawa, H.; Wakamatsu, T., Tilting functors and stable equivalences for self-injective algebras, J. Algebra, 109, 138-165 (1987) · Zbl 0616.16012
[13] Wakamatsu, T., On modules with trivial self-extensions, J. Algebra, 114, 106-114 (1988) · Zbl 0646.16025
[14] Wakamatsu, T., Stable equivalences for self-injective algebras and a generalization of tilting modules, J. Algebra, 134, 298-325 (1990) · Zbl 0726.16009
[15] Wakamatsu, T., Tilting theory and self-injective algebras, (Finite Dimensional Algebras and Related Topics (1994), Kluwer), 361-390 · Zbl 0814.16009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.