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\(\omega\)-Gorenstein modules. (English) Zbl 1153.16009
Auslander (1966/1967) introduced the notion of the Gorenstein dimension of a finitely generated module over a Cohen-Macaulay (commutative and Noetherian) ring. In the paper the author considers a ring with a generalized tilting module \(\omega\) in the sense of Wakamatsu (1988) and over such a ring defines what he calls an \(\omega\)-Gorenstein module. These would correspond to Auslander’s modules with Gorenstein dimension \(0\), or what are sometimes called Gorenstein projective modules. He then generalizes many of the basic results about Gorenstein projective modules to his situation. When \(\omega\) is moreover tilting and cotilting he introduces several classes of modules associated with \(\omega\) and exhibits the relationship between them. Finally he proves the existence of certain approximations connected with some of these classes.
This article is another instance of the convergence of certain ideas from commutative ring theory (e.g. dualizing modules and complexes) and from noncommutative ring theory (e.g. the various tilting and cotilting notions).

16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16G10 Representations of associative Artinian rings
16D90 Module categories in associative algebras
16E10 Homological dimension in associative algebras
18G10 Resolutions; derived functors (category-theoretic aspects)
18G20 Homological dimension (category-theoretic aspects)
Full Text: DOI
[1] DOI: 10.1006/jabr.1999.8154 · Zbl 0949.16012
[2] DOI: 10.2140/pjm.2003.212.1 · Zbl 1056.16001
[3] Auslander M., Anneaux de Gorenstein, et torsion en algèbre commutative (1966)
[4] Auslander M., Mem. AMS. pp 94– (1969)
[5] DOI: 10.1016/0001-8708(91)90037-8 · Zbl 0774.16006
[6] Auslander M., Progress in Math. 95 pp 221– (1991)
[7] DOI: 10.1007/BF02572634 · Zbl 0845.16005
[8] DOI: 10.1080/00927879508825254 · Zbl 0823.16003
[9] Enochs E., Tsukuba J. Math. 19 pp 1– (1995)
[10] DOI: 10.1006/jabr.1996.0121 · Zbl 0847.16003
[11] Enochs E., Tsukuba J. Math 20 pp 487– (1996)
[12] DOI: 10.1007/BF02872195 · Zbl 1009.16008
[13] DOI: 10.1007/BF01163359 · Zbl 0578.16015
[14] DOI: 10.1007/BF02571521 · Zbl 0725.16011
[15] DOI: 10.1016/0021-8693(88)90215-3 · Zbl 0646.16025
[16] DOI: 10.1016/0021-8693(90)90055-S · Zbl 0726.16009
[17] DOI: 10.1080/00927870500288085 · Zbl 1107.16012
[18] DOI: 10.1016/S0021-8693(03)00143-1 · Zbl 1043.16007
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