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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).

MSC:
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)
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