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Dimension estimates for representable equivalences of module categories. (English) Zbl 0884.16005

The author investigates various generalizations of the classical theorems for tilting theory of Artin algebras. His generalizations go in two directions. He investigates what happens when one drops the restriction of the algebra being artinian, and also generalizes the notion of tilting modules. He investigates the notion of *-modules, which was introduced by Menini and Orsatti, as a generalization of the notion of tilting modules, introduced by Brenner-Butler, and the notion of quasi-progenerators, introduced by Fuller.
The definition of a *-module is the following: Let \(P\in\text{Mod-}R\), \(S=\text{End}_R(P)\), \(Q\) an injective cogenerator for \(\text{Mod-}R\) and \(P^*=\operatorname{Hom}_R(P,Q)\). A module \(P\) is a *-module if \(\operatorname{Hom}_R(P,-)\) and \(-\otimes_SP\) induce category equivalences between \(\text{Gen }P\) and \(\text{Cogen }P^*\). The author also investigates what happens in the cases where \(P\) is quasi-tilting, almost tilting, etc.…
Almost all the nonclassical notions are defined in the paper, and there are always examples showing that the hypotheses in the theorems are necessary. In the first section various relations between the global dimension of the original ring \(R\) and \(S=\text{End}(P)\) are obtained for the cases where \(P\) is *-module, almost tilting, quasi-tilting. Also relations between the global dimensions of \(S\) and \(R/\text{Ann}_R(P)\) are obtained. In the second section the author investigates the relation of various notions of Grothendieck groups of the various rings. He generalizes to several cases the isomorphism of Grothendieck groups of an algebra and a tilted algebra.

MSC:

16D90 Module categories in associative algebras
16E10 Homological dimension in associative algebras
16G10 Representations of associative Artinian rings
16D10 General module theory in associative algebras
16E20 Grothendieck groups, \(K\)-theory, etc.
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References:

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