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Representable equivalences are represented by tilting modules. (English) Zbl 0706.16011
It was shown by C. Menini and A. Orsatti that, if R, S are two rings, Y a full subcategory of left R-modules closed under direct sums and factor modules, D a full subcategory of left S-modules containing \({}_ SS\) and closed under submodules, such that there exists an inverse pair of additive equivalences F: \(Y\to D\) and G: \(D\to Y\) then there exists a module \({}_ RM\) such that \(End_ RM=S\), \(F\overset \sim \rightarrow Hom_ R(_ RM\),-), Y is the full subcategory of all R- modules generated by \({}_ RM\), while \(G\overset \sim \rightarrow M_ S\otimes\)-, and D is the full subcategory of all S-modules cogenerated by \(Hom_ R(_ RM_ S,_ RQ)\) where \({}_ RQ\) is an injective cogenerator of the category of R-modules [Rend. Semin. Mat. Univ. Padova 82, 203-231 (1989; Zbl 0701.16007)]. Thus, examples of modules with these properties are the tilting modules in the sense of D. Happel and C. M. Ringel [Trans. Am. Math. Soc. 274, 399- 443 (1982; Zbl 0503.16024)]. The aim of this paper is to show that, if R is a finite dimensional algebra over a field, and \({}_ RM\) is a faithful finite dimensional module with the above properties, then \({}_ RM\) is a tilting module.
Reviewer: I.Assem

MSC:
16P10 Finite rings and finite-dimensional associative algebras
16D90 Module categories in associative algebras
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References:
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