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Constructions of stable equivalences of Morita type for finite‐dimensional algebras III
Author(s) -
Liu Yuming,
Xi Changchang
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm065
Subject(s) - morita equivalence , mathematics , endomorphism , functor , pure mathematics , type (biology) , endomorphism ring , morita therapy , injective function , equivalence (formal languages) , invariant (physics) , generator (circuit theory) , discrete mathematics , physics , ecology , power (physics) , quantum mechanics , mathematical physics , biology
In this paper, we provide a new method to produce stable equivalences of Morita type. Our main results can be stated as follows. Let A and B be two finite‐dimensional k ‐algebras over a field k . Suppose that two bimodules A M B and B N A define a stable equivalence of Morita type between A and B and that R is a generator for A ‐modules. Then there is a stable equivalence of Morita type defined by X and Y between the endomorphism algebra End A ( R ) of the module R and the endomorphism algebra End B ( N ⊗ A R ) of the module N ⊗ A R . If M and N satisfy the property that both ( N ⊗ A −, M ⊗ B −) and ( M ⊗ B −, N ⊗ A −) are adjoint pairs of functors, then so do the modules X and Y . Moreover, we show that the self‐injective dimension and the Gorenstein property are invariant under stable equivalences of Morita type with the above‐mentioned adjoint property.