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On modules with trivial self-extensions. (English) Zbl 0646.16025

Let \({}_ BT_ A\) be a bimodule over connected artin algebras A and B with the properties (1) \(T_ A\) and \({}_ BT\) finitely generated (2) \(B=End(T_ A)\) and \(A=End(_ BT)\), and (3) \(Ext\) \(i_ B(T,T)=0=Ext\) \(i_ A(T,T)\) for all integers \(i\geq 1\). It is well known that such modules appear in the tilting theory and in the study of the generalized Nakayama conjecture. The author proves that the projective (injective) dimensions of \(T_ A\) and \({}_ BT\) coincide, if they are finite; in this case \(T_ A\) and \({}_ BT\) have the same number of nonisomorphic indecomposable direct summands.
Reviewer: W.Müller

MSC:

16Gxx Representation theory of associative rings and algebras
16P20 Artinian rings and modules (associative rings and algebras)
16E10 Homological dimension in associative algebras
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References:

[1] Auslander, M.; Reiten, I., On a generalized version of the Nakayama conjecture, (Proc. Amer. Math. Soc., 52 (1975)), 69-74 · Zbl 0337.16004
[2] Happel, D.; Ringel, C. M., Tilted algebras, Trans. Amer. Math. Soc., 274, 399-443 (1982) · Zbl 0503.16024
[3] Tachikawa, H., Quasi-Frobenius Rings and Generalizations, (Lecture Notes in Math., Vol. 351 (1973), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0177.05901
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