Gorenstein algebras of finite Cohen-Macaulay type. (English) Zbl 1184.16011
Summary: An Artin algebra \(A\) is said to be CM-finite if there are only finitely many isomorphism classes of indecomposable finitely generated Gorenstein-projective \(A\)-modules. Inspired by Auslander’s idea on representation dimension, we prove that for \(2\leqslant n<\infty\), \(A\) is a CM-finite \(n\)-Gorenstein algebra if and only if there is a resolving Gorenstein-projective \(A\)-module \(E\) such that \(\text{gl.dim\,End}_A(E)^{op}\leqslant n\).
MSC:
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16E10 | Homological dimension in associative algebras |
| 16G10 | Representations of associative Artinian rings |
| 16G50 | Cohen-Macaulay modules in associative algebras |
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