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τ-Tilting finite cluster-tilted algebras
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 4 / November 2020
- Published online by Cambridge University Press:
- 21 July 2020, pp. 950-955
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TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 1 / February 2013
- Published online by Cambridge University Press:
- 28 February 2013, pp. 133-144
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- February 2013
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Let
$\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If 
$\Lambda $ admits a trivial maximal 1-orthogonal subcategory of 
$\text{mod } \Lambda $, then, for any indecomposable module 
$M\in \text{mod } \Lambda $, the projective dimension of 
$M$ is equal to one if and only if its injective dimension is also equal to one, and 
$M$ is injective if the projective dimension of 
$M$ is equal to two. In this case, we further get that 
$\Lambda $ is a tilted algebra.
