arXiv · 1902.08169
On modules $M$ with $\tau(M) \cong \nu \Omega^{d+2}(M)$ for isolated singularities of Krull dimension $d$
Abstract
A classical formula for the Auslander-Reiten translate $\tau$ says that $\tau(M)\cong \nu \Omega^2(M)$ for every indecomposable module $M$ of a selfinjective Artin algebra. We generalise this by showing that for a $2d$-periodic isolated singularity $A$ of Krull dimension $d$, we have for the Auslander-Reiten translate of an indecomposable non-projective Cohen-Macaulay $A$-module $M$, $\tau(M)\cong \nu \Omega^{d+2}(M)$ if and only if $Ext_A^{d+1}(M,A)=Ext_A^{d+2}(M,A)=0$. We give several applications for Artin algebras.
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Rene Marczinzik. 2019-02-21. On modules $M$ with $\tau(M) \cong \nu \Omega^{d+2}(M)$ for isolated singularities of Krull dimension $d$. https://arxiv.org/abs/1902.08169
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