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On modules $ M$ with $ \tau(M) \cong\nu\Omega^{d+2}(M)$ for isolated singularities of Krull dimension $ d$


Author: René Marczinzik
Journal: Proc. Amer. Math. Soc. 148 (2020), 527-534
MSC (2010): Primary 16G10, 16E10
DOI: https://doi.org/10.1090/proc/14738
Published electronically: August 7, 2019
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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 nonprojective Cohen-Macaulay $ A$-module $ M$, $ \tau (M)\cong \nu \Omega ^{d+2}(M)$ if and only if $ \operatorname {Ext}_A^{d+1}(M,A)=\operatorname {Ext}_A^{d+2}(M,A)=0$. We give several applications for Artin algebras.


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Additional Information

René Marczinzik
Affiliation: Institute of algebra and number theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: marczire@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/proc/14738
Keywords: Artin algebra, isolated singularity, reflexive modules, Auslander--Reiten translation
Received by editor(s): March 6, 2019
Received by editor(s) in revised form: June 14, 2019
Published electronically: August 7, 2019
Communicated by: Sarah Witherspoon
Article copyright: © Copyright 2019 American Mathematical Society