Stable categories of higher preprojective algebras |
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Authors: | Osamu Iyama Steffen Oppermann |
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Institution: | 1. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan;2. Institutt for matematiske fag, NTNU, 7491 Trondheim, Norway |
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Abstract: | We introduce (n+1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+1)-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)-preprojective algebra is (n+1)-Calabi–Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)-cluster tilting object. We show that even if the (n+1)-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}) the results above still hold for the stable category of Cohen–Macaulay modules. |
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Keywords: | Higher preprojective algebra Stable category n-cluster tilting" target="_blank">gif" overflow="scroll">n-cluster tilting n-representation finite algebra" target="_blank">gif" overflow="scroll">n-representation finite algebra n-cluster category" target="_blank">gif" overflow="scroll">n-cluster category |
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