Stable categories of higher preprojective algebras

TL;DR

This paper introduces (n+1)-preprojective algebras for algebras of global dimension n, showing their stable categories are (n+1)-Calabi-Yau and contain cluster tilting objects, with results extending to Cohen-Macaulay modules under certain conditions.

Contribution

It defines (n+1)-preprojective algebras and establishes their stable categories as (n+1)-Calabi-Yau with cluster tilting objects, extending known results to Cohen-Macaulay modules.

Findings

Stable categories are (n+1)-Calabi-Yau for n-representation-finite algebras.

Stable categories contain (n+1)-cluster tilting objects.

Results extend to Cohen-Macaulay modules under specific assumptions.

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 \in {1,2}) the results above still hold for the stable category of Cohen-Macaulay modules.