Stable categories of higher preprojective algebras

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.

The introduction has been revised. Minor corrections throughout. 38 pages

References in corpus (6)

• On triangulated orbit categories
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• Deformed Calabi-Yau Completions
• Perverse equivalences, BB-tilting, mutations and applications
• Trivial Maximal 1-Orthogonal Subcategories For Auslander's 1-Gorenstein Algebras
• n-representation-finite algebras and n-APR tilting

Cited by in corpus (13)

• $n$-angulated categories
• n-representation-finite algebras and twisted fractionally Calabi-Yau algebras
• Perverse equivalences, BB-tilting, mutations and applications
• On Generalized Cluster Categories
• On derived equivalences of lines, rectangles and triangles
• ${\tau }^2$-stable tilting complexes over weighted projective lines
• Cluster tilting for higher Auslander algebras
• n-representation-finite algebras and n-APR tilting
• A non-simply laced version for cluster structures on 2-Calabi-Yau categories
• Higher dimensional cluster combinatorics and representation theory
• On m-cluster tilted algebras and trivial extensions
• APR tilting modules and graded quivers with potential
• Realizing stable categories as derived categories