Stable categories of higher preprojective algebras

By Osamu Iyama, Steffen Oppermann. 2009

142 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/24e95ddab276fc9c13705ed9e5942bfb9e7c06ae

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday (2011, methodology): “i09] from 2 to n ≥ 2 is described in [Guo10]. The functor π : Db(Λ) → C n(Λ) is also described by a universal property (cf [Kel05, Ami09]). Here is the version we will use in this paper (see appendix [IO09]). Proposition 1.10. [Kel05, Ami09],[IO09, Thm A.20] Let Λ be a ﬁnite dimensional algebra of g”
Auslander-Gorenstein algebras and precluster tilting (2016, result): “(We do not assume gldimΛ ≤ n in this paper in contrast with several earlier papers [18, 24, 25].”
Representations of Algebras and Related Topics (2011, background): “ : Db(Λ) Res //D b(Λ) − L ⊗ΛM //D b(Π 2(kQ)) induced by these algebra morphisms and check that it satisﬁes the hypotheses of Theorem 3.25. Remark 4.3. This theorem has been very nicely generalized in [62] for higher cluster categories as follows: For n ≥2, let A be a (n −1)-representation-ﬁnite alge”