On the tilting complexes for the Auslander algebra of the truncated polynomial ring

Abstract

We give a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of K[T]/T n and Z × Bn where Bn is the Artin braid goup of type A with n − 1 generators. The tilting complexes have mutation components parametrized by Z and each component has a natural faithful and transitive operation of Bn. This also implies that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and Z × Bn. This work is to be seen as a continuation of the work of Geuenich and an application of the work of Aihara and Mizuno on tilting complexes of preprojective algebras of Dynkin type.We denote by ε 1 , . . . , ε n the complete set of primitive orthogonal idempotents corresponding to the vertices in the quiver and by Q i := e i Λ n the indecomposable projective right Λ n -module.