Tilting modules for the Auslander algebra of $K[x]/(x^n)$

Abstract

We construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of K[x]/(x n ) and the interval of rational permutation braids in the braid group on n strands. Hence, there are only finitely many tilting modules.

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

“…Furthermore, in the work [IZ16] the support τ -tilting modules and classical tilting modules are studied (and shown to be in bijection to support τ -tilting modules of Π n ). In recent work of [Geu18] all tilting modules are studied. It is surprising that it has only finitely many.…”

“…Corollary 4.3. In fact, this is explained using a factorization ρ : B n → DPic(Λ n ), x → T x of the map ρ over the derived Picard group (see [Geu18,Prop. 6.3] or section 3).…”