Derived equivalence classification of twisted and generalized multifold extensions of affine type

Abstract

Let $B$ be a piecewise hereditary algebra of affine type over an algebraically closed field. In this paper we give a derived and stable equivalence classification of twisted multifold extensions of $B$. The types $\widetilde D$ and $\widetilde E$ are already covered by the classification for tree type. We therefore consider the remaining types $\widetilde{A}_{p,q}$. We show that the $n$th power of the Nakayama functor on the full subcategory of a transjective component determines the twist, including its action on morphisms, up to conjugacy in the derived Picard group of a hereditary algebra. Together with the hereditary derived type and the folding number, this gives a complete invariant for derived equivalence, and also for stable triangle equivalence. We then compute these conjugacy classes in the unbalanced, balanced, bypass, and Kronecker cases, keeping the finite rotation data. Finally, by the normalization theorem of Asashiba--Kimura--Nakashima--Yoshiwaki, we obtain the same classification for generalized multifold extensions with nonzero integer jump.