Perverse Equivalences in the Dg-Stable Category
Let A be a finite-dimensional self-injective algebra, graded in non-positive degree. We define A-dgstab, the differential graded stable category of A, to be the quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We prove that A-dgstab is the triangulated hull of the orbit category A-grstab/ Ω(1), which allows computations in the dg-stable category to be performed in the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A-dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. When A is a symmetric algebra with socle concentrated in degree -d < 0, we show that A-dgstab has Calabi-Yau dimension -d-1.
Chuang and Rouquier (2017) describe an action by perverse equivalences on the set of bases of a triangulated category of Calabi-Yau dimension -1. We develop an analogue of their theory for Calabi-Yau categories of arbitrary negative dimension and apply this theory to the dg-stable category.
As an example, we analyze the dg-stable category of a Brauer tree algebra, with an arbitrary non-positive grading. We compute the Auslander-Reiten quiver, then develop a combinatorial model for A-dgstab, which we use to describe the action of perverse equivalences. Using our model, we show that perverse equivalences act transitively on the set of bases.