Jeremy Brightbill | Searcharxiv

arXiv2020

Perverse Equivalences and Dg-stable Combinatorics

Chuang and Rouquier 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 dimension $w<0$ and show it is equivalent to the mutation theory of $w$-simple-minded systems. Given a non-positively graded, finite-dimensional symmetric algebra $A$, we show that the differential graded stable category of $A$ has negative Calabi-Yau dimension. When $A$ is a Brauer tree algebra, we construct a combinatorial model of the dg-stable category and show that perverse equivalences act transitively on the set of $|w|$-bases.

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arXiv2019

The Differential Graded Stable Category of a Self-Injective Algebra

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 express A-dgstab as the triangulated hull of the orbit category A-grstab/$Ω$(1). This result allows computations in the dg-stable category to be performed by reducing to 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. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.

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