The Derived Auslander-Iyama Correspondence

TL;DR

This paper generalizes the Derived Auslander Correspondence to higher dimensions, linking twisted periodic algebras with algebraic triangulated categories possessing unique differential graded enhancements.

Contribution

It extends the correspondence to relate twisted (d+2)-periodic algebras with dZ-cluster tilting triangulated categories, providing recognition theorems and applications to cluster categories.

Findings

Established a higher-dimensional Derived Auslander Correspondence.

Proved the uniqueness of differential graded enhancements for these categories.

Applied results to the Donovan-Wemyss Conjecture in the Homological Minimal Model Program.

Abstract

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.