The derived Auslander–Iyama correspondence

We work over a perfect field. In this article, given an integer d ≥ 1 d≥ 1 , we establish a bijection between twisted ( d + 2 ) (d+2) -periodic algebras and suitable algebraic triangulated categories with a d Z dZ -cluster tilting object. Furthermore, we show that these 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 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.