We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space. 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 $d\mathbb$ is the moduli stack of Gieseker-semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. Recent work of the third-named author established a Triangulated Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type.
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