Tate-Hochschild cohomology, the singularity category and applications

Following work of Buchweitz, one defines Tate-Hochschild cohomology of an algebra A to be the Yoneda algebra of the identity bimodule in the singularity category of bimodules. We show *if the bounded derived category of A is smooth* (hypothesis added on 03/06/2020) then Tate-Hochschild cohomology is canonically isomorphic to the ordinary Hochschild cohomology of the singularity category of A (with its canonical dg enrichment). In joint work with Zheng Hua, we apply this to prove a weakened version of a conjecture by Donovan-Wemyss which states that a complete local isolated compound Du Val singularity is determined by the derived equivalence class of the contraction algebra associated with a smooth model.