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Tate-Hochschild cohomology, the singularity category and applications Keller, Bernhard


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 that 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 isolated cDV singularity is determined by the derived equivalence class of the contraction algebra associated with a resolution.

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