The little 2-disc operad action on Tate-Hochschild cohomology

The singularity category was introduced by Buchweitz in 1986 and then rediscovered by Orlov in algebraic geometry. Following Buchweitz, we define the Tate-Hochschild cohomology of an associative algebra as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. 

 

In this talk, we show that Tate-Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology. Namely, there is a Gerstenhaber algebra structure in cohomology and a little 2-disc operad action at the cochain level. We will also talk about Keller’s recent result that Tate-Hochschild cohomology is isomorphic to Hochschild cohomology of the dg singularity category (i.e. the canonical dg enhancement of the singularity category).