Gerstenhaber algebra and Deligne’s conjecture on the Tate–Hochschild cohomology

Abstract

Using noncommutative differential forms, we construct a complex called the singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate–Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of singular chains of the little 2 2 -discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a B ∞ B_{\infty } -algebra structure and its cohomology ring is a Gerstenhaber algebra. Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate–Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in nonnegative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate–Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin–Vilkovisky algebra.