Hochschild (Co)homologies of DG K-algebras and Their Koszul Duals

Abstract

We compare the Hochschild (co)homologies of a complete typical DG K-algebra and its Koszul dual. We show that the Koszul dual of a finite dimensional complete typical symmetric DG K-algebra is a Calabi–Yau DG K-algebra whose Hochschild cohomology is a Batalin–Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical symmetric DG K-algebra and its Koszul dual are isomorphic as Batalin–Vilkovisky algebras.