We use the Dolbeault cohomology to investigate Koszul–Brylinski homology on holomorphic Poisson manifolds. obtain Leray–Hirsch theorem for Hochschild and Mayer–Vietoris sequence, Künneth homology. In particular, we show some relations of homologies around a blow-up transformation general case (not necessarily compact) by our previous works cohomology.

A notion of Hochschild cohomology HH∗(A) of an abelian category A was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism χ from the Hochschild cohomology of A into the graded centre Z∗(Db(A)) of the bounded derived category of A. An element c ∈ HH2(A) corresponds to a first order deformation Ac of A (Lowen and Van den Bergh, 2006). The problem o...

We show that for some finite group block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial. Along way we obtain methods to investigate nontriviality of twisted algebras.