Noncommutative differential calculus structure on secondary Hochschild (co)homology

Abstract

Let B be a commutative algebra and A be a B-algebra (determined by an algebra homomorphism ). M. D. Staic introduced a Hochschild like cohomology called secondary Hochschild cohomology, to describe the non-trivial B-algebra deformations of A. J. Laubacher et al later obtained a natural construction of a new chain complex in the process of introducing the secondary cyclic (co)homology. In this paper, we establish a connection between the two (co)homology theories for B-algebra A. We show that the pair forms a noncommutative differential calculus, where denotes the homology of the complex