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 $\epsilon :B\to A$). M. D. Staic introduced a Hochschild like cohomology $H^{\bullet }((A,B,\epsilon );A)$ 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 (and cochain) complex ${\overline{C}}_{\bullet }(A,B,\epsilon )$ (resp. ${\overline{C}}^{\bullet }(A,B,\epsilon )$) in the process of introducing the secondary cyclic (co)homology. It turns out that unlike the classical case of associative algebras (over a field), there exist different (co)chain complexes for the $B$-algebra $A$. In this paper, we establish a connection between the two (co)homology theories for $B$-algebra $A$. We show that the pair $(H^{\bullet }((A,B,\epsilon );A),H{H}_{\bullet }(A,B,\epsilon ))$ forms a non-commutative differential calculus, where $H{H}_{\bullet }(A,B,\epsilon )$ denotes the homology of the complex ${\overline{C}}_{\bullet }(A,B,\epsilon )$.

Any comments are welcome