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 $H^{•}((A,B,\epsilon );A)$ 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 complex ${\overline{C}}_{•}(A,B,\epsilon )$ 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 $(H^{•}((A,B,\epsilon );A),H{H}_{•}(A,B,\epsilon ))$ forms a noncommutative differential calculus, where $H{H}_{•}(A,B,\epsilon )$ denotes the homology of the complex ${\overline{C}}_{•}(A,B,\epsilon ).$

Keywords:

• Gerstenhaber algebra
• comp module
• noncommutative differential calculus
• secondary Hochschild (co)homology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16E40

Acknowledgements

We thank the anonymous referee for his useful suggestions and remarks. The research of S. K. Mishra is supported by the NBHM postdoctoral fellowship. The author thanks NBHM for its support.