2021/02/14 by Apurba Das, Das, Apurba, Satyendra Mishra +3
Mathematics · #16E40 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2102.07095
openalex publication_date 2021/02/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let B be a commutative algebra and A be a B-algebra (determined by an\nalgebra homomorphism \ε:B\→ A). M. D. Staic introduced a\nHochschild like cohomology H bullet((A,B,\ε);A) called secondary\nHochschild cohomology, to describe the non-trivial B-algebra deformations of\nA. J. Laubacher et al later obtained a natural construction of a new chain\n(and cochain) complex \C bullet(A,B,\ε) (resp.\n\C bullet(A,B,\ε)) in the process of introducing the\nsecondary cyclic (co)homology. It turns out that unlike the classical case of\nassociative algebras (over a field), there exist different (co)chain complexes\nfor the B-algebra A. In this paper, we establish a connection between the\ntwo (co)homology theories for B-algebra A. We show that the pair\n\(H bullet((A,B,\ε);A),HH bullet(A,B,\ε)\)\nforms a non-commutative differential calculus, where\nHH bullet(A,B,\ε) denotes the homology of the complex\n\C bullet(A,B,\ε).\n