Bar Simplicial Modules and Secondary Cyclic (Co)homology

TL;DR

This paper explores the simplicial structure of secondary Hochschild cohomology complexes, introduces secondary cyclic (co)homology, and establishes foundational properties for these new invariants.

Contribution

It introduces the concept of secondary cyclic (co)homology and studies its properties, extending classical Hochschild and cyclic theories to a secondary setting.

Findings

Defined the simplicial object (A,B,) and its role

Established properties of secondary cyclic (co)homology

Linked secondary Hochschild cohomology with simplicial structures

Abstract

In this paper we study the simplicial structure of the complex $C^{\bullet}((A,B,\varepsilon); M)$, associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object $\mathcal{B}(A,B,\varepsilon)$, which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 3.9 and 4.11).