# Secondary Hochschild cohomology and derivations

## TL;DR

This paper introduces secondary derivations and an analogue of Connes' sequence to compute secondary Hochschild and cyclic cohomologies, establishing a universal property linking secondary Kähler differentials with these derivations.

## Contribution

It presents a novel generalization of derivations called secondary derivations and develops an analogue of Connes' sequence for secondary cohomologies.

## Key findings

- Computed low-dimensional secondary Hochschild and cyclic cohomologies.
- Established a universal property relating secondary Kähler differentials and secondary derivations.
- Provided a framework for understanding secondary cohomologies in commutative triples.

## Abstract

In this paper, we introduce a generalization of derivations. Using these so-called secondary derivations, along with an analogue of Connes' Long Exact Sequence, we are able to provide computations in low dimension for the secondary Hochschild and cyclic cohomologies associated to a commutative triple. We then establish a universal property, which paves the way to relating secondary K\"ahler differentials with the aforementioned secondary derivations.