# Properties of the Secondary Hochschild Homology

## TL;DR

This paper investigates properties of secondary Hochschild homology for triples, establishing invariance under Morita equivalence, connecting it with usual Hochschild homology, and discussing its functoriality.

## Contribution

It introduces a Morita invariance for secondary Hochschild homology and links it to classical Hochschild homology through exact sequences.

## Key findings

- Morita equivalence invariance of secondary Hochschild homology
- Existence of an exact sequence relating secondary and usual Hochschild homology
- Functoriality properties of the secondary Hochschild homology

## Abstract

In this paper we study properties of the secondary Hochschild homology of the triple $(A,B,\varepsilon)$ with coefficients in $M$. We establish a type of Morita equivalence between two triples and show that $H_\bullet((A,B,\varepsilon);M)$ is invariant under this equivalence. We also prove the existence of an exact sequence which connects the usual and the secondary Hochschild homologies in low dimension, allowing one to perform easy computations. The functoriality of $H_\bullet((A,B,\varepsilon);M)$ is also discussed.