Jacobi-Zariski long nearly exact sequences for associative algebras

TL;DR

This paper develops a Jacobi-Zariski long nearly exact sequence for associative algebra extensions, relating Hochschild homologies and revealing partial exactness properties with a converging spectral sequence.

Contribution

It introduces a new long nearly exact sequence connecting Hochschild homologies of algebra extensions and analyzes the sequence's exactness properties.

Findings

The sequence is exact twice in three.

A spectral sequence converges to the sequence's gap of exactness.

Provides new tools for studying Hochschild homology in algebra extensions.

Abstract

For an extension of associative algebras $B\subset A$ over a field and an $A$-bimodule $X$, we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of $A$ and $B$, and the relative Hochschild homology, all of them with coefficients in $X$. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness.