Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology

TL;DR

This paper establishes long exact sequences in Hochschild and cyclic (co)homology for algebra inclusions, generalizing the Jacobi-Zariski sequence, under flatness and H-unitality conditions.

Contribution

It introduces a Jacobi-Zariski type long exact sequence for Hochschild and cyclic (co)homology applicable to non-commutative algebra inclusions with flatness assumptions.

Findings

Proves long exact sequences in Hochschild and cyclic (co)homology for algebra inclusions.

Generalizes the Jacobi-Zariski sequence to non-commutative algebras.

Expresses the Wodzicki excision sequence as a single long exact sequence.

Abstract

We prove that for an inclusion of unital associative but not necessarily commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in Andr\'e-Quillen homology, provided that the quotient $B$-bimodule $A/B$ is flat. We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an H-unital kernel, one can express the Wodzicki excision sequence and the corresponding Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.