Jacobi-Zariski exact sequence for Hochschild homology and cyclic (co)homology

Abstract

We prove that for an inclusion of unital associative but not necessarily commutative $\mathbb{k}$-algebras $\mathcal{B}\subseteq \mathcal{A}$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $\mathcal{B}$-module $\mathcal{A}/\mathcal{B}$ is flat. We also prove that for an arbitrary $r$-flat morphism $\varphi\colon\mathcal{B}\to\mathcal{A}$ with an H-unital kernel, one can express the Wodzicki excision sequence and our Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.