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

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.

💡 Research Summary

The paper establishes Jacobi‑Zariski‑type long exact sequences for Hochschild homology and cyclic (co)homology of associative unital algebras, extending the classical André‑Quillen Jacobi‑Zariski sequence to a non‑commutative setting. The main hypothesis is that for an inclusion of unital algebras (B\subseteq A) the quotient (A/B) is flat as a (B)-bimodule; such extensions are called “r‑flat”. This condition is weaker than full flatness of (A) over (B) and includes many natural examples such as polynomial algebras (both commutative and free non‑commutative) and group algebras (B