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# Homology, Homotopy and Applications

## Volume 14 (2012)

### Number 1

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

Pages: 65 – 78

DOI: http://dx.doi.org/10.4310/HHA.2012.v14.n1.a4

#### Author

#### Abstract

We prove that for an inclusion of unital associative but not necessarily commutative $\Bbbk$-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 orphism $\varphi\colon\mathcal{B}\to\mathcal{A}$ 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.

#### Keywords

Jacobi-Zariski sequence, excision, Hochschild homology, cyclic cohomology

#### 2010 Mathematics Subject Classification

16W70, 18G25, 18G40, 19D55

Published 13 July 2012