Jacobi-Zariski exact sequence for Hochschild homology and cyclic (co)homology. (English) Zbl 1259.19003

Kaygun, Atabey

Homology Homotopy Appl. 14, No. 1, 65-78 (2012); erratum ibid. 21, No. 2, 301-303 (2019).

Summary: We prove that for an inclusion of unital associative but not necessarily commutative \(k\)-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 \(\varphi: \mathcal B \rightarrow \mathcal A\) 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.

MSC:

19D55	\(K\)-theory and homology; cyclic homology and cohomology
18G25	Relative homological algebra, projective classes (category-theoretic aspects)
16W70	Filtered associative rings; filtrational and graded techniques
18G40	Spectral sequences, hypercohomology

Keywords:

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