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In our paper “Jacobi–Zariski exact sequence for Hochschild homology and cyclic (co)homology” [Homology Homotopy Appl., 14(1):65–78, 2012], in Theorem 2.3 and in all remaining results based on this theorem, it is stated that the Jacobi–Zariski long exact sequence works for the range p⩾0. The range should be corrected to p⩾1. [ABSTRACT FROM AUTHOR]

We prove that for an inclusion of unital associative but not necessarily commutative k-algebras B ⊆ 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 B-module A=B is flat. We also prove that for an arbitrary r-flat morphism φ : B → 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. [ABSTRACT FROM AUTHOR]

We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with a few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category of (special) towers of super complexes, or the derived category of mixed complexes. [ABSTRACT FROM AUTHOR]

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show that this (co)homology, called Hopf-Hochschild (co)homology, can also be defined as a derived functor on the category of representations of an equivariant analogue of the enveloping algebra of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf-Hochschild (co)homology. [ABSTRACT FROM AUTHOR]