Cyclic Homologies of Crossed Modules of Algebras

The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki’s excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.

💡 Research Summary

The paper develops a comprehensive homological framework for crossed modules of associative algebras, without assuming the presence of a unit. A crossed module consists of a pair of algebras (R) and (S) together with a homomorphism (\partial\colon R\to S) and an action of (S) on (R) satisfying the usual Peiffer identities. The authors first adapt the classical Hochschild complex to this setting by forming a total complex that combines the Hochschild complexes of (R) and (S) linked through (\partial). The homology of this total complex is denoted (HH_\ast(R,S)) and extends ordinary Hochschild homology of a single algebra.

Next, the cyclic homology theory is lifted to crossed modules. By adapting Connes’ mixed complex (the (B) and (S) operators) to the total complex, the authors obtain a cyclic homology (HC_\ast(R,S)) together with a long exact sequence that mirrors the familiar Hochschild‑to‑cyclic exact triangle. A major achievement is the extension of Wodzicki’s excision theorem: for an inclusion crossed module ((I\hookrightarrow A)) with (I) H‑unital, the natural map (HH_\ast(I)\to HH_\ast(A)) (and similarly for cyclic homology) is an isomorphism after passing to the crossed‑module level. The proof relies on a filtration of the total complex and a careful analysis of Tor‑vanishing conditions, showing that the excision property survives the additional crossed‑module structure.

The paper then introduces a cotriple (or comonad) approach. Using the free‑algebra functor as a cotriple (\mathbb{G}), one builds a (\mathbb{G})‑resolution of a crossed module and defines cotriple cyclic homology (HC^{\operatorname{cot}}_\ast(R,S)). This homology is often easier to compute because the resolution is built from free objects. The authors prove a comparison theorem: there is a natural long exact sequence \