[Paper Review] Cyclic Homologies of Crossed Modules of Algebras

Гурам Донадзе, Nick Inassaridze+2 more|ArXiv.org|Dec 3, 2008

Rings, Modules, and Algebras12 references3 citations

TL;DR

This paper develops cotriple cyclic homology for crossed modules of non-unital associative algebras and establishes a long exact homology sequence comparing cyclic and cotriple cyclic homology, generalizing relative cyclic homology exact sequences. It extends Wodzicki's excision theorem to crossed modules and proves that the homology of the nerve of a crossed module fits into a long exact sequence with its cotriple cyclic homology via a spectral sequence argument.

ABSTRACT

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.

Motivation & Objective

• To develop cotriple cyclic homology theory in the category of crossed modules of non-unital associative algebras.
• To compare cyclic homology of crossed modules with their cotriple cyclic homology using a long exact homology sequence.
• To generalize the relative cyclic homology exact sequence to the setting of crossed modules.
• To extend Wodzicki's excision theorem to inclusion crossed modules of algebras.
• To establish a spectral sequence argument linking the homology of the nerve of a crossed module to its cotriple cyclic homology.

Proposed method

• The authors use simplicial methods and define the cyclic homology of a crossed module via the nerve construction.
• They apply the Barr-Beck cotriple resolution to construct a simplicial resolution of a crossed module in the category of crossed algebras.
• A bicomplex structure is formed from the Hochschild and cyclic chain complexes of the resolution, leading to a spectral sequence.
• The spectral sequence is analyzed to show degeneration at the E²-page, yielding an isomorphism between the total homology and the cotriple cyclic homology.
• The acyclicity of certain augmented simplicial vector spaces is established using the asphericity of semidirect product algebras.
• The long exact sequence is derived from a short exact sequence of complexes and the induced homology long exact sequence.

Experimental results

Research questions

• RQ1 How can cotriple cyclic homology be defined for crossed modules of non-unital associative algebras?
• RQ2 What is the relationship between the cyclic homology of a crossed module and its cotriple cyclic homology?
• RQ3 Does Wodzicki's excision theorem extend to inclusion crossed modules of algebras?
• RQ4 Can the relative cyclic homology exact sequence be generalized to the setting of crossed modules?
• RQ5 What spectral sequence structure underlies the comparison between cyclic and cotriple cyclic homology of crossed modules?

Key findings

• The cotriple cyclic homology of a crossed module (R,A,ρ) is isomorphic to the homology of the total complex of a bicomplex constructed from its simplicial resolution.
• A long exact homology sequence is established between the cyclic homology of the crossed module, its cotriple cyclic homology, and a shifted version of the latter.
• The spectral sequence associated to the bicomplex degenerates at the E²-page, yielding the isomorphism H_{n+1}(γ(M,R,μ)) ≅ ξHC_n(M,R,μ) for n ≥ 0.
• The excision theorem for Hochschild and cyclic homology is extended to inclusion crossed modules of algebras.
• The homology of the nerve of a crossed module fits into a long exact sequence with its cotriple cyclic homology, generalizing the classical relative cyclic homology sequence.
• The acyclicity of the augmented simplicial vector spaces β_n and γ_n is proven under the assumption that the underlying simplicial algebras are aspherical.