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More on the Hochschild and cyclic homologies of crossed modules of algebras. (English) Zbl 1259.18004

Summary: We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras \((I, A, \rho )\) is isomorphic to \(HC_*(\rho ): HC_*(I)\to HC_*(A)\), provided \(I\) is \(H\)-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras \((R, \, 0, \, 0)\) for each algebra \(R\) with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.

MSC:

18G10 Resolutions; derived functors (category-theoretic aspects)
18G50 Nonabelian homological algebra (category-theoretic aspects)
18G60 Other (co)homology theories (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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