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تسجيل مستخدم جديدFree products, cyclic homology, and the Gauss-Manin connection
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We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce and study `free-product deformations of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.
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Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Co
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$HC_*(A rtimes G)$ is the cyclic homology of the crossed product algebra $A rtimes G.$ For any $g epsilon G$ we will define a homomorphism from $HC_*^g(A),$ the twisted cylic homology of $A$ with respect to $g,$ to $HC_*(A rtimes G).$ If $G$ is the f
inite cyclic group generated by $g$ and $|G|=r$ is invertible in $k,$ then $HC_*(A rtimes G)$ will be isomorphic to a direct sum of $r$ copies of $HC_*^g(A).$ For the case where $|G|$ is finite and $Q subset k$ we will generalize the Karoubi and Connes periodicity exact sequences for $HC_*^g(A)$ to Karoubi and Connes periodicity exact sequences for $HC_*(A rtimes G)$ .
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