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تسجيل مستخدم جديدPeriodic cyclic homology of certain nuclear algebras
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Relying of properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n).
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We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $phi: Xto Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism
of cohomology groups $H^n(phi): H^n(X) to H^n(Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $hat{otimes}$-algebras: the tensor algebra $E hat{otimes} F$ generated by the duality $(E, F, < cdot, cdot >)$ for nuclear Frechet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$; nuclear biprojective K{o}the algebras $lambda(P)$ which are Frechet spaces or $DF$-spaces; the algebra of distributions $mathcal{E}^*(G)$ on a compact Lie group $G$.
We study the periodic cyclic homology groups of the cross-product of a finite type algebra $A$ by a discrete group $Gamma$. In case $A$ is commutative and $Gamma$ is finite, our results are complete and given in terms of the singular cohomology of th
e strata of fixed points. These groups identify our cyclic homology groups with the dlp orbifold cohomologydrp of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. We provide examples of Azumaya algebras for which this identification is, however, no longer valid. As an example, we discuss some affine Weyl groups.
Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E relative to
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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 $HC_*(A rtimes G)$ .
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
nnes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.
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