The paper proves completion theorems identifying completed equivariant K-theory and cyclic homology of quotient stacks with ordinary K-theory and homology of bar or Borel constructions, and gives explicit formulas for finite-stabilizer actions.
The homotopy fixed points of the circle action on Hochschild homology
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abstract
We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
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Thomason's completion for K-theory and cyclic homology of quotient stacks
The paper proves completion theorems identifying completed equivariant K-theory and cyclic homology of quotient stacks with ordinary K-theory and homology of bar or Borel constructions, and gives explicit formulas for finite-stabilizer actions.