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A Localization Theorem for Derived Loop Spaces and Periodic Cyclic Homology

  • Author(s): Chen, Harrison I-Yuan
  • Advisor(s): Nadler, David
  • et al.
Abstract

Motivated by a theorem in the K-theoretic setting relating the localization of K_0(X/T) over a closed point z \in Spec(K_0(BT)) to the Borel-Moore homology of the fixed points H.^{BM}(X^z; C), we prove an equivariant localization theorem for smooth quotient stacks by reductive groups G in the setting of derived loop spaces and periodic cyclic homology, realizing a Jordan decomposition of loops described by Ben-Zvi and Nadler. We show that the derived loop space L(X/G) is a family of twisted unipotent loop spaces over Aff(L(BG)) = G//G; more precisely, the fiber over a formal neighborhood of a semisimple orbit [z] \in G//G is the unipotent loop space of the classical fixed points with a twisted S^1-action. We further study the relationship between unipotent loop spaces and formal loop spaces, and prove that their Tate S^1-invariant functions are isomorphic. Applying a theorem of Bhatt identifying derived de Rham cohomology with Betti cohomology, we obtain an equivariant localization theorem for periodic cyclic homology in the smooth case, identifying the completion of HP(Perf(X/G)) at z \in G//G with the 2-periodic equivariant singular cohomology of the z-fixed points H*(X^z/G^z; k)((u)).

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