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Equivariant localization and completion in cyclic homology and derived loop spaces
Advances in Mathematics ( IF 1.494 ) Pub Date : 2020-02-03 , DOI: 10.1016/j.aim.2020.107005
Harrison Chen

We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups X/G in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack X/G and its classical z-fixed point stack π0(Xz)/Gz become equivalent after completion along a semisimple parameter [z]G//G, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes Perf(X/G). We then prove an analogue of the Atiyah-Segal completion theorem in the setting of periodic cyclic homology, where the completion of the periodic cyclic homology of Perf(X/G) at the identity [e]G//G is identified with a 2-periodic version of the derived de Rham cohomology of X/G. Together, these results identify the completed periodic cyclic homology of a stack X/G over a parameter [z]G//G with the 2-periodic derived de Rham cohomology of its z-fixed points.

更新日期:2020-02-03

 

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