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 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 and its classical z-fixed point stack become equivalent after completion along a semisimple parameter , implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes . 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 at the identity is identified with a 2-periodic version of the derived de Rham cohomology of . Together, these results identify the completed periodic cyclic homology of a stack over a parameter with the 2-periodic derived de Rham cohomology of its z-fixed points.